text:
\section{Introduction}
Convex optimization problems arise in many
areas of engineering, including machine learning~\cite{sra12},
communications~\cite{luo06}, robotics~\cite{verscheure09}, and others.
Fundamentally, regardless of the application area, the goal is to design
an algorithm that will converge to a minimum of an objective function,
possibly under some constraints. Recently,
there has been increased interest in studying
optimization algorithms
in continuous time using
tools from dynamical systems to establish convergence
to minimizers; see~\cite{su16,rahili2017,garg20}.

In this paper, we develop a hybrid optimization algorithm for the analysis
of multi-agent systems with continuous-time
updates and intermittent discrete-time communication events. This is motivated
by two factors. First, we wish to leverage the large collection of
tools from dynamical systems to analyze multi-agent optimization.
Second, there exist many multi-agent controllers
that operate in continuous time to minimize some objective function,
e.g., in consensus~\cite{olfati07} and coverage control~\cite{cortes04},
and our analyses will apply to such systems.
However, while
individual agents' computations occur in continuous time, communication between them inherently occur in discrete time because
communicated information arrives at isolated time instants.
This mixture of continuous- and
discrete-time elements naturally leads us to a hybrid system model.

The algorithm we propose is essentially a hybrid version
of parallelized block coordinate descent~\cite{bertsekas89}, in which each agent
updates only a small subset of all decision variables
in continuous time, and agents
communicate these updates to others in discrete time. In the proposed model, communication between agents
occurs when a decreasing timer reaches zero, at which point the timer
is reset to some value within a specified range.
Agents use a sample-and-hold strategy in which gradients
are computed at the communication times and then held constant
and used continuously until the next communication event.
This approach is inspired by recent work~\cite{PHILLIPS2019} that has
successfully applied it to synchronization problems.
We consider objective functions that satisfy
typical, mild assumptions for distributed optimization, namely
strongly convex objective functions with Lipschitz gradients.

We leverage the theory of hybrid systems to prove that
the proposed hybrid algorithm has several desirable properties.
First, we define a hybrid system model for this algorithm
and show that, under these hybrid dynamics, every maximal
solution is complete, with domain allowing arbitrarily large ordinary time. As a result, there are no
theoretical obstructions to running this algorithm
for arbitrarily long periods of time.
Second, we use Lyapunov analysis to show that, even under intermittent information sharing,
the hybrid optimization algorithm exponentially converges to the minimizer of an objective function. Furthermore, we derive
an explicit convergence rate in terms of system parameters.

The developments in this paper can be regarded as continuous-time counterparts to
``classical'' discrete-time algorithms in multi-agent
optimization~\cite{bertsekas89}.
Related research in multi-agent continuous-time optimization includes~\cite{gharesifard2014,lu2012,rahili2017}, though
those works all use a consensus-based update law that executes computations
and communications both in continuous time. However, we avoid continuous-time
communications to account for cases in which they are not possible
or simply undesirable,
e.g., over long distances or when power is limited.

The most similar works are~\cite{PHILLIPS2019},~\cite{KIA2015}, which also study continuous-time optimization
with discrete-time communication. However, those works also use consensus-based optimization
algorithms in which each agent updates all decision variables.
In contrast, we consider agents with a common objective function and require that
each agent update only a small subset of decision variables. This has
the advantage that an individual agent's computational burden can be small, even
when solving high-dimensional problems.

The rest of the paper is organized as follows. Section~\ref{sec:prob} includes our problem statement, assumptions, and algorithm. Section~\ref{sec:prelim} provides background
on hybrid systems. We present our hybrid system model in Section~\ref{sec:hsm} and establish the existence of complete solutions. Section~\ref{sec:convergence} proves
that the hybrid multi-agent update law exponentially converges to the minimizer of an objective function.
We include numerical results as validation in Section~\ref{sec:sim}.

\section{Problem Statement and Algorithm Overview} \label{sec:prob}
In this section, we state the class of problems that we consider and give an overview of the proposed hybrid optimization algorithm.

\subsection{Problem Formulation}
We consider a group of~$N$ agents jointly solving an optimization problem of the following form:
\begin{problem} \label{prob:first}
Given an objective function~$L : \mathbb{R}^n \to \mathbb{R}$,
\begin{equation}
\textnormal{minimize } L(x), \qquad x \in \mathbb{R}^n
\end{equation}
using~$N$ distributed agents while requiring that (i) only one agent updates any entry of the decision variable~$x$, and (ii) agents
require only intermittent information sharing from others.
\end{problem}

Each agent executes computations locally and then shares the results of those computations. Criterion~(i) is there for scalability, only a single agent will update each decision variable. This reduces the computation load on agents and removes duplicated efforts. Criterion~(ii) ensures that the algorithm performs even in environments where communications may be limited. In many practical settings, we expect bandwidth to be limited and/or agents to
have limited onboard power available, which means communications should not be constant.

We assume the following about the objective function~$L$.

\begin{assumption} \label{as:L}
The function~$L$ is twice continuously
differentiable,~$\beta$-strongly convex for some~$\beta > 0$, and~$K$-smooth (namely,~$\nabla L$ is~$K$-Lipschitz).
\hfill $\triangle$
\end{assumption}

Assumption~\ref{as:L} allows a large number of convex problems to be considered, such as strongly convex quadratic programs. It is a standard assumption in multi-agent
optimization~\cite{bertsekas89}.
It implies that~$K \geq \beta$.

We solve Problem~\ref{prob:first} by applying gradient descent in continuous time using data received intermittently in discrete time. The proposed hybrid optimization algorithm uses jumps to characterize the discrete-time communication events and flows to represent the continuous-time dynamics.
Analogously to past research that has developed distributed versions of the discrete-time gradient
descent law, our update law during flows is based on the following
(centralized)
first-order dynamical system:
\begin{equation} \label{eq:gdall}
    \dot{x} + \nabla L(x) =0.
\end{equation}
This is motivated by the use of gradient-based controllers in multi-agent systems, e.g., in consensus~\cite{olfati07}, as well
as the simplicity of distributing gradient-based updates and the robustness
to asynchrony that results from doing so~\cite{bertsekas89}.
Next, we distribute this across a team of agents.

\subsection{Algorithmic Framework}
We seek to distribute~$\eqref{eq:gdall}$ across a team of agents in
accordance with the parallelization requirement
in Problem~\ref{prob:first}.
We consider~$N$ agents indexed over~$i \in [N]:=\lbrace 1, \ldots , N\rbrace$ and divide~$x \in \mathbb{R}^n$ into~$N$ blocks. Then agent~$i$ is responsible for updating and
communicating values of the~$i$-th block,~$x_{i} \in \mathbb{R}^{n_i}$,
where~$n_i \in \mathbb{N}$ and~$\sum_{i \in [N]} n_i = n$.
Thus, the variable~$x$ may be written as the vertical concatenation of all agents' blocks. Each agent performs gradient descent on their own block during flows
but does not update any others.

Agents' updates occur in continuous time while communication of these updates occurs in discrete time. Communications
are coordinated using a decreasing timer,~$\tau$, that is shared by all agents.
When the timer reaches zero, all agents communicate their current values to all of the other agents and the timer resets to a value within a specified interval~$[\tau_{min}, \tau_{max}]$. We assume that communicated data are received at the same time they are sent.
These communicated blocks are gathered into the vector~$\eta \in \mathbb{R}^n$
with the current value of~$x_i$ being assigned to~$\eta_i$ at communication events.
The value of~$\eta$ is used in each agent's continuous-time computations in a sample-and-hold
manner between communication events. That is, each agent uses the previously communicated data in their updates rather than the continuously evolving values of the other agents. Formally, we write~$\nabla_i L = \frac{\partial L}{\partial x_i}$,
and during flows agent~$i$ executes
\begin{equation}
\dot{x}_i = -\nabla_i L(\eta).
\end{equation}
This sample-and-hold method is common in the literature~\cite{PHILLIPS2019} and
is used to demonstrate the feasibility of the hybrid approach in multi-agent
optimization.

The complete algorithm is summarized in Algorithm~\ref{alg:first}.

\begin{algorithm} \label{alg:first}
\SetAlgoLined
Initialization: set~$x_o, \eta_o \in \mathbb{R}^n$ and~$\tau_o \in [0, \tau_{max}]$\;
\While{$\tau \geq 0$}{
$\dot{x}_i = - \nabla_i L(\eta)$, for all~$i\in\lbrace 1, \dots, N \rbrace $\;
$\dot{\tau} = -1$\;
\If{$\tau=0$}{
reset~$\eta_i$ to $x_i$, for all~$i\in\lbrace 1, \dots, N \rbrace $\;
reset~$\tau$ to a value in~$ [\tau_{min},\tau_{max}]$\;}
}
\caption{Distributed Hybrid Gradient Descent}
\end{algorithm}

The next section provides the tools that will be
used to analyze Algorithm~\ref{alg:first}.

\section{Hybrid System Preliminaries} \label{sec:prelim}
In this section, we recount the background material necessary for the hybrid system
modeling and analysis in the remainder of the paper.

\subsection{Preliminaries on Hybrid Systems}
For the purposes of this paper, a hybrid system~$\mathcal{H}$ has data~$(C,f,D,G)$ that takes the general form
\begin{align}
    \mathcal{H} = \begin{cases}
    \dot{x} = f(x) &x\in C \\
    x^+ \in G(x) &x\in D
    \end{cases}, \label{eq:hybriddef}
\end{align}
where~$x\in \mathbb{R}^n$ is the system's state, and~$f$ defines the flow map and continuous dynamics for which~$C$ is the flow set. The
set-valued
jump map~$G$ captures the system's discrete behavior for the jump set~$D$. More information on this definition and hybrid systems can be found in~\cite{goebel12}.
\begin{definition}[Hybrid Basic Conditions,~\cite{goebel12}] \label{def:hybridbasic} A hybrid system~$\mathcal{H}$ as in~\eqref{eq:hybriddef} with data~$(C,f,D,G)$ satisfies the~\emph{hybrid basic conditions} if
\begin{itemize}
    \item~$C$ and~$D$ are closed subsets of~$\mathbb{R}^n$;
    \item~$f$ is a continuous function from~$\mathbb{R}^n \to \mathbb{R}^n$;
    \item~$G:\mathbb{R}^n \rightrightarrows \mathbb{R}^n$ is outer semicontinuous and locally bounded relative to~$D$, and~$D \subset \textnormal{dom } G$.
\end{itemize}
\end{definition}
If a hybrid system meets the hybrid basic conditions, then we say that the system is~\emph{well-posed} (Theorem 6.30,~\cite{goebel12}).

We denote solutions to~$\mathcal{H}$ by~$\phi$, which we parameterize by~$(t,j) \in \mathbb{R}_{\geq 0} \times \mathbb{N}$, where~$t$ denotes the ordinary (continuous) time, and~$j$ denotes the jump (discrete) time. Per Definition 2.3 in~\cite{goebel12},~$\textnormal{dom } \phi \subset \mathbb{R}_{\geq 0} \times \mathbb{N}$ is a~\emph{hybrid time domain} if for all~$(T,J) \in \textnormal{dom } \phi$, the~$\textnormal{dom } \phi \cap ([0,T]\times \{0,1,\dots,J\}$ can be written as~$\bigcup_{j=0}^{J-1} ([t_j,t_{j+1}],j)$ for some finite sequence of times~$0 = t_0 \leq t_1 \leq \dots \leq t_J$. We say that a solution~$\phi$ is~\emph{complete} if~$\textnormal{dom } \phi$ is unbounded. A solution~$\phi$ to~$\mathcal{H}$ is called maximal if it cannot be extended further.

\section{Hybrid System Model} \label{sec:hsm}
In this section, we define a hybrid system model that encompasses all agents' current states and their most recently communicated state values.
Towards defining this model, we first formally define the timer that governs communication events. This allows us to define the hybrid subsystems that are distributed across the agents. Building on this, we present a definition of the hybrid system modelling the~$N$ agents, their algorithm, and the mechanism governing the communication events. Finally, we show the existence of solutions and conclude that all maximal solutions are complete.

\subsection{Mechanism Governing the Communication Events}
We seek to account for intermittent communication events that occur only at some time instances~$t_j$, for~$j\in \mathbb{N}$, that are not known \emph{a priori}. We assume that the sequence~$\lbrace t_j\rbrace_{j=1}^\infty$ is strictly increasing and unbounded. Between consecutive time events, some amount of time elapses which we upper and lower bound with positive scalars~$\tau_{min}$ and~$\tau_{max}$:
\begin{align}
    0<\tau_{min} \leq t_{j+1} - t_j \leq \tau_{max} \quad \forall j \in \mathbb{N} \setminus \{ 0 \}. \label{eq:timer}
\end{align}
The upper bound~$\tau_{max}$ prevents infinitely long communication delays and ensures convergence, while the lower bound~$\tau_{min}$ rules out Zeno behavior.

To generate events at times~$t_j$ satisfying~\eqref{eq:timer}, let~$\tau$ be the timer that governs when agents exchange data, where~$\tau$ is defined by
\begin{align}
    \dot{\tau} &= -1   &&\tau \in [0, \tau_{max}], \\
    \tau^+ &\in [ \tau_{min}, \tau_{max} ] &&\tau = 0 ,
\end{align}
for~$\tau_{min}, \tau_{max} \in \mathbb{R}_{> 0}$. The timer~$\tau$ steadily decreases until it reaches zero. At this point, it is reset to a value within~$[ \tau_{min}, \tau_{max} ]$.

There is indeterminacy built into the timer in that the reset
map is only confined to a compact interval,~$[\tau_{min}, \tau_{max}]$, where~$\tau_{min}$ and~$\tau_{max}$ are both positive real numbers.

\subsection{Hybrid Subsystems}
Recall that agent~$i$ stores and updates its own state variable~$x_i \in \mathbb{R}^{n_i}$, and the variable~$x$ is the vertical concatenation of all agents' states. Data from all agents are collectively stored in~$\eta \in \mathbb{R}^{n}$ at communication events. We define the state of agent~$i$'s hybrid system as~$\xi_i = (x_{i}, \eta, \tau)$, where~$x_{i}$ is agent~$i$'s state (the one it is responsible for updating),~$\eta$ is the memory state storing the states of the agents measured at communication events, and~$\tau$ is defined as above.
This leads to the hybrid subsystem given by
\begin{align}
    \dot{\xi_i} & = \begin{bmatrix}
    -\nabla_{i} L(\eta) \\
    0 \\
    -1
    \end{bmatrix} &&\xi_i \in \mathbb{R}^{n_i} \times \mathbb{R}^n \times [0,\tau_{max}]  \\
    \xi_i^+ &\in \begin{bmatrix}
    x_{i} \\
    x \\
    [\tau_{min}, \tau_{max}]
    \end{bmatrix} &&\xi_i \in \mathbb{R}^{n_i} \times \mathbb{R}^n \times \{0\},
\end{align}
where~$x=(x_1^T, \ldots, x_N^T)^T$.

\subsection{Combined Hybrid System}
We are now ready to combine the distributed subsystems into one hybrid system for analysis. First, we define a variable~$z = (z_1, z_2) \in \mathbb{R}^n \times \mathbb{R}^n$ such that
\begin{align}
    z_1 &= \textnormal{col} (x_1, \dots, x_N) \\
    z_2 &= \eta,
\end{align}
where~$\textnormal{col} (x_1, \dots, x_N) = (x_1^T, \ldots, x_N^T)^T$.

We define the state of the combined hybrid system as~$\xi = (z_1, z_2, \tau) \in \mathcal{X}$, where~$z_1$,~$z_2$, and~$\tau$ are defined as above, and~$\mathcal{X}:=\mathbb{R}^n \times \mathbb{R}^n \times [0,\tau_{max}]$.
This leads to the combined hybrid system~$\mathcal{H}=(C,f,D,G)$ given by
\begin{align}
    \dot{\xi} & = \begin{bmatrix}
    -\nabla L(z_2) \\
    0 \\
    -1
    \end{bmatrix}:= f(\xi)  && \xi \in C, \label{eq:hfirst} \\
    \xi^+ &\in \begin{bmatrix}
    z_1 \\
    z_1 \\
    [\tau_{min}, \tau_{max}]
    \end{bmatrix}:= G(\xi) && \xi \in D, \label{eq:hlast}
\end{align}
where~$C:= \mathcal{X}$ and~$D:=\mathbb{R}^n \times \mathbb{R}^n \times \lbrace 0 \rbrace$.

\subsection{Hybrid Basic Conditions }
We now demonstrate that~$\mathcal{H}$ meets the hybrid basic conditions and is well-posed.
\begin{lemma}
Let~$L$ satisfy Assumption~\ref{as:L}. Then, the hybrid system given by~$\mathcal{H}$ with data~$(C,f,D,G)$ defined in~\eqref{eq:hfirst}--\eqref{eq:hlast} satisfies the hybrid basic conditions from Definition~\ref{def:hybridbasic} and is nominally well-posed as a result.
\end{lemma}
\noindent \emph{Proof: } \ifbool{Report}{The sets~$C$ and~$D$ are closed subsets of~$\mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}$ by definition. Due to our assumption that~$\nabla L$ is continuous,~$f$ is a continuous function from~$\mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}$ to~$\mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}$. By construction,~$G$ is outer semicontinuous and locally bounded relative to~$D$. Finally,~$D \subset \textnormal{dom } G$ because~$\textnormal{dom } G$ is~$\mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}$. \hfill $\blacksquare$}{See~\cite{hendrickson21}.\hfill $\blacksquare$}

\subsection{Existence of Solutions}
In addition to being well-posed, there exists a nontrivial solution to~$\mathcal{H}$ from each point in~$C\cup D$, and all maximal solutions are complete and not Zeno under mild conditions on problem
parameters. Complete solutions cannot be extended further and their domains are unbounded. Practically, this means that the proposed algorithm may run for an arbitrarily long period of time and does not reach a point where it can neither flow nor jump.
\begin{lemma}[Existence of Solutions] \label{lem:solutions}
Let Assumption~\ref{as:L} hold. Let~$\tau_{min}$ and~$\tau_{max}$ be such that~$0 < \tau_{min} \leq \tau_{max} < \frac{\beta^2}{3K^3}$, where~$\beta$ is the strong convexity constant of~$L$ and~$K$ is the Lipschitz constant of~$\nabla L$.
Then there exists a nontrivial solution to~$\mathcal{H}=(C,f,D,G)$ from every initial point in~$C \cup D$. Additionally, every maximal solution~$\phi$ to the hybrid system~$\mathcal{H}$ is complete and not Zeno.
\end{lemma}
\noindent \emph{Proof: } \ifbool{Report}{See the appendix. \hfill $\blacksquare$}{See~\cite{hendrickson21}.\hfill $\blacksquare$}

\section{Convergence Analysis} \label{sec:convergence}
In this section, we define the set for solutions to converge to and present some useful properties of the hybrid system~$\mathcal{H}$ in Lemmas~\ref{lem:contract} and~\ref{lem:gradprop}. We then propose a Lyapunov function in Lemma~\ref{lem:comparison}. As an interim result, we show that for a solution~$\phi = (\phi_{z_1},\phi_{z_2},\phi_{\tau})$ to~$\mathcal{H}$ in~\eqref{eq:hfirst}--\eqref{eq:hlast}, if~$\phi_{z_1}(0,0) = \phi_{z_2}(0,0)$, we are able to bound the distance from the minimizer of~$L$ for all~$(t,j) \in \textnormal{dom }\phi$. Finally, we present our main result, exponential convergence to the minimizer of~$L$, in Theorem~\ref{thm:ges}.

\subsection{Convergence Set}
 Let~$\textbf{0}_n$ be the vector of zeros in~$\mathbb{R}^n$; similarly, let~$\textbf{0}_{n_i}$ be the vector of zeros in~$\mathbb{R}^{n_i}$. Convergence using gradient descent occurs when the gradient of~$L$ is~$\textbf{0}_n$. Given a complete solution~$\phi = (\phi_{z_1},\phi_{z_2},\phi_{\tau})$ to the hybrid system~$\mathcal{H}$, we seek to assure that~$\lim_{t+j \to \infty} \nabla_i L(\phi_{z_2}(t,j)) = \textbf{0}_{n_i},$ for~$i=1, \dots, N$. This is equivalent to a set convergence problem where the set to converge to for the hybrid system~$\mathcal{H}$ is given by \ifbool{Report}{
 \begin{align}
    \mathcal{A} &:= \lbrace \xi = (z_1, z_2, \tau) \in \mathcal{X} : \nabla L(z_2) = \textbf{0}_n, z_2 = z_1, \tau \in [0, \tau_{max}] \rbrace \\
    & =\{x^*\}\times\{x^*\}\times[0, \tau_{max}], \label{eq:Adef}
\end{align}}{
\begin{align}
    \mathcal{A} &:= \lbrace \xi = (z_1, z_2, \tau) \in \mathcal{X} : \\
    &\quad \quad \nabla L(z_2) = \textbf{0}_n, z_2 = z_1, \tau \in [0, \tau_{max}] \rbrace \\
    & =\{x^*\}\times\{x^*\}\times[0, \tau_{max}], \label{eq:Adef}
\end{align}}
where~$x^*$ is the unique fixed point of~$\nabla L$. Equivalence of the expression for~$\mathcal{A}$ stems from Assumption~\ref{as:L}: because~$L$ is strongly convex, it has a unique minimum (denoted by~$x^*$) and this unique minimum is the unique stationary point of~$\nabla L$. Given a vector~$\xi=(z_1,z_2, \tau) \in \mathcal{X}$, the squared distance from~$\mathcal{A}$ is given by~$|\xi|^2_\mathcal{A} := \|z-z^*\|^2 = \|z_1 - x^*\|^2 + \|z_2 - x^*\|^2$, where~$\|\cdot\|$ denotes the Euclidean norm throughout this paper.

\subsection{Useful Properties of~$\mathcal{H}$}
Combining gradient descent  with a bound on~$\tau_{\max}$ allows us to establish relationships that prove useful during Lyapunov analysis.

\begin{lemma} \label{lem:contract}
Let Assumption~\ref{as:L} hold. Consider the hybrid system given by~$\mathcal{H}$ with data~$(C,f,D,G)$ defined
in~\eqref{eq:hfirst}--\eqref{eq:hlast}.
Let~$\tau_{min}$ and~$\tau_{max}$ be such that~$0 < \tau_{min} \leq \tau_{max} < \frac{\beta^2}{3K^3}$, where~$\beta$ is the strong convexity constant of~$L$ and~$K$ is the Lipschitz constant of~$\nabla L$. Denote the unique fixed point of~$\nabla L$ by~$x^*$. Pick a solution~$\phi = (\phi_{z_1},\phi_{z_2},\phi_{\tau})$ to~$\mathcal{H}$ such that $\phi_{z_1}(0,0) = \phi_{z_2}(0,0)$. For each~$I^j := \{ t: (t,j) \in \textnormal{dom } \phi \}$ with nonempty interior and with~$t_{j+1} > t_j$ such that~$[t_j, t_{j+1}] = I^j$, we have
 \begin{align}
     \phi_{z_1}(t,j) &= \phi_{z_2}(t_j,j) - (t-t_j) \nabla L(\phi_{z_2}(t_j,j)) \label{eq:z1update} \\
     \phi_{z_2}(t,j) &= \phi_{z_2}(t_j,j), \label{eq:z2update}
 \end{align}
for all~$t \in (t_j, t_{j+1})$.
 Additionally, for all~$(t,j) \in \textnormal{dom } \phi$, the following are satisfied:
 \begin{align}
     \|\phi_{z_1}(t,j) - x^* \|^2 &\leq q(t,t_j) \|\phi_{z_2}(t_j,j) - x^*\|^2; \quad \label{lem:conpart1}\\
     \|\phi_{z_1}(t,j) - \phi_{z_2}(t,j) \| &\leq \tau_{max} \| \nabla L(\phi_{z_2}(t_j,j)) \|; \label{lem:conpart2} \\
     \|\phi_{z_1}(t,j) - x^* \|^2 &\geq B \|\phi_{z_2}(t_j,j) - x^*\|^2; \label{lem:conpart3}
 \end{align}
 where~$q (t,t_j):= (1 - 2(t-t_j) \beta + (t-t_j)^2 K^2) \in (0,1)$ and~$B:=(1-2\tau_{max} K) \in (0,1)$.
\end{lemma}
\noindent \emph{Proof: } \ifbool{Report}{ Given~$t\in(t_j, t_{j+1})$, the solution~$\phi$ has flowed some distance given by~$(t-t_j)\dot{\phi}$
where~$\dot{\phi}$ is constant due to the sample-and-hold
methodology.
Applying our definition of~$f$ in~\eqref{eq:hfirst} gives~\eqref{eq:z1update}
and~\eqref{eq:z2update}. \\
\emph{Proof of}~\eqref{lem:conpart1}: Using~\eqref{eq:z1update} and the fact that~$\nabla L(x^*) = 0$, we can rewrite~$\|\phi_{z_1}(t,j) - x^*\|^2$ as
\begin{align}
    \|\phi_{z_1}(t,j) - x^*\|^2 &= \|\phi_{z_2}(t_j,j) - (t-t_j) \nabla L(\phi_{z_2}(t_j,j)) - x^* + (t-t_j) \nabla L(x^*)\|^2 \\
    &= \|\phi_{z_2}(t_j,j) - x^*\|^2 -2(t\!-\!t_j) (\nabla L(\phi_{z_2}(t_j,j))\!-\!\nabla L(x^*))^T (\phi_{z_2}(t_j,j)\!-\!x^*) \\
    &\quad + (t-t_j)^2 \|\nabla L(\phi_{z_2}(t_j,j)) - \nabla L(x^*)\|^2,
\end{align}
where the second equality follows from expanding the norm squared. Using the~$\beta$-strong convexity of~$L$ and the Lipschitz property of~$\nabla L$, we upper bound this with
\begin{align}
    \|\phi_{z_1}(t,j) - x^*\|^2 &\leq \|\phi_{z_2}(t_j,j) - x^*\|^2\! -\!2 (t\!-\!t_j) \beta \|\phi_{z_2}(t_j,j) - x^*\|^2 + (t-t_j)^2 K^2 \|\phi_{z_2}(t_j,j) - x^*\|^2 \\
    &= (1\!-\!2(t\!-\!t_j) \beta\!+\!(t\!-\!t_j)^2 K^2) \|\phi_{z_2}(t_j,j)\!-\!x^*\|^2.
\end{align}

For contraction, we must show~$(1\!-\!2(t\!-\!t_j) \beta\!+\!(t\!-\!t_j)^2 K^2) < 1$.
To derive a sufficient condition for this,
note that~$(1\!-\!2(t\!-\!t_j) \beta\!+\!(t\!-\!t_j)^2 K^2) < 1$
may be rewritten as~$(t-t_j) K^2 < 2 \beta$ by subtracting~$1$ from both sides, dividing by~$t-t_j$, and then adding~$2\beta$ to both sides. Using~$\beta \leq K$ and~$\tau_{max} < \frac{\beta^2}{3K^3}$, we have~$(t-t_j) \leq \tau_{max} < \frac{\beta^2}{3K^3} < \frac{2\beta}{K^2}$, and therefore~$(1\!-\!2(t\!-\!t_j) \beta\!+\!(t\!-\!t_j)^2 K^2) < 1$. To show that this term is also positive, it is sufficient to show that~$(1 - 2(t-t_j) \beta) \geq 1 - 2 \tau_{max} K > 0$. This is satisfied for $\tau_{max} < \frac{\beta^2 }{3 K^3} \leq \frac{1}{3 K} < \frac{1}{K}$. Thus,~$q(t,t_j) = (1 - 2(t-t_j) \beta + (t-t_j)^2 K^2) \in (0,1)$.

\emph{Proof of}~\eqref{lem:conpart2}: From~\eqref{eq:z1update}--\eqref{eq:z2update}, we have~$\| \phi_{z_1}(t,j)\!-\!\phi_{z_2}(t,j) \| = \|\phi_{z_2}(t_j,j)\!-\!(t\!-\!t_j) \nabla L(\phi_{z_2}(t_j,j))\!-\!\phi_{z_2}(t_j,j) \|$. Simplifying and bounding~$t-t_j$ above by~$\tau_{max}$ gives the final result.

\emph{Proof of}~\eqref{lem:conpart3}: Using~\eqref{eq:z1update} and the fact that~$\nabla L(x^*) = 0$, we can rewrite~$\|\phi_{z_1}(t,j) - x^*\|^2$ as $\|\phi_{z_2}(t_j,j)\!-\!(t\!-\!t_j) \nabla L(\phi_{z_2}(t_j,j))\!-\!x^*\!+\!(t\!-\!t_j) \nabla L(x^*)\|^2$. Expanding the norm squared gives
\begin{align}
    \|\phi_{z_1}(t,j) - x^*\|^2 &= \|\phi_{z_2}(t_j,j)\!-\!x^*\|^2\! +(t\!-\!t_j)^2 \|\nabla L(\phi_{z_2}(t_j,j))\!-\!\nabla L(x^*)\|^2\\
    &\quad \!-\!2(t\!-\!t_j)(\phi_{z_2}(t_j,j)\!-\!x^*)^T(\nabla L(\phi_{z_2}(t_j,j))\!-\!\nabla L(x^*)).
\end{align}
Dropping the middle term (which is positive)
and using~$(\phi_{z_2}(t_j,j) - x^*)^T(\nabla L(\phi_{z_2}(t_j,j))- \nabla L(x^*)) \leq K \|\phi_{z_2}(t_j,j) - x^*\|^2$, which follows from~$\nabla L$ being~$K$-Lipschitz, we can derive a lower bound:
\begin{align}
    \|\phi_{z_1}(t,j) - x^* \|^2 &\geq (1 -2(t-t_j) K) \|\phi_{z_2}(t_j,j) - x^*\|^2\\
    &\geq (1 -2\tau_{max} K) \|\phi_{z_2}(t_j,j) - x^*\|^2. \label{eq:z1geqz2}
\end{align}
Let~$B := 1 - 2\tau_{max} K$. Then using~$\beta \leq K$, we have $\tau_{max} < \frac{\beta^2 }{6 K^3}  < \frac{1}{2K}$. Therefore,~$B \in (0,1)$.
\hfill $\blacksquare$}{See~\cite{hendrickson21}. \hfill $\blacksquare$}

In preparation for establishing the convergence
properties of~$\mathcal{H}$, we also show that the angle between the gradient of the current state and the gradient of the previously communicated state is never greater than~$90$ degrees as a result of the bound on~$\tau_{max}$. This is formally stated in Lemma~\ref{lem:gradprop}.
\begin{lemma} \label{lem:gradprop}
Let Assumption~\ref{as:L} hold. Consider the hybrid system given by~$\mathcal{H}$ with data~$(C,f,D,G)$ defined
in~\eqref{eq:hfirst}--\eqref{eq:hlast}. Let~$\tau_{min}$ and~$\tau_{max}$ be such that~$0 < \tau_{min} \leq \tau_{max} < \frac{\beta^2}{3K^3}$, where~$\beta$ is the strong convexity constant of~$L$ and~$K$ is the Lipschitz constant of~$\nabla L$. Denote the unique fixed point of~$\nabla L$ by~$x^*$. Pick a solution~$\phi$ such that~$\phi_{z_1}(0,0) = \phi_{z_2}(0,0)$. For each~$I^j := \{ t: (t,j) \in \textnormal{dom } \phi \}$ with nonempty interior and with~$t_{j+1} > t_j$ such that~$[t_j, t_{j+1}] = I^j$, we have
\begin{align}
    \nabla L(\phi_{z_1}(t,j))^T \nabla L(\phi_{z_2}(t,j)) \geq A \|\phi_{z_2}(t_j,j)-x^*\|^2,
\end{align}
for all~$t \in (t_j, t_{j+1})$, where~$A:=\beta^2(1 - 2 \tau_{max} K) - \tau_{max} K^3 > 0$.
\end{lemma}
\noindent \emph{Proof: }\ifbool{Report}{We begin by expanding~$\|\nabla L(\phi_{z_1}(t,j))\|^2$:
\begin{align}
    \|\nabla L(\phi_{z_1}(t,j))\|^2 &= \nabla L(\phi_{z_1}(t,j))^T\Big(\nabla L(\phi_{z_2}(t,j)) + \nabla L(\phi_{z_1}(t,j)) - \nabla L(\phi_{z_2}(t,j))\Big) \\
    &\leq \nabla L(\phi_{z_1}(t,j))^T \nabla L(\phi_{z_2}(t,j)) + \|\nabla L(\phi_{z_1}(t,j))\| \|\nabla L(\phi_{z_1}(t,j)) - \nabla L(\phi_{z_2}(t,j))\| \\
    &\leq \nabla L(\phi_{z_1}(t,j))^T \nabla L(\phi_{z_2}(t,j)) + K \|\nabla L(\phi_{z_1}(t,j))\| \|\phi_{z_1}(t,j) - \phi_{z_2}(t,j)\| \\
    &\leq \nabla L(\phi_{z_1}(t,j))^T \nabla L(\phi_{z_2}(t,j)) + \tau_{max} K \|\nabla L(\phi_{z_1}(t,j))\| \|\nabla L(\phi_{z_2}(t_j,j))\|, \label{eq:z1exp}
\end{align}
where the last two inequalities are from the~$K$-Lipschitz property of~$\nabla L$ which states $\|\nabla L(x) - \nabla L(y)\| \leq K \|x-y\|,$ for all~$x,y \in \mathbb{R}^n$ and~\eqref{lem:conpart2} in Lemma~\ref{lem:contract}. Note that the~$K$-Lipschitz property also gives the inequalities~$\|\nabla L(\phi_{z_1}(t,j))\| \leq K \|\phi_{z_1}(t,j)-x^*\|$ and~$\|\nabla L(\phi_{z_2}(t_j,j))\| \leq K \|\phi_{z_2}(t_j,j)-x^*\|$ because~$\nabla L(x^*) = 0$. Thus,~\eqref{eq:z1exp} becomes
\begin{align}
    \|\nabla L(\phi_{z_1}(t,j))\|^2 &\leq \nabla L(\phi_{z_1}(t,j))^T \nabla L(\phi_{z_2}(t_j,j)) + \tau_{max} K^3 \|\phi_{z_1}(t,j)-x^*\| \|\phi_{z_2}(t_j,j)-x^*\| \\
    &\leq \nabla L(\phi_{z_1}(t,j))^T \nabla L(\phi_{z_2}(t_j,j)) + \tau_{max} K^3 \|\phi_{z_2}(t_j,j)-x^*\|^2, \label{eq:z1Upper}
\end{align}
where the last inequality follows from~\eqref{lem:conpart1} in Lemma~\ref{lem:contract}. Because~$L$ is~$\beta$-strongly convex, we have~$\|\nabla L(\phi_{z_1}(t,j)) - \nabla L(x^*) \|^2 \geq \beta^2 \|\phi_{z_1}(t,j) - z^*\|^2$. To lower bound~\eqref{eq:z1Upper} in terms of~$\|\phi_{z_2}(t_j,j) - x^*\|^2$, we combine this with~$\nabla L(x^*) = 0$ and~\eqref{lem:conpart3} in Lemma~\ref{lem:contract}. This gives the inequality
\begin{align}
\|\nabla L(\phi_{z_1}(t,j))\|^2 &\geq \beta^2 \|\phi_{z_1}(t,j)-x^*\|^2 \\
&\geq \beta^2 B \|\phi_{z_2}(t_j,j) - x^*\|^2, \label{eq:z1Lower2}
\end{align}
where~$B:=(1 - 2 \tau_{max} K)$. Then using~\eqref{eq:z1Lower2} in conjunction with~\eqref{eq:z1Upper} allows us to lower bound~$\nabla L(\phi_{z_1}(t,j))^T \nabla L(\phi_{z_2}(t,j))$:
\begin{align}
    \nabla L(\phi_{z_1}(t,j))^T \nabla L(\phi_{z_2}(t,j)) &\geq \beta^2 B \|\phi_{z_2}(t_j,j) - x^*\|^2  - \tau_{max} K^3 \|\phi_{z_2}(t_j,j)-x^*\|^2 \\
    &=(\beta^2(1 - 2 \tau_{max} K) - \tau_{max} K^3) \|\phi_{z_2}(t_j,j)-x^*\|^2 .
\end{align}
This lower bound is positive for~$\tau_{max} < \frac{\beta^2 }{2 \beta^2K + K^3}$.
Because~$\beta \leq K$, the denominator~$2\beta^2K + K^3 \leq 3K^3$. This lower bound is positive since~$\tau_{max} < \frac{\beta^2 }{3 K^3} \leq \frac{\beta^2 }{2 \beta^2K + K^3}$. \hfill $\blacksquare$}{See~\cite{hendrickson21}. \hfill $\blacksquare$}

\subsection{Bound on the Lyapunov Function}
Central to proving our main result is a Lyapunov function that is bounded above and below by~$\mathcal{K}_\infty$ comparison functions~$\alpha_1, \alpha_2$ given in Lemma~\ref{lem:comparison}.

\begin{lemma} \label{lem:comparison}
Let Assumption~\ref{as:L} hold. Let~$\tau_{min}$ and~$\tau_{max}$ be such that~$0 < \tau_{min} \leq \tau_{max} < \frac{\beta^2}{3K^3}$, where~$\beta$ is the strong convexity constant of~$L$ and~$K$ is the Lipschitz constant of~$\nabla L$. Let~$V : \mathcal{X} \to \mathbb{R}_{\geq 0}$ be a Lyapunov function candidate for the hybrid system~$\mathcal{H}= (C,f,D,G)$ defined
in~\eqref{eq:hfirst}--\eqref{eq:hlast}, given by
\begin{align}
    V(\xi) &= (L(z_1) - L(x^*))^2 + (L(z_2) - L(x^*))^2,
\end{align}
for all~$\xi=(z_1, z_2, \tau) \in \mathcal{X}$, where~$L$ is the objective function and~$x^*$ is the unique fixed point of~$\nabla L$. Then there exist~$\alpha_1, \alpha_2 \in \mathcal{K}_\infty$ such that
\begin{align}
    &\alpha_1(|\xi|_\mathcal{A}) \leq V(\xi) \leq \alpha_2(|\xi|_\mathcal{A})
\end{align}
for all~$\xi \in C \cup D \cup G(D)$. In particular,~$\alpha_1$ and~$\alpha_2$ may be given by, for each~$s\geq 0$,
\begin{align}
    \alpha_1(s) = \frac{\beta^2}{16} s^4 \quad \textnormal{and} \quad \alpha_2(s) = \frac{K^2}{2} s^4.
\end{align}
\end{lemma}
\noindent \emph{Proof: } \ifbool{Report}{The minimizer of~$L$ is~$\xi^* = (z_1^*, z_2^*, \tau) = (x^*, x^*, \tau)$
for any~$\tau$,
and, by construction,~$V(\xi)$ is zero only for~$\xi = \xi^*$ and is positive otherwise.

Because~$\nabla L$ is~$K$-Lipschitz,~$L(x)-L(x^*) \leq \nabla L(x^*)^T (x-x^*) + \frac{K}{2} \|x-x^*\|^2 =  \frac{K}{2} \|x-x^*\|^2,$ for all~$x\in \mathbb{R}^n$. Thus,~$V(\xi)$ may be bounded as
\begin{align}
    V(\xi) \leq \frac{K^2}{4} \|z_1 - x^*\|^4 + \frac{K^2}{4} \|z_2 - x^*\|^4. \label{eq:lxi}
\end{align}
First consider the case where~$\|z_1 - x^*\| \leq \|z_2 - x^*\|$. Then~\eqref{eq:lxi} becomes
\begin{align}
    V(\xi) \leq \frac{K^2}{2} \|z_2 - x^*\|^4 \leq \frac{K^2}{2} |\xi|^4_\mathcal{A}, \label{eq:compupper}
\end{align}
where the last inequality follows from~$\|z_2 - x^*\|^2 \leq |\xi|_\mathcal{A}^2$ by definition of~$|\xi|_\mathcal{A}$. If~$\|z_1 - x^*\| > \|z_2 - x^*\|$ instead, then~\eqref{eq:lxi} is bounded above by
\begin{align}
    V(\xi) < \frac{K^2}{2} \|z_1 - x^*\|^4 \leq \frac{K^2}{2} |\xi|^4_\mathcal{A},
\end{align}
where the definition of~$|\xi|_\mathcal{A}$ is again used in the last inequality. Thus,~$V(\xi) \leq \frac{K^2}{2} |\xi|^4_\mathcal{A}$ for all~$\xi \in \mathcal{X}$ and we set~$\alpha_2(s) = \frac{K^2}{2} s^4 \in \mathcal{K}_\infty$ for all~$s \geq 0$.

The~$\beta$-strong convexity of~$L$ and~$\nabla L(x^*) = 0$ allow us to write~$L(z_1) - L(x^*) \geq \frac{\beta}{2} \|z_1-x^*\|^2$
and~$L(z_2) - L(x^*) \geq \frac{\beta}{2} \|z_2-x^*\|^2$.
Applying both inequalities to the definition of~$V$ gives
\begin{align}
    V(\xi) &\geq \frac{\beta^2}{4} \|z_1-x^*\|^4 + \frac{\beta^2}{4} \|z_2-x^*\|^4. \label{eq:vupper}
\end{align}
First, consider the case where~$\|z_1 - x^*\| \geq \|z_2 - x^*\|$. Then dropping the second term in~\eqref{eq:vupper} and using~$|\xi|^2_\mathcal{A} = \|z_1 - x^*\|^2 + \|z_2 - x^*\|^2 \leq 2 \|z_1 - x^*\|^2$,
\begin{align}
    V(\xi) &\geq \frac{\beta^2}{4}\|z_1-x^*\|^4 \geq \frac{\beta^2}{16} |\xi|_\mathcal{A}^4. \label{eq:complower}
\end{align}
Now consider the case that~$\|z_1 - x^*\| < \|z_2 - x^*\|$. Then the same steps apply and~$V(\xi) > \frac{\beta^2}{16} |\xi|_\mathcal{A}^4$. Thus,~$V(\xi) \geq \frac{\beta^2}{16} |\xi|_\mathcal{A}^4$ for all~$\xi \in \mathcal{X}$.
Accordingly,~$\alpha_1(s) = \frac{\beta^2}{16} s^4$ for all~$s\geq 0$. \hfill $\blacksquare$}{See~\cite{hendrickson21}.\hfill $\blacksquare$}

\subsection{Exponential Convergence}
Using Lemmas~\ref{lem:contract},~\ref{lem:gradprop}, and~\ref{lem:comparison}, we are able to bound the distance to the minimizer of~$L$ over time for a class of initial conditions in Proposition~\ref{prop:conv}. This result will then be expanded to include all possible solutions and initial conditions in Theorem~\ref{thm:ges}, thus showing exponential convergence to the minimizer of~$L$.
\begin{proposition} \label{prop:conv}
Let Assumption~\ref{as:L} hold and consider the hybrid system~$\mathcal{H}$ defined in~\eqref{eq:hfirst}-\eqref{eq:hlast}. Let~$\mathcal{A}$ be as defined in~\eqref{eq:Adef} and let~$\tau_{min}$ and~$\tau_{max}$ be such that~$0 < \tau_{min} \leq \tau_{max} < \frac{\beta^2}{3K^3}$, where~$\beta$ is the strong convexity constant of~$L$ and~$K$ is the Lipschitz constant of~$\nabla L$. For each solution~$\phi$ to~$\mathcal{H}$ such that~$\phi_{z_1}(0,0) = \phi_{z_2}(0,0)$, for all~$(t,j) \in \textnormal{dom } \phi$, the following is satisfied:
\begin{align}
    |\phi(t,j)|_\mathcal{A} &\leq \sqrt{\frac{K}{\beta}} \sqrt[4]{8} \exp \Big( -\frac{\beta A B}{8 K^2} t\Big) |\phi(0,0)|_\mathcal{A},
\end{align}
where~$A=\beta^2(1 - 2 \tau_{max} K) - \tau_{max} K^3 > 0$ and~$B=(1-2\tau_{max} K) \in (0,1)$.
\end{proposition}
\noindent \emph{Proof: } \ifbool{Report}{We first consider~$\xi \in C$ and the Lyapunov function~$V$ defined in Lemma~\ref{lem:comparison}. The partial derivatives of~$V$ with respect to~$z_i$ are given by~$\nabla_{z_i} V(\xi) = 2 \nabla L(z_i) (L(z_i) - L(x^*)) \in \mathbb{R}^n$, for~$i=1,2$, where~$x^*$ is the unique fixed point of~$\nabla L$.
This leads to
\begin{align}
    \langle \nabla V(\xi), f(\xi) \rangle =\!-\!2(L(z_1)\!-\!L(x^*)) \nabla L(z_1)^T \nabla L(z_2). \quad \label{eq:innerdef}
\end{align}
We now pick a solution~$\phi$ such that~$\phi_{z_1}(0,0) = \phi_{z_2}(0,0)$. For each~$I^j := \{t:(t,j) \in \textnormal{dom } \phi \}$ with nonempty interior and with~$t_{j+1} > t_j$ such that~$[t_j, t_{j+1}] = I^j$, we have from Lemma~\ref{lem:gradprop},
\begin{align}
    \nabla L(\phi_{z_1}(t,j))^T \nabla L(\phi_{z_2}(t,j)) &\geq A \|\phi_{z_2}(t_j,j)-x^*\|^2,
\end{align}
where~$A=\beta^2(1 - 2 \tau_{max} K) - \tau_{max} K^3 >0$. Combining this with~$(L(\phi_{z_1}(t,j)) - L(x^*)) \geq \frac{\beta}{2} \|\phi_{z_1}(t,j) - x^*\|^2$ from the~$\beta$-strong convexity of~$L$, and using~\eqref{eq:innerdef} we can write
\begin{align}
    \langle \nabla V(\phi(t,j)), f(\phi(t,j)) \rangle
    &\leq - \beta\|\phi_{z_1}(t,j) - x^*\|^2 \nabla L(\phi_{z_1}(t,j))^T \nabla L(\phi_{z_2}(t,j)) \\
    &\leq - \beta A \|\phi_{z_1}(t,j) - x^*\|^2  \|\phi_{z_2}(t_j,j) - x^*\|^2 \\
    &\leq - \beta A B \|\phi_{z_2}(t_j,j)-x^*\|^4, \label{eq:innerz2}
\end{align}
where the last inequality is from applying~\eqref{lem:conpart3} in Lemma~\ref{lem:contract}.
Additionally, for each~$(t,j)$ in the interval of flow~$[t_j, t_{j+1}] \times \{j\}$ in~$\textnormal{dom } \phi$, we have~$\|\phi_{z_1}(t,j) - x^*\| \leq \|\phi_{z_2}(t_j,j) - x^*\|$ from~\eqref{lem:conpart1} in Lemma~\ref{lem:contract}. Applying this to the definition of~$|\phi(t,j)|_\mathcal{A}^2$ gives the relationship~$|\phi(t,j)|_\mathcal{A}^2 \leq 2 \|\phi_{z_2}(t_j,j) -x^*\|^2$. Squaring both sides allows us to rewrite~\eqref{eq:innerz2} as
\begin{align}
    \langle \nabla V(\phi(t,j)), f(\phi(t,j)) \rangle &\leq - \frac{\beta A B}{4} |\phi(t,j)|_\mathcal{A}^4. \label{eq:dotVupper}
\end{align}
This bound may be related back to~$V(\phi (t,j))$ using the comparison function~$\alpha_2$ from Lemma~\ref{lem:comparison}, which leads to
\begin{align}
    \langle \nabla V(\phi(t,j)), f(\phi(t,j)) \rangle &\leq - \frac{\beta A B}{2K^2} V(\phi(t,j)). \label{eq:tinbound}
\end{align}

We now consider the change of~$V$ at jumps. For each~$\xi \in D,$ and~$g=(g_{z_1},g_{z_2},g_{\tau}) \in G(\xi)$,~$V(g) = (L(g_{z_1}) - L(x^*))^2 + (L(g_{z_2}) - L(x^*))^2$. Thus, we can write the change of~$V$ at jumps as~$V(g) - V(\xi) = (L(g_{z_1}) - L(x^*))^2 - (L(z_2) - L(x^*))^2$ because~$g_{z_1}=z_1$ and~$g_{z_2} = z_1$ from~\eqref{eq:hlast}.

For a solution~$\phi$ such that~$\phi_{z_1}(0,0) = \phi_{z_2}(0,0)$, this is equivalent to
\begin{align}
    V(G(\phi(t_{j+1},j))) - V(\phi(t_{j+1},j)) = (L(\phi_{z_1}(t_{j\!+\!1},j\!+\!1))\!-\!L(x^*))^2\! -\!(L(\phi_{z_2}(t_{j\!+\!1},j))\!-\!L(x^*))^2,
\end{align}
for all~$(t_{j+1},j), (t_{j+1},j+1) \in \textnormal{dom }\phi$. For this quantity to be nonpositive, it is sufficient to show that~$L(\phi_{z_1}(t_{j+1},j+1)) \leq L(\phi_{z_2}(t_{j+1},j))$.

Towards doing this, we leverage~$L(x) \leq L(y) + \nabla L(y)^T(x - y) + \frac{K}{2}\|x - y\|^2$, for all~$x,y\in \mathbb{R}^n$ from the~$K$-Lipschitz property of~$\nabla L$. Applying this for~$x=\phi_{z_1}(t_{j+1},j+1), y=\phi_{z_2}(t_{j+1},j)$, we have
\begin{align}
L(\phi_{z_1}(t_{j+1},j+1)&) \leq L(\phi_{z_2}(t_{j+1},j)) +\nabla L(\phi_{z_2}(t_{j+1},j))^T(\phi_{z_1}(t_{j+1},j+1)\!-\!\phi_{z_2}(t_{j+1},j)) \\
& \qquad \qquad + \frac{K}{2}\|\phi_{z_1}(t_{j+1},j+1)-\phi_{z_2}(t_{j+1},j)\|^2 \\
&= L(\phi_{z_2}(t_{j+1},j))\!+\!\nabla L(\phi_{z_2}(t_{j+1},j))^T(\phi_{z_2}(t_j,j)\!-\!(t_{j+1}\!-\!t_j) \nabla L(\phi_{z_2}(t_j,j))\!-\!\phi_{z_2}(t_{j+1},j)) \\
& \qquad \qquad   +\frac{K}{2}\|\phi_{z_2}(t_j,j)\!-\!(t_{j+1}\!-\!t_j) \nabla L(\phi_{z_2}(t_j,j))\!-\!\phi_{z_2}(t_j,j)\|^2 \\
&= L(\phi_{z_2}(t_{j+1},j)) - \Big(1- \frac{K}{2}(t_{j+1} - t_j)\Big) (t_{j+1} - t_j)\|\nabla L(\phi_{z_2}(t_j,j))\|^2,
\end{align}
where the first equality uses~$\phi_{z_1}(t_{j+1},j+1) = \phi_{z_1}(t_{j+1},j) = \phi_{z_2}(t_j,j) - (t_{j+1} - t_j) \nabla L(\phi_{z_2}(t_j,j))$ from~\eqref{eq:hlast} and~\eqref{eq:z1update}, and the last equality uses~$\phi_{z_2}(t_{j+1},j) =\phi_{z_2}(t_j,j)$. Since~$t_{j+1}\!-\!t_j \leq \tau_{max} \leq \frac{\beta^2}{3K^3} \leq \frac{1}{3K} \leq \frac{2}{K}$, we have the desired property:~$L(\phi_{z_1}(t_{j+1},j+1)) \leq L(\phi_{z_2}(t_{j+1},j))$. Thus,
\begin{align}
    V(G(\phi(t_{j+1},j))) &- V(\phi(t_{j+1},j)) \leq 0. \label{eq:jinbound}
\end{align}

Following the work done in~\cite{chai17} and~\cite{goebel12}, we are able to perform direct integration in order to upper bound~$V(\phi(t,j))$ in terms of~$V(\phi(0,0))$ using~\eqref{eq:tinbound} and~\eqref{eq:jinbound} as bounds. Thus,
\begin{align}
    V(\phi(t,j)) \leq \exp \Big( -\frac{\beta A B}{2 K^2} t\Big) V(\phi(0,0)).
\end{align}
Using the comparison functions given in Lemma~\ref{lem:comparison}, we get a bound for~$|\phi(t,j)|^4$:
\begin{align}
    |\phi(t,j)|^4_\mathcal{A} &\leq \frac{16}{\beta^2} \exp \Big( -\frac{\beta A B}{2 K^2} t\Big) V(\phi(0,0)) \\
    &\leq \frac{8 K^2}{\beta^2} \exp \Big( -\frac{\beta A B}{2 K^2} t\Big) |\phi(0,0)|^4_\mathcal{A}.
\end{align}
Taking the fourth root gives the final answer. \hfill $\blacksquare$}{See~\cite{hendrickson21}. \hfill $\blacksquare$}

In practice, this preliminary result is useful when agreeing on initial values is easy to implement. However, it does not show our desired result, namely, exponential convergence to the minimizer of~$L$, regardless of initialization. By examining all possible scenarios at the first jump, we show in Theorem~\ref{thm:ges} below that exponential convergence to the minimizer of~$L$ still applies after the first jump.

\begin{theorem}[Exponential Convergence] \label{thm:ges}
Let Assumption~\ref{as:L} hold and consider the hybrid system~$\mathcal{H}$ defined in~\eqref{eq:hfirst}-\eqref{eq:hlast}. Let~$\mathcal{A}$ be as defined in~\eqref{eq:Adef} and choose~$\tau_{min}$ and~$\tau_{max}$ such that~$0 < \tau_{min} \leq \tau_{max} < \frac{\beta^2}{3K^3}$, where~$\beta$ is the strong convexity constant of~$L$ and~$K$ is the Lipschitz constant of~$\nabla L$. For each solution~$\phi$ and for all~$(t,j) \in \textnormal{dom } \phi$ such that~$j \geq 1$, the following is satisfied:
\begin{align}
    |\phi(t,j)|_\mathcal{A} &\leq \frac{8}{3} \sqrt[4]{2} \sqrt{\frac{K}{\beta}}   \exp \Big( -\frac{\beta A B}{8 K^2} t\Big) |\phi(0,0)|_\mathcal{A},
\end{align}
where~$A=\beta^2(1 - 2 \tau_{max} K) - \tau_{max} K^3 > 0$ and~$B=(1-2\tau_{max} K) \in (0,1)$.
\end{theorem}
\noindent \emph{Proof: } \ifbool{Report}{ Two initialization scenarios must be considered: $\phi_{z_1}(0,0) = \phi_{z_2}(0,0)$ and~$\phi_{z_1}(0,0) \neq \phi_{z_2}(0,0)$.

For the first case,~$\phi_{z_1}(0,0) = \phi_{z_2}(0,0)$, Proposition~\ref{prop:conv} applies in its original form. This is the best-case scenario that results in the smallest upper bound.

Now consider the second case, when~$\phi_{z_1}(0,0) \neq \phi_{z_2}(0,0)$. We note that after the first jump, all assumptions of Proposition~$\ref{prop:conv}$ hold. Thus, for any solution~$\phi$ where~$(t,j) \in \textnormal{dom } \phi$ such that~$j\geq 1$, we have the following:
\begin{align}
    |\phi(t,j)|_\mathcal{A} &\leq \sqrt{\frac{K}{\beta}} \sqrt[4]{8} \exp \Big( -\frac{\beta A B}{8 K^2} t\Big) |\phi(t_1,1)|_\mathcal{A}, \label{eq:jgeq1}
\end{align}
where~$t_1$ denotes the time of the first jump.

We now seek to bound~$|\phi(t_1,1)|_\mathcal{A}$ in terms of~$|\phi(0,0)|_\mathcal{A}$. We begin by expanding and simplifying~$|\phi(t_1,1)|_\mathcal{A}^2$ using that~$\phi_{z_2}(t_1,1) = \phi_{z_1}(t_1,1)$ after the first jump:
\begin{align}
    |\phi(t_1,1)|_\mathcal{A}^2 &= \|\phi_{z_1}(t_1,1) - x^*\|^2 + \|\phi_{z_2}(t_1,1) - x^*\|^2= 2 \|\phi_{z_1}(t_1,1) - x^*\|^2, \label{eq:jumpnorm}
\end{align}
where~$x^*$ is the unique fixed point of~$\nabla L$.
Applying the flow dynamics and using~$\phi_{z_1}(t_1,1) = \phi_{z_1}(t_1,0)$ allows us to rewrite the distance of~$\phi_{z_1}$ from the minimizer of~$L$ after the first jump as
\begin{align}
    \|\phi_{z_1}(t_1,1) - x^*\|^2\!&=\!\|\phi_{z_1}(0,0)\!-\!t_1 \nabla L(\phi_{z_2}(0,0))\!-\!x^* \|^2 \\
    &\leq\|\phi_{z_1}(0,\!0)\!-\!x^* \|^2\!+\!2 t_1 \|\phi_{z_1}(0,\!0)\!-\!x^*\| \|\nabla L(\phi_{z_2}(0,\!0))\| + t_1^2 \|\nabla L(\phi_{z_2}(0,0)) \|^2 \\
    &\leq\|\phi_{z_1}(0,\!0)\!-\!x^* \|^2\!+\!2 t_1 K \|\phi_{z_1}(0,\!0)\!-\!x^*\| \|\phi_{z_2}(0,\!0)\!-\!x^*\| + t_1^2 K^2 \|\phi_{z_2}(0,0) - x^*\|^2, \label{eq:jumpexp}
\end{align}
where the last inequality comes from the~$K$-Lipschitz property of~$\nabla L$ and the fact that~$\nabla L(x^*) = 0$.
If~$\|\phi_{z_1}(0,0) - x^* \| \leq \|\phi_{z_2}(0,0) - x^* \|$, then~\eqref{eq:jumpexp} becomes
\begin{align}
    \|\phi_{z_1}(t_1,1)\!-\!x^*\|^2 &\leq (1\!+\!2t_1K\!+\!t_1^2K^2) \|\phi_{z_2}(0,0)\!-\!x^*\|^2 \\
    &\leq (1+t_1K)^2 |\phi (0,0)|_\mathcal{A}^2.
\end{align}
Similarly, if~$\|\phi_{z_1}(0,0) - x^* \| > \|\phi_{z_2}(0,0) - x^* \|$,  then~\eqref{eq:jumpexp} becomes~$\|\phi_{z_1}(t_1,1)\!-\!x^*\|^2 < (1+t_1K)^2 |\phi (0,0)|_\mathcal{A}^2$.
Thus, in both cases,
\begin{align}
    \|\phi_{z_1}(t_1,1)\!-\!x^*\|^2 \leq (1+t_1K)^2 |\phi (0,0)|_\mathcal{A}^2 \label{eq:firstflow}
\end{align}
Applying this to~\eqref{eq:jumpnorm} and using~$t_1 \leq \tau_{max} < \frac{\beta^2}{3K^3} < \frac{1}{3K}$, we get the bound
\begin{align}
    |\phi(t_1,1)|^2_\mathcal{A} &\leq 2 \left(\frac{4}{3}\right)^2 |\phi (0,0)|_\mathcal{A}^2. \label{eq:phit1bound}
\end{align}
Taking the square root and applying to~\eqref{eq:jgeq1} gives the final result for any~$(t,j) \in \textnormal{dom } \phi$ such that~$j\geq 1$. \hfill $\blacksquare$}{See~\cite{hendrickson21}. \hfill $\blacksquare$}

\section{Numerical Validation} \label{sec:sim}
We consider~$N=n$ agents for various values of~$n$. Each agent updates a scalar and they minimize
\begin{align}
    L(x) = \frac{1}{2}x^T Q x + b^Tx,
\end{align}
where~$x\in \mathbb{R}^n$,~$Q$ is a~$n \times n$ symmetric, positive definite matrix, and~$b\in \mathbb{R}^n$.
To form Q, we decompose a random~$n \times n$ matrix into an unitary orthogonal matrix~$U$ and a matrix~$D$ that contains only our desired eigenvalues on the diagonal. We use these two resulting matrices to set~$Q = U^T D U$. Our choice of eigenvalues varies by trial (discussed below) with the minimum eigenvalue corresponding to~$\beta$ and the maximum eigenvalue corresponding to~$K$. The entries of~$b$ are
set to random values between 1 and 5.
Simulations used
the HyEq Toolbox (Version 2.04)~\cite{HyEqToolbox}\footnote{Simulation code for this section may be found at www.github.com/kathendrickson/DistrHybridGD.}.

\begin{figure}[t!]
\centering
\includegraphics[width=8.8cm, trim={.25cm, .25cm, .25cm, .25cm},clip]{fig/FirstJump_combined.png}
\caption{Effect of initial values on convergence for two trials, where flows are denoted with solid lines and jumps with stars and dashed lines. Trial 1 sets~$\phi_{z_1} (0,0) = \phi_{z_2} (0,0)$, while Trial 2 sets~$\phi_{z_2} (0,0) = x^*$ instead. The first jump does not increase the distance from the minimizer in Trial 1 but does increase this distance in Trial 2. However, after the first jump, progress continues toward the optimum and differences between trials diminish.}
\label{fig:jump}
\end{figure}

We first compare convergence results for different initial values of~$\phi_{z_1}$ and~$\phi_{z_2}$ for five agents. For the first trial, we consider the case where~$\phi_{z_1} (0,0) = \phi_{z_2} (0,0) = (2,2,2,2,2)^T$. In Trial 2, we consider the ``worst-case'' initialization scenario: when~$\phi_{z_1} (0,0) = (2,2,2,2,2)^T$ is some distance from the optimum but~$\phi_{z_2} (0,0) = x^*$, resulting in an increase in the distance from the minimizer of~$L$ before the first jump. We consider~$\beta = K = 5$ and set~$\tau_{max} = \frac{\beta^2}{3K^3 + 1}$ and~$\tau_{min} = \frac{1}{2}\tau_{max}$. Figure~\ref{fig:jump} shows the distance from optimum through the first twenty jumps for both trials. There is a consistent decrease in the distance to the minimizer, even at jumps, for the first trial. In contrast, when initial values for~$\phi_{z_1}$ and~$\phi_{z_2}$ are not equal, there is an increase in distance to the minimizer after the first jump in the second trial. However, as expected, distance to the optimum
decreases exponentially thereafter, with the difference between the two trials decreasing over time.

\begin{figure}[t!]
\centering
\includegraphics[width=8.8cm, trim={.25cm, .25cm, .25cm, .25cm},clip]{fig/networksize.png}
\caption{Effect of network size on convergence. Convergence results still hold even for very large network sizes, demonstrating the scalability of our algorithm.}
\label{fig:networksize}
\end{figure}
\balance
We then examined the effects of varying the network size from 5 agents to 100, 500, 1000, and 5000 agents. We set~$\beta = 2$ and~$K = 4$ and chose to initialize~$\phi_{z_1}$ and~$\phi_{z_2}$ with vectors of twos in~$\mathbb{R}^n$. For each network size, the matrix~$Q$ and vector~$b$ were randomly generated. As shown in Figure~\ref{fig:networksize}, drastically expanding the network size does not have a significant impact on convergence. This demonstrates our algorithm's scalability and convergence results that hold regardless of network size.

\section{Conclusion} \label{sec:concl}
This paper presented a hybrid systems framework for analyzing continuous-time multi-agent optimization with discrete-time communications. Using this framework, we established that every maximal solution is complete, as well as the exponential convergence of a block coordinate descent law to the minimizer of a strongly convex and smooth objective function. Future work in this area includes the use of heterogeneous timers and exploration of other update laws, as well as constrained problems.

\ifbool{Report}{

redpajama_set_name: RedPajamaArXiv


text:
Tribute on Mars to Alan Shepard and the flight of Freedom 7
Posted by EarthSky in Space | May 5, 2011
The Freedom 7 crater on Mars is 82 feet in diameter, matching the height of Shepard's Redstone rocket.
The team exploring Mars with NASA's Opportunity rover for the past seven years has informally named a Martian crater for the Mercury spacecraft that astronaut Alan Shepard christened Freedom 7. Shepard piloted Freedom 7 in America's first human spaceflight on May 5, 1961.
Alan Shepard after landing. Shepard's flight was the first of six Project Mercury missions piloted by solo astronauts. Image Credit: NASA
The team is using Opportunity this week to acquire images of a cluster of small, relatively young craters along the rover's route toward a long-term destination. The cluster's largest crater, spanning about 25 meters (82 feet), is Freedom 7. The diameter of Freedom 7 crater is equivalent to the height of the Redstone rocket that launched Shepard's flight.
Scott McLennan of the State University of New York at Stony Brook, who is this week's long-term planning leader for the rover science team, said:
Many of the people currently involved with the robotic investigations of Mars were first inspired by the astronauts of the Mercury Project who paved the way for the exploration of our solar system.
The Freedom 7 crater on Mars. Image Credit: NASA/JPL-Caltech
Rover team member James Rice of NASA Goddard Space Flight Center, Greenbelt, Maryland, said:
The first 50 years of American manned spaceflight have been built upon immeasurable courage, dedication, sacrifice, vision, patriotism, teamwork and good old-fashioned hard work, all terms that embody and define the United States and her people. Alan Shepard's brave and historic 15-minute flight in Freedom 7 put America in space, and then a scant eight years later, Americans were standing upon the surface of the moon.
Shepard himself would later walk on the moon when he commanded the Apollo 14 mission in early 1971, fewer than 10 years after his Freedom 7 flight. He died on July 21, 1998.
By observing craters of diverse ages, the Opportunity mission is documenting how impact craters change with time. The cluster that includes Freedom 7 crater formed after sand ripples in the area last migrated, which is estimated to be about 200,000 years ago.
Matt Golombek, rover team member at NASA's Jet Propulsion Laboratory in Pasadena, California, said:
This cluster has about eight craters, and they're all the same age. They're from an impactor that broke up in the atmosphere, which is quite common.
Artist's concept of Mars exploration rover. Image Credit: NASA/JPL/Cornell University
Opportunity and its twin, Spirit, completed their three-month prime missions on Mars in April 2004. Both rovers continued for years of bonus, extended missions. Both have made important discoveries about wet environments on ancient Mars that may have been favorable for supporting microbial life. Spirit has not communicated with Earth since March 2010. Opportunity remains active. It has driven 28.6 kilometers (17.8 miles) total on Mars, including 1.9 kilometers (1.2 miles) since leaving Santa Maria crater on March 24, 2011, having studied that crater for three months.
Bottom line: To commemorate the 50th anniversary of Alan Shepard's flight on May 5, 1961, the team exploring Mars via NASA's Opportunity rover for the past seven years has informally named a Martian crater Freedom 7.
Via NASA JPL
Mars Exploration Rovers
Paul Niles: Rocks may represent habitable environment on ancient Mars

redpajama_set_name: RedPajamaCommonCrawl


Bamboo Cheese Board with Handle – Matchmade Designs LLC.
Look out cheese, there's a new cutting board in town! Handmade in our woodworking shop this cheese board is beautiful and compliments your kitchen and get togethers. Available in Bamboo only.


text:
Home The Yarn It's all about respect
It's all about respect
By Siddharth Suresh
In November, the Australian Catholic University (ACU) held its fifth Parliamentary Interfaith Breakfast at Parliament House in Melbourne. This year, parliamentarians had the opportunity to meet with leaders from more than 17 faiths including various Christian, Muslim, Jewish, Hindu, Buddhist, Sikh and other communities—with the intention of celebrating shared values. The leaders of five different faiths delivered readings on the importance of religious freedom, leadership, service and community harmony, and prayed for wise deliberations and good governance. Archbishop of Melbourne Peter A Comensoli, who was also a guest at the event called it "a simple and joyful celebration of freedom in Australia to practice one's religion".
Meetings like the Interfaith dialogue are a powerful tool in building a respectful understanding of cultural differences, and therefore meetings like this are more than welcomed in today's world. It's no wonder Victorian Deputy Premier the Hon James Merlino MP said one of Victoria's greatest strengths was its history of multiculturalism and religious diversity.
November 25 marked the International Day for the Elimination of Violence against Women and it is towards this end goal that the Stop it at the Start campaign is working tirelessly.
Their current campaign advertising and supporting activities encourage communities to ask themselves—could we be teaching our children disrespect? It shows how young people can misinterpret words and actions, and how small changes can make a big difference. To support the Indian community to continue this important conversation, online tools have been developed and are available in many languages on the campaign website. Read more about it here.
Our cover story this issue turns the spotlight on power couple KD and Nitu Ahuja. KD is among the leading realtors, while Nitu is an artist. KD has extensive knowledge of the Western Suburbs and has over 15 years of experience in sales and business development. Nitu's paintings are a "play of light and shadows" in a monochromatic palette. Both have been incredibly successful in their fields of expertise—KD having won numerous awards in the real estate business, while Nitu was recently commissioned by the Consulate General of India—Birmingham to make a custom designed painting on the Kartarpur Corridor. An inspiring story to read.
Diversity, inclusion, key themes at Parliament House interfaith breakfast
Vic Police celebrate tenth annual multi-faith dinner
Religious leaders serve Christmas lunch to disadvantaged community
Syro-Malabar Church to plant roots in Australia
Celebrating Israel's Independence Day
Previous articleTwo Australians in MIT Technology Review's 20 emerging innovators under 35 for Asia Pacific region
Next articleCelebrating Haryana
Work begins on historic Bridge Hotel, pub and restaurant in the offing
it-s-all-about-respect-the-indian-sun
Work begins on historic Bridge Hotel, pub and restaurant in the...


Craig Livingston, a history professor at Lone Star College-Montgomery, performs bagpipe ballads on campus to commemorate the 100th anniversary of the end of World War I.
For a few weeks now, students at Lone Star College-Montgomery may have heard the sound of bagpipes resonating throughout campus intermittently.
The man behind the pipes, professor Craig Livingston, has been performing on campus to commemorate the 100th anniversary of the end of World War I. The so-called "war to end all wars" ended in an armistice on Nov. 11, 1918.
While donning his traditional kilt, Livingston played songs such as "The Bloody Fields of Flanders," "King George V's Army" and "Battle of the Somme," all of which are remembrances of the sacrifices made in WWI.
Sacrifices which were indeed great: millions of people, both military and civilian, died during the fight involving more than 30 countries.
"World War I was a huge tragedy," said Linvingston, a U.S. History professor.
That aspect, he added, is important for students to realize.
"For the Americans who died there, they went there for an ideal to restore harmony to the nations. The Americans died for France, and France remembers it that way," Livingston said.
Livingston said that though lives were lost, the war proved that the sacrifices were not made in vain.
"It showed that there were things worth fighting for." he added.
In 2014, Livingston organized a similar commemoration on the start of WWI in which he taught a class on the war. This Sunday closes out the four-year remembrance.
In addition to the piping and the class, Livingston has also organized special lectures, movie nights and a slate of students to man an information table about the war around campus in the week or so leading up to the anniversary.
While short student-made videos played on a television next to her, Jessica Caruth interacted with students walking by. She's taking a history class covering the war, taught by Livingston, during this semester.
"It's nice to be able to inform people about World War I," Caruth said.
Caruth and Jocelyn Haro, another student at the information table, were also making felt sticky-backed poppies to hand out to students.
Poppies, usually red or white flowers, have become a symbol of remembrance for the soldiers who lost their lives during wartime, as well as the coming of peace.
Haro said that a few students knew some details about the war, but she was able to tell them about the importance of the felt flowers. Several students took flowers with them to wear on their shirts.


Ask our leasing office about our preferred employer discounts! Qualifying new residents receive $300 off the first month and 1/2 off the security deposit.
Two pets per apartment. No weight limits for dogs however there are breed restrictions. Contact a leasing professional for details. Deposits and pet rents vary by size and count.


text:
using System.Reflection;
using System.Runtime.CompilerServices;
using System.Runtime.InteropServices;
using Starcounter.Internal;

// General Information about an assembly is controlled through the following
// set of attributes. Change these attribute values to modify the information
// associated with an assembly.
[assembly: AssemblyTitle("KitchenSink")]
[assembly: AssemblyDescription("")]
[assembly: AssemblyConfiguration("")]
[assembly: AssemblyCompany("")]
[assembly: AssemblyProduct("KitchenSink")]
[assembly: AssemblyCopyright("")]
[assembly: AssemblyTrademark("")]
[assembly: AssemblyCulture("")]

// Setting ComVisible to false makes the types in this assembly not visible
// to COM components.  If you need to access a type in this assembly from
// COM, set the ComVisible attribute to true on that type.
[assembly: ComVisible(false)]

// The following GUID is for the ID of the typelib if this project is exposed to COM
[assembly: Guid("526edd06-b4f0-469f-a4d4-e95b3c94f5ac")]

// Version information for an assembly consists of the following four values:
//
//      Major Version
//      Minor Version
//      Build Number
//      Revision
//
// You can specify all the values or you can default the Build and Revision Numbers
// by using the '*' as shown below:
// [assembly: AssemblyVersion("1.0.*")]
[assembly: AssemblyVersion("1.0.0.0")]
[assembly: AssemblyFileVersion("1.0.0.0")]

// Assures the current assembly has a reference to the Starcounter
// assembly. A reference to Starcounter is currently required for
// Starcounter to detect that an assembly should be hosted.
[assembly: StarcounterAssembly()]

redpajama_set_name: RedPajamaGithub


text:
Original Equipment Suppliers Association (OESA)
25925 Telegraph Road
Automotive Trade Associations
The Original Equipment Suppliers Association (OESA) represents business interests of more than 400 members that provide parts to OEM vehicle manufacturers. Since 1998, OESA has been committed to focusing on issues throughout the OEM parts supply chain, and advocating on behalf of automotive suppliers. OESA is a non-profit trade association and therefor complies with antitrust guidelines.
OESA is one of the four main divisions of the Motor & Equipment Manufacturer's Association (MEMA).
Become a member of OESA
Companies that join OESA are part of a network of more than 430 companies driving the content of events and council meetings. OESA members represent every tier, every component and every size of organization in both the public and private sectors. The organization is run by a board of directors consisting of CEOs and other senior management from member companies who echo the diversity of members. The board give direction and strategy for all current and future OESA initiatives. The organization's staff is dedicated to producing timely and relevant programing, information, and thought leadership to members. This team is assisted and supported by the MEMA government affairs staff in Washington, D.C. who works diligently in Washington, D.C., and a the state level on behalf of OESA members.
To join OESA follow this link.
Please note this website is not an official representative of OESA. "OESA", the OESA logo, "MEMA" and the MEMA logo are registered trademarks or trademarks of MEMA, its subsidiaries, affiliates or licensees in the United States and worldwide.
info@oesa.org
www.oesa.org


More than 4,500 accredited institutions make up U.S. higher education in the United States. Unlike many countries, U.S. higher education institutions are not centrally organized or managed but are accredited on a national or regional level by independent accrediting bodies. Studying in the United States and abroad doesn't just give you tangible degrees and certificates. Your experience says something about who you are. Living and studying in another country—especially where your language isn't spoken—is challenging, requires courage and a positive attitude. Sometimes, these characteristics are more influential than your degree.
Canada is the home of some of the world's top research facilities and academic institutions. Imagine learning in supportive academic environments where professors are approachable and classrooms reflect the cultural diversity of our land. Another reason that makes Canada a dream education destination is that some international students who have graduated from Canadian Institution are allowed to apply for permanent residency visa without moving out of the country. Also, it's a fact that over 90% of students who graduate from Canadian institutions get employed in their field of studies within 6 months after graduation, which is why studying in Canada is said to be an all-encompassing experience.
Ready to Study in the US and Canada?
Canada is a North American country stretching from the U.S. in the south to the Arctic Circle in the north.
The U.S. is a country of 50 states covering a vast swath of North America.


This is a new feature in TabSRMM 3, however, it needs some clarification when and how it works, because some people reported that they cannot get it to work, so I investigated a bit further.
First, it requires Windows 7 (obviously). Second, it requires that the task bar is configured to show large icons (right click the taskbar, choose Properties... and uncheck "Use small icons"). And third, it must be enabled in TabSRMM under Message Sessions->Advanced tweaks->Use Windows 7 task bar features. The setting is enabled by default and you can leave it on even when you are not running Miranda on Windows 7 in which case the code is disabled, regardless of the setting.
After checking that these 3 prerequisites are fulfilled, it should work.
If it doesn't, then because you have pinned Miranda to the task bar AND are using the "icon only" mode ("Always combine, hide labels") checked in the aforementioned properties dialogue. In this case, Miranda itself (the contact list to be more specific) will "take over" the task bar button and the message window will not be able to change the icon. It can still set the overlay icon though, but it cannot replace the big Miranda icon with the user's avatar. So, in this case, you will always see the big Miranda icon on the task bar and the status icon of the current message tab will be visible as an overlay.
This is how it should look like (note that text labels on the task bar are enabled - in this case, you can pin Miranda to the task bar and still get the avatars).
To sum it up: Miranda pinned to the task bar AND using the "Always combine" mode will prevent TabSRMM from showing avatars on the task bar. In any other configuration it should work.
In addition, if the avatar still doesn't show up, try pinning then unpinning the message window, this will temporarily allow the avatars to show (not a permanent fix, but it might be a clue). Alternatively, creating a new profile also solved the issue for me (this is a permanent solution, but took a lot of work re-customizing everything).


I am using the following scenario, My switch 2 is connected with router R6's e1/0, I am creating two sub interfaces on e1/0.1 and e1/0.2 on the router both sub-interfaces have the ip addresses assigned, int e1/0 with no ip and is up. R6's e1/0.1 is in vlan 14 and e1/0.2 is in vlan 15, the link between R6 and SW2 is configured as trunk on switch and vlan 15 is set as native vlan. On R6 side I set the native vlan 15 on e1/0.2 interface. The problem I am facing that I can't ping e1/0.2 sub-interface but I can ping e1/0.1 sub-interface. I created a SVI for e1/0.2 on switch but still I am unable to ping it from switch. Could anyone let me know that what am i missing?
I imagine that you can't configure trunk on routers ETHERNET subs.
d]"deb ip icmp" & "ping <Switch SVI>"
Outputs for giving any further advice.
Just a quick note, if I don't change native vlan to 15 then the e0/1.2 sub-interface works as well. Below are the Configs of the interfaces.
There probably a small thing I am missing.
okay. nice thing here. I've just spent half an hour playing with it in dynamips and got it working. I admit there seems to be an order of operation problem.
the difference? the native vlan is configured on the FIRST subinterface and even after that I had to shut both end of the link and unshut switch than router.
It is still not working for me. Same thing is happening. I changed the config and put the native vlan 15 on e0/1.1 and vlan 14 on e0/1.2, saved the config. reloaded both router and switch but same result. vlan 14 can communicate i.e e0/1.2 but native vlan 15 e0/1.1 cannot. Below are the configs. Any help greatly appreciated.
On switch side the config remained the same.
Is vlan 15 created on SW2?
Please can you paste in a "show vlan" from the SW2 device.
- is it real switch or dynamips? Which one?
- is 'vlan dot1q tag native' enabled globaly on switch?
- did you configure all vlans on your switch?
- what IOS are you using?
- did you try to do the same config on FASTETHERNET link?
The limitation of ethernet 10Mb/s link is that native vlan must be 1.
Go and try the configuration on fast ethernet.
Actually, you can configure it as trunk port but it will only works for native vlan 1.


text:
Back Vanessa Hua: Kindred spirits in the heroines of Little Women and the Little House books Print
Influences October 4, 2016
Vanessa Hua: Kindred spirits in the heroines of Little Women and the Little House books
Deceit and Other Possibilities by Vanessa Hua (Willow Books, 2016)
Our series of guest posts by contemporary American writers discussing their influences continues today with a post by Vanessa Hua, whose debut short story collection, Deceit and Other Possibilities, has just been released by Willow Books.
Hua has playfully summarized Deceit to Flavorwire as "model minorities behaving badly"; its stories focus on members of various U.S. immigrant communities as they contend, not always successfully or admirably, with major life choices. Early admirer Gary Shteyngart calls Deceit "fast-paced, dazzling, smart, and fun" for its depictions of "the insanities and heartbreaks on both sides of the Pacific."
Below, Hua recounts the early inspiration she found in two classic American authors.
My parents don't have any mementos from their childhood. No toys, no yearbooks. Born in China, they grew up during the war against Japan and, later on, the Communists. Their families fled to Taiwan and in the early 1960s my parents were among the wave of engineering and science students who arrived in this country on graduate school fellowships. My mother landed in Chicago, my father in Iowa. They brought a few clothes but no keepsakes, nothing that offered clues to their lives before coming to America.
I knew that, and yet I still believed the copy of Little Women I found on the bookshelf—with the yellowing pages, tattered green cover, and musty smell—had to have been my mother's, something that she'd read in college, even though she was a scientist and not an English major, even though I'd never once seen her read a work of fiction.
Somehow, the book showed up in our house, and I invented its past, a past that I wanted to believe in. As the daughter of immigrants, I often pieced together explanations my parents couldn't—or wouldn't—provide. Through observation, through imagination and through books, I filled in the gaps of lost or hidden history.
1871 portrait of Louisa May Alcott by George Healy. (Wikimedia Commons)
From childhood through high school, I read Little Women at least twice a year, inspired by Jo March who wrote her own fate by selling her stories. I read Laura Ingalls Wilder's Little House books almost as avidly. Laura Ingalls (Wilder's alter ego in the series) also excelled as a writer, as we see in the chapter where she wins the teacher's praise when she writes an essay about ambition. Though we never learn in the series how and when she wrote her books, her name on the cover was proof enough that a feisty girl like her—like me—could get published someday.
In them, I found a mirror, even though we didn't share the same ethnicity or geography or era.
We were all strivers: my striving that of an immigrant family's, Jo's born out of fallen fortunes, and Laura's from pioneer necessity. We were outsiders, too: my family was among a handful of Chinese Americans in my suburban hometown, Jo was a whistling tomboy, and Laura chafed at comparisons to her obedient, golden-haired older sister.
Laura Ingalls Wilder c. 1887. (Fine Art Images/Heritage Images/Getty Images)
Many characters in my short story collection are strivers, too, their ambitions leading them into calamities of their own making. After re-reading Little Women recently, I found it more old-fashioned than I remembered—at times sanctimonious. I've also come to understand that the Little House novels emerged from an often-tortured collaboration between Laura Ingalls Wilder and her daughter, Rose Wilder Lane, and that the books skipped over the more painful aspects of the family's history.
That doesn't change the impact those books had upon my development, how they made me feel more at home because I recognized characters who shared my dreams. What I remembered, what I held onto, is what I needed then to make my way in the world.
The cover has fallen off my copy of Little Women, and the pages are even more yellowed, the color of a pirate's treasure map. With each generation, my family's roots deepen in this country. Like me, my five-year-old twins love books, pulling the ones that intrigue them off the shelf and demanding we read to them.
They're curious about what I did at their age. "Mama, what toys did you play with, when you were a girl? What books did you read, when you were a girl?" Soon enough, they'll discover Little Women—and the book will become an heirloom at last.
Vanessa Hua is a columnist for the San Francisco Chronicle and the recipient of a 2015 Rona Jaffe Foundation Writers' Award and a Steinbeck Fellowship in Creative Writing from San José State University. Her journalism and fiction have appeared in the New York Times, FRONTLINE/World, The Atlantic, ZYZZYVA, Guernica, and other outlets. Hua's first novel, A River of Stars, is forthcoming from Ballantine Books. She lives in the San Francisco Bay Area with her husband and twins.
Classic Hollywood Little Women captures the story's appeal for all time
Susan Cheever on Louisa May Alcott: "A writer who refused conventional ideas of women's roles"
How Louisa May Alcott revolutionized the "book for girls" with Little Women
Related Writers: Laura Ingalls Wilder Louisa May Alcott
Related Volumes: Louisa May Alcott: Little Women, Little Men, Jo's Boys
Cara Blue Adams: Denis Johnson's "lush minimalism" and other influences on You Never Get It Back
Shangyang Fang: Hearing the "meticulous music" in contemporary American poetry
Ray Bradbury: Prophetic visionary, "word-wizard," and next-door neighbor
Shane McCrae: My war with John Ashbery


We launched Proyecto Up! a new initiative to develop Project Management skills in kids and young.
They learn problem-solving, team-working, global citizenship, leadership and communication skills through a game dinamic.
We will be gratefull to receive your comments and opinions!


Plenty Valley Christian College is a well-established, independent, co-educational, K-12 College of around 790 students. It is situated in a magnificent semi-rural setting at the edge of the northern growth corridor of Melbourne.
This is an exciting opportunity to lead an already strong Christian educational community into a new phase of development.
We are currently seeking Diploma Qualified Early Childhood Teachers/Assistants for casual relief work.
An innovative leader with a heart for and an understanding of Christian Education.


This is a brand page for the MEAN GENE'S BURGERS trademark by ORION FOOD SYSTEMS, L.L.C. in SIOUX FALLS, SD, 57118.
Write a review about a product or service associated with this MEAN GENE'S BURGERS trademark. Or, contact the owner ORION FOOD SYSTEMS, L.L.C. of the MEAN GENE'S BURGERS trademark by filing a request to communicate with the Legal Correspondent for licensing, use, and/or questions related to the MEAN GENE'S BURGERS trademark.
On Tuesday, April 14, 1998, a U.S. federal trademark registration was filed for MEAN GENE'S BURGERS. The USPTO has given the MEAN GENE'S BURGERS trademark serial number of 75467769. The current federal status of this trademark filing is EXPIRED. The correspondent listed for MEAN GENE'S BURGERS is Gregory C. Golla of Merchant & Gould P.C., P.O. BOX 2910, MINNEAPOLIS, MN 55402 . The MEAN GENE'S BURGERS trademark is filed in the category of Advertising, Business & Retail Services . The description provided to the USPTO for MEAN GENE'S BURGERS is Franchising services, namely, offering technical assistance in the establishment and/or operation of restaurants or carry-out food shops.
Last Applicant/Owner: ORION FOOD SYSTEMS, L.L.C.
2009-05-14 REGISTERED - SEC. 8 (6-YR) ACCEPTED & SEC. 15 ACK.
Trademarkia-Network law firms can help you incorporate a business around your MEAN GENE'S BURGERS trademark in less than 5 minutes. Trademarkia makes the process easy and convenient, so start now!
Gregory C. Golla is a correspondent of MEAN GENE'S BURGERS trademark.
MEAN GENE'S BURGERS is providing Franchising services, namely, offering technical assistance in the establishment and/or operation of restaurants or carry-out food shops.


text:
Bargaining Survey 2023
January 13, 2023 in Contract News
We have a path to power and it starts with members! While we won a strong two-year contract last year, steps were not implemented as recommended by the Mercer study. We have an opportunity to re-open the contract this year to seek an increase in wages and do what it takes to secure a more equitable salary structure for Support Staff.
We kicked off the New Year with a series of Town Hall webinars this week to share CSUEU's Path to Power plan. Step 2 is to roll out a bargaining survey to assess members' priorities and commitment to winning in 2023.
Please take time to complete the important survey. Some questions center on immediate versus long-term gains. Members' input will help guide the Bargaining Team's work in the coming months.
We will bargain in good faith, engaging frequently with members to provide updates. Most importantly, we will organize support actions to show that CSUEU members are united in our demands.
Deadline to submit survey is Jan. 30. A link to the survey in Spanish will be distributed next week.


This is how it works: Diane Alber, the Arizona artist who created Kip and Gary, creates a cartoon, and we challenge our readers to submit the funniest, most clever caption they think will be a fit. Then we ask our readers to vote for the best submission and the winner receives a signed print of the cartoon. Submit your caption in the comments below.
Congratulations to our reader with the username Rollinggraves for winning October's edition of the caption contest. The reader's caption was: "Sorry about that power surge, folks. All good now."


Lord Kuber is the bestower of wealth and richness. He is also the fourth Lokpal as appointed by Lord Brahma, the creator of the universe. He is responsible for granting richness and wealth to an individual. He is also regarded as the guardian of the riches present in the entire universe. He is considered to be the wealthiest of all Gods and Goddesses. Hence, the brass Kuber, energized and presented as a Live Vastu remedy, has the capability of translating one's actions into riches and growth and even in assisting one to achieve bliss and abundance of wealth.
Lord Kuber should be placed on the working table but remember never to use incense sticks to worship this deity. Having installed this in your work place, one would soon see wealth and happiness flowing in from all directions in his life.
Yaksh raj Kuber is the Lord of Wealth and guardian of all the treasures of the universe. Hence, while formulating a Live Vastu remedy for dealing with wealth relating problems, Lord Kuber was the only deity which could be revered for meeting desires relating to wealth and richness.
He is considered to be the wealthiest devta or God.Kuber has the ability to translate your actions into riches and growth, helping you to achieve bliss and abundant wealth. However, make sure never to worship Lord Kuber with incense sticks. The north direction is owned by Lord Kuber and hence he should be kept on the northern side of your working table to experience immediate and sudden flow of wealth, richness and prosperity.
Yaksh raj Kuber is the Lord of Wealth and guardian of all the treasures of the universe.
Effects: He is considered to be the wealthiest devta or God.Kuber has the ability to translate your actions into riches and growth, helping you to achieve bliss and abundant wealth.


Now that I'm more than two years in to my career as a freelance writer, I think it's safe to talk about my share of struggles. I'm feeling good about this, guys.
But there are also some not-so-great realities we need to talk about.
Let's get real—what are the struggles freelance writers have to face?
Working on your own (and from your home, if you're like me) means dealing with loneliness. Switching from an office environment where people would pop in and out of my office all day long to a quiet room in my house turned out to make for a much more efficient workday, but there's no interaction with other humans. Sorry dog and cat, you don't count.
There are some days where I don't say a word out loud until around 5:00 at night. I've also found that spending so much time alone means that your social skills aren't as sharp when you need them. It gets more difficult to make small talk and actively engage in listening when you don't practice doing that.
That means you have to be deliberate about getting out of your sweatpants and back in the real world. I've been doing this by going to a yoga class twice a week, regularly visiting friends for coffee, and keeping communication open with fellow freelancers online.
Speaking of: Want to join the Slack group I created for fellow freelance writers/content marketers? Email me. It's an awesome (small) group that asks/answers questions, shares job opps, and helps each other on data collection/quotes.
Now back to struggle talk.
I've found this uncertainty can actually be a benefit rather than a struggle, as it's a great motivator for me to beat past months. However, it certainly means taking on more risk—which takes some getting used to.
It's taken me a long time to figure out how to explain what I do to others, and unfortunately, I've found that people really only seem to express understanding and an interest when I can touch on something they already know—like writing for a publication or for a business they've heard of. It's a work in progress.
It feels like this: I'm so busy! I'm stressed. I'm not busy enough! I'm stressed.
There's a lot of stress born from an irregular workload, but if you can be efficient and manage projects effectively, having that flexible schedule is actually a really wonderful thing.
Overall, do the benefits of freelance writing outweigh the struggles? YES. 100 percent, in my opinion.
What struggles do you face?
I think it's time we have a real conversation about the struggles and obstacles we face as freelancers and writers. Care to share some of your own? You can reply to this message or tweet me @kaleighf.


Before I say anything, I just want to give you a little public service announcement: please remember to vote today!! It's really super important.
Anyway... onto the post. A few weeks ago, I read this post by Kaelah, and she talked about a few business books that inspired her. I decided that Creative, Inc. sounded like a fun choice, and promptly ordered it from Amazon. Then when I received my book, I found myself a cozy corner in the little coffee shop/ candy store that I blogged about last week, and had myself skim through.
Truth be told, it intimidated me just a bit.
But for the bigger picture- I've been thinking about myself as a brand for a long time now. I always figured that after awhile, I'd branch out. I've pondered the different ways to market my various skills. I've thought about all the directions I could take my talents, and how many avenues I could pursue as a creative individual. But now I'm thinking that this is a mistake.
This book got me to thinking.... and it made me think that if you are too many different things, if you have too much to offer, people get confused. It seems like most people interviewed in this book fell into one career path, had one skill, and did it really well.
So I can be a writer, photographer, designer, musician, blogger... but it's hard to present that in a way which makes sense and comes together tidily in people's brains. I feel like everyone would just be more comfortable if I presented a package that said "Chantilly: Singer-songwriter." "Chantilly: Music instructor." "Chantilly: Blogger." But I don't feel like that at all. There are too many things to get excited about in this world.
I guess I'm confused and a little excited about all the different options out there. But I'm (still) fuzzy on where my path is headed.
Any thoughts? Are you, too confused out there in the world?
the more i live and breathe and talk to people, the more i realize that almost no one has a concrete feeling of what they're "supposed" to do. it's such a fantasy that should really stop being perpetuated.
i guess it's a useful book to have... i've personally just been more inspired by other ones out there.
I've been having this problem lately too, at 25 I feel like I should know where my life is going, but at the moment I feel like I am still just muddling along. I'd love to write professionally, but I don't have any of the necessary qualification to make any money doing it now, and trying to fit that into my spare time along with blogging, learning to sew and all my other commitments leaves me with no time to actually live my life (let alone sleep and eat).
it's good that you're watching the election! i read a LOT of comments and tweets from people from outside the us that were pretty invested in it as well.
Well, i can't say i've ever feel confused about this, i work as a librarian and teacher in a highschool but that doesn't define me as a person, I'm only ME, i don't let a tittle tell me what path i have to choose. I enjoy taking pics, but i don't consider myself a photographer, it's just a hobby, i starting to love blogging, but i'm not only a blogger, i love writing, and finally finished my first novel, but that didn't make me a writer, my point is i love so many things and i really enjoy them, but at the end this things don't make me who i am, don't define who i am, because i could never choose only one.
you're right. i think people get so caught up in letting the things they do define them. it's only part of them. people have so much more to offer.
I have this book on my Amazon wishlist. It's also pretty cray cray that the post on my blog today is quite similiar to this, wow!
I struggled with something similar when I first started blogging, having to pick a topic, a theme, a brand, an identity, there was so much to chose from. In the end, I had to go with my passion, events. But I know the feeling of being overwhelmed.
im a 16 yr old going on 50 this year and i still cant figure out my nitch,, i was an athlete,, an artist,, a singer,, a dancer, a serious shopper & hippy. Grew up in the Big City and moved away to the country. Now i drive tractors, raise horses, still attend fashion shows but also farm shows,, lol am a happier & more content person but will always be confused as to what exactly my calling was & is,, so im just blogging about it now.
Chantilly: Renaissance Woman! I think as long as you can market yourself in a way that makes sense to the consumer you're fine. Do what makes you happy, that's what my Mom always told me! Right now, I'm not doing what makes me happy... I want to be more creative and I want more freedom. One day when I'm more financially stable I'll be able to do that. I love that you're one step closer to working at home.
thank you for your vote of confidence, kristen!
you know what? I feel you. but I think that if there is a perfect time for people to understand and embrace a polivafacetic professional profile it's just right now. Also, five years ago I enrolled in changing course by Valerie Young and i there I met Barbara Winter and Barbara Sher, this last one is know for being a "scanner profile" defender, and so I thought maybe you would enjoy the reading of her book, "refuse tho choose", whch is about all the worries you were talking about in this post. here0s the link: http://www.amazon.com/gp/product/1594863032/qid=1137700399/sr=8-1/ref=pd_bbs_1/103-4524712-8792631?n=507846&s=books&v=glance, ( I haven't read it, thought, but I thought you might be interested in).
Anyways, have a nice day, and keep up the good work.
I feel you! I often feel totally confused by the choices that lay ahead of me. Be I artist, or be I blogger? Be I scientist, or writer, or ugh, the list goes on. I keep trying to focus on just the one thing, but it's very tough. I generally end up feeling a bit "jack of all trades, master of none"!
Yeah, having private students is not an easy thing. I only did it for a brief time when I first started teaching. My biggest problem was that most of the people I got (not many at all) were people I knew and I felt bad asking for money for the classes. Stupid, I know, but I have a huge problem demanding to be paid ... Even when I had employers! I ended up working as a teacher for institutes which sent me to big companies and I found it fun and fulfilling! I got to learn things from my students and I built up self-confidence being faced with challenges and having to deal with people of all hierarchies.
One of my closest friends, in the end, chose to stop working in companies and kept working as a teacher at home. It was very difficult for her to build her clientele but she's told me that now she can finally say her project is finally paying off. It took her about 4 years to get to this point. The key I guess is to do a great job, obviously, and word of mouth will eventually pay off. Offering competitive prices can be of importance, too, but at the same time, you have to set prices that show your work and input are valuable. Plus, if you don't charge enough, chances are you won't be able to feel fulfilled by this job in the long run.
I'm sorry if you don't care for what I've said :/. Just wanted to tell you about my brief experience in teaching.
Have you consider teaching in schools and music institutes? In Argentina, a few of my peers worked in bilingual schools as "English drama teachers". They were not actors, they took singing classes (not sure how great singers they were, though) and they were English teachers so all that entitled them to lead and organize the musicals most bilingual schools there organize. If you're interested, you can also probably take up classes to become a music therapist. I bet that must be fulfilling ;). There are several schools here in Cali which offer relatively short courses and once you get your degree they help you get hired. There's one which even claims they help you pay bills if they can't find you a job within X amount of months, hahaha!
I apologize for this extremely long comment. I should've emailed you :/.
Anyway, hope you figure things out. It's amazing how things can work out in the end! I mean, look at me! After applying for the crappiest jobs, I finally got the one I'd wanted for so long! Crazy.
i started college majoring in meteorology. then switched to math major. then went to FIT for fashion design (which lasted *almost a week...couldn't hang in the big city:)...then got married...had my babies...returned to school and got a degree in civil engineering.
i either have really bad A>D>D. or i like a lot of things:)...point is, part of finding out your path and what you love is trying all different pieces of what you LIKE.
i think you are creatively very inspirational...So you could just throw all of your gifts and interests under Professional Inspire-er. Problem solved:):).


Satur Farms Profile: What Makes Vegetables Better Quality and/or More Expensive?
Do you ever wonder why some baby spinach is tastier and more expensive than others? The answer, Mindful Readers, is care and freshness, as I learned at Satur Farms. I'd first noticed Satur Farms produce proudly listed on many restaurant menus in NYC. When I mentioned them to private chef Stephane Lemagenen of Zen Can Cook, he enthused "Satur Farms grows the best produce I've seen anywhere on the East Coast. From various types of lettuce to baby vegetables and zuchinni blossoms in the summer, the quality and variety of their produce is just outstanding."
So when Emma, one of my favorite foodies at FreshDirect, suggested that I meet the owners of Satur Farms, I jumped at the chance because 1) I wanted to see what went behind the vegetables that are so well established among NYC chefs and 2) Emma has good ideas.
After a couple conversations, I hopped in a Mint Car and drove out to the North Fork of Long Island to meet Paulette Satur. Having never been to a produce farm before, I didn't know what to expect. (Well, I knew one thing - they have a strict food safety program since I wasn't allowed to bring my dog.) After rolling through miles of flat land, I turned down a charming street and ended up at a large, gorgeous farm. Low slung wooden buildings blend into the surroundings, there are trucks and portapotties, and the entire farm and facility hums with an organized energy.
I found Paulette responding to customers via email in the office. After a few minutes, her husband, Eberhard Muller, burst through the door. While Paulette seems cool and collected, Eberhard is voluble and warm. They give off the intellectually curious vibe of explorers - a couple that stumbled onto something they were good at, then worked really hard to become great at it with a lot of learning and interesting discoveries on the way.
Have they always been farmers? Fascinatingly, no. Paulette was a wine distributor. Eberhard was a celebrated chef whose last post was as Executive Chef at Lutece before he became a full time farmer. Having grown up on a farm, Paulette always wanted to own one. So naturally, when they bought a weekend house on Long Island, they started a garden. Enamoured of the freshness and quality of the produce Eberhard brought it back to use at Lutece. Soon, chef friends were clamoring for their produce, and the couple started farming four acres to satisfy that need. At some point, they had to hire someone to manage their "farm", and as demand grew, they farmed 11 acres, then 35. Now they farm 180 acres with 4 crop rotations per year, or 510 acres.
Chefs (and now retail customers) appreciate Satur Farms for the high quality of their produce. It is obvious that their crops are meticulously and painstakingly cared for. When Paulette and Eberhard decided that growing specialty vegetables on the North Fork was going to be their full-time vocation, they were told by 2-3 people it was impossible. They dug their heels in and persisted, tending their land and buying the equipment that would allow them to become efficient.
As we toured the farm, it was clear that everything was planned with great care. In my unexperienced eyes, everything looked cleaner, more orderly and better cared for than I would hope for. The crops are rotated to make sure the land stays fertile, and the minimum amount of chemicals are used. Paulette explained that extreme weather conditions like humidity made it necessary to use chemicals to keep out fungus and other pests. In addition to the production acres they farm, they keep experimental beds for new crops that chefs request or they are interested in growing.
Equally important as the care that goes into growing their produce is the freshness of it. It is of utmost importance to Paulette and Eberhard that they get produce picked and to customers within 24 hours since fresh food has the best taste. Satur Farms harvests, washes and packages its produce in the morning, then trucks it to customers in the afternoon. This freshness is just not comparable to produce that is grown far away (for example, in California, as it can take 10 days to get to us on the East Coast). Due to the time differential, produce that is shipped long distances is not picked ripe. You can imagine the taste difference. To satisfy their customers' needs for produce in the winter, Eberhard moves to Florida from October through the spring, where they lease 170 acres. He ships to Paulette and it arrives within 24 hours for her to package out to customers.
Paulette explained that plants continue to produce energy after being picked, so if you harvest some greens and put them in a box overnight, then plunge your hands in the box the next morning, it will be a warm 90+ degrees. She explained this right before she showed me the quick chiller, where all harvested produce was brought down to a chilly temp within a couple minutes. Immediately after being chilled, it is washed and packed and sent on its way. Paulette commented several times about the aroma of fresh lettuce in the morning. You don't think of lettuce as fragrant, do you? However it's easy to imagine the heady fresh scent - probably not unlike freshly mown grass. I think I'd like a lettuce candle.
In addition to producing great food, Paulette and Eberhard believe in doing things as environmentally as possible, from growing to packaging. They also dream of reviving the agricultural community in Long Island to create more jobs. In recognition of their community efforts, Paulette was recenty appointed to the Governor's Regional Economic Council. Is there anything else these two can do? Oh, they recently acquired a new larger packaging facility so they can continue to grow and be more accessible to all of us. In addition to supplying top restaurants in the region, Satur Farms can now be purchased by retail consumers at FreshDirect and Whole Foods. If you are in the New York region, try their vegetables for a quality and flavor boost. If you're not, get out and meet your local growers!


text:
Q: What use are constructors in abstract classes? Since we can't instantiate an abstract class, then what is the necessity of having constructors in abstract class?

A: Abstract classes are designed to be extended, each constructor from the child must perform a call to a constructor from the base class, thus you need constructors in your abstract class.
The abstract class is a skeleton and thus makes no sense to instantiate it directly since it is still incomplete (children will provide the rest).

A: An example:
public abstract class BaseClass
{
    private String member;

    public BaseClass(String member)
    {
        this.member = member;
    }

    ... abstract methods...
}

public class ImplementingClass extends BaseClass
{
    public ImplementingClass(String member)
    {
        /* Implementing class must call a constructor from the abstract class */
        super(member);
    }

    ... method implementations...
}

A: We can use a abstract class constructor to execute code that is relevant for every subclass. This way preventing duplicate code

A: Abstract classes can have fields and non-abstract methods(what makes it an abstract class rater than an interface). The fields probably need to be initialized when a class that extends it is instantiated.
Having a constructor in the abstract class allows you to call super(foo); to initialize them as opposed to doing it directly

redpajama_set_name: RedPajamaStackExchange