Industrial biotechnology relies on nature's tool set and converts bacteria, yeast and fungi into microbial cell factories to generate value-added products such as biological detergents, chemical building blocks, food additives, pharmaceuticals and biofuels.
In the past decades themes such as synthetic biology and metabolic engineering gained momentum and were also introduced in industrial biotechnology, opening doors for synthesis of compounds non-natural to the host.
Yet, here some limitations are met; in many cases problems are encountered to get the compound out of the cell resulting in toxic effects, low yields and additional processing costs.
To intensify industrial biotechnological processes, it's important to understand and exploit the mechanisms of transport into and out of cells. This is where the BioPort team of Prof. Van Bogaert is focusing on.
Most known biosurfactants are glycolipids. They are carbohydrates in combination with long-chain aliphatic acids or hydroxyalyphatic acids. One of the most promising glycolipids are sophorolipids produced by the yeasts Starmerella bombicola. We performed pioneering work regarding the development of molecular tools for this unconventional yeast and elucidated the core sophorolipid biochemical pathway. This opened doors for advanced engineering of the yeast and turning it into a versatile production platform for tailored glycolipids and other biomolecules.


Manuel Riemer is an associate professor of community and sustainability science at Wilfrid Laurier University and the Director of the Viessmann Centre for Engagement and Research in Sustainability (VERiS), Environment and Justice Research Group. He received his PhD in Psychology with a major in Quantitative Methods from Peabody College at Vanderbilt University (USA). He is a recipient of the Early Researcher Award of the Society for Community Research and Action / Division 27 of the American Psychological Association. He is also a founding partner and academic advisor of the Sustainable Societies Consulting Group (www.sscg.org), a social enterprise that offers research, evaluation and organizational development services to environmental, health and social organizations.


text:
There's No Stopping America's Political Realignment
Published Dec. 4, 2020
With a month in the rearview mirror since November's presidential election, we now have sufficient perspective to offer some broad conclusions about the state and nature of our politics.
This election confirmed and crystallized many already-ascendant political trends. Partisans of multiculturalism and the Orwellian "anti-racism" movement suggest an America increasingly balkanized along racial lines, but the emerging reality is quite a bit different.
President Donald Trump, despite the left's four-year effort to tar him as a uniquely menacing bigot, made substantial inroads among both young Black men and Hispanics. On the other hand, donor-class elites — Wall Street, Silicon Valley and corporate America writ large — broke overwhelmingly for the Democratic challenger, Joe Biden.
To those who, Rip Van Winkle-like, may have been asleep for the past 20 years, this may come as a bit of a surprise. And for those octogenarians who grew up in the New Deal era, the bemusement may be even more stark. Long, long gone are the days where the Democratic Party was the political vehicle for the downtrodden, oppressed "little guy" and the Republican Party the vehicle for conniving, Mr. Burns-esque plutocrats.
The reality is nonetheless something roughly resembling the precise opposite. Today, it is the Democratic Party that disproportionately comprises and represents the college-educated elite — purveyors of the addlebrained "woke" catechism, dispensers of redistributionist noblesse oblige. And it is the Republican Party that disproportionately comprises and represents a multiethnic, non-college-educated working class — the Rust Belt and Sun Belt "deplorables" of Hillary Clinton's ire who, pace Barack Obama, still cling to guns and religion. Though it is true that exit polling data suggest Blacks and Hispanics did not quite mirror whites in breaking so sharply along educated lines, the conclusion is nonetheless apt that our partisan duopoly now fractures more along lines of class than along lines of race.
Elites are often puzzled by this. Democratic elites, prone to word-vomit the full litany of social justice "woke"-isms, are often in denial as to how unpopular "defund the police" and the Green New Deal are with the public. And Republican elites, stubbornly moored to laissez-faire fundamentalism and limited government as intrinsic ends, rather than instrumental means, are often in denial as to how miniscule the political constituency is for a corporatist right-liberalism detached from the day-to-day needs of the working class.
But there is no putting this genie back in the bottle. Our now-rapidly accelerating political realignment will not slow down any time soon.
Presuming the Electoral College formalizes Joe Biden as our next president, the next four years are potentially ripe for crossover opportunity. The beauty of an unfolding political realignment in which so much is in flux is that there may well be more Venn diagram overlap than meets the eye. Conservatives must remain constantly vigilant about a far-left capture of the frail and deteriorating Biden, but it would also be a mistake to preemptively write off the possibility of any bipartisan cooperation whatsoever.
On economic matters, there is room for legislative deal-making on the need for an industrial policy that reshores strategic supply chains and spurs greater domestic investment in advanced manufacturing, pharmaceuticals and semiconductors. The GOP's more populist wing, which was already vocal about hawkishness on China, the downsides of free trade absolutism and the need for a greater economic resilience on the home front, was buoyed by a global pandemic that saw its worst fears realized.
Biden has expressed support for some variety of a pro-manufacturing industrial policy — and it is incumbent upon Republican watchdogs to ensure such a policy, if enacted, does not manifest itself in the form of venal, Solyndra-like crony capitalist boondoggles.
On Big Tech, too, there is strong bipartisan outrage at the status quo. Conservatives and progressives may be angry at Facebook's Mark Zuckerberg and Twitter's Jack Dorsey for different reasons, but it is still easily foreseeable that there might be room to collaborate on antitrust reform or on reform of Section 230 — the 1990s-era legal immunity giveaway to the Big Tech giants. Partisans from both sides are increasingly coming around to the view that these goliaths must be reined in.
It is deeply unfortunate that Trump appears headed for an Electoral College defeat. The next four years could be rough. But there are always silver linings to be found amid the rubble.

redpajama_set_name: RedPajamaCommonCrawl


The Rocket Post is based on the true story of Gerhard Zucher and his experiments with rocket delivered mail between the western isles of Scotland during the 1930's. The Rocket Post was filmed on the islands of Harris and Taransay and should be released late 2002. Or maybe not! "£10m Hebridean rocket post film up in smoke as 'tartan turkey' nose-dives" reads the headline in the Sunday Herald newspaper of 1st December 2002.
Finally released 12th March 2007 on Region 2 DVD.


• The economical, geopolitical, ecological, systemic and societal paradigm shifts increasingly require from Leaders and their organizations to play a pivotal transformational role at the service of Humankind, Society and the World.
• Organizations of the future will be those where the leaders inspire all co-workers to self-actualize through why they do what they do really matters and serves the higher purpose the organization stands for. That development needs different organizational cultures, where consciousness, harmony and full engagement will become the natural path to exceptional sustainable positive impact and great results.
• Inspiring authentic leadership and the very subtle yet complex system dynamics in organizations have therefore to be integrated again, because the leader and the system can't operate in a sustainable successful way when separated from each other and when not serving a higher purpose.
fully authentic with their higher purpose, who know how to get inspiration and information from the 'universal wisdom' and from Nature, who are visionary architects able to transform organizations in positively impacting systems, and who have the capacity to transform themselves, show humanity and embrace vulnerability.
• In that context ALSDO©, the two-day intense, purposeful and highly interactive Authentic Leadership and System Dynamics in Organizations program©, was created by Olivier Onghena-'t Hooft and will address both the individual leadership dimension and the organizational system dynamic dimension.


"Imagine a vibrant market in a village in Sub-Saharan Africa, filled with vendors selling lush tomatoes, hearty ears of corn, ripe mangos, and a myriad of other fruits, vegetables, and grains. Where did all that food come from? Where did the farmers get the financing to buy the seeds and fertilizer they needed? What research institutions developed the seed varieties that thrived in local agro-ecological conditions? How did farmers learn the agriculture techniques to produce high-quality crops? And how did farmers get those high-quality crops from their farms to the market?
People often talk as if the private sector, nonprofits, and farmer organizations in Africa are at odds, with extremely different motivations, ways of working, and goals. In reality, we are more similar than different. We are all seeking innovative ways to build a strong agriculture sector in which smallholder farmers are profitable businesspeople and responsible environmental stewards. And we all recognize that the public sector has the responsibility to set certain conditions that foster an environment where innovation and growth can occur."
Read more from The Aspen Journal of Ideas.


Startling Snail Pictures To Color Puppy And Coloring Page For Kids Animal Pages is one of best wallpaper image reference about coloring pages. This wallpaper has been created for kids, toddler and kindergarten. Startling Snail Pictures To Color Puppy And Coloring Page For Kids Animal Pages was posted in May 30, 2018 at 12:32 pm and has viewed by 221 users. Startling Snail Pictures To Color Puppy And Coloring Page For Kids Animal Pages reference also have Tag ;Startling Snail Pictures To Color Puppy And Coloring Page For Kids Animal Pages, snail pictures to color, for your convenience in searching this reference more specific.
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HAWTHORNE, Calif.—SpaceX said Jan. 16 that it will build test versions of its Mars spaceship in south Texas instead of the Port of Los Angeles in another blow to the local economy that comes days after the company announced massive layoffs.
On Jan. 11, SpaceX announced it would lay off 10 percent of its roughly 6,000 workers, most of them at its Hawthorne headquarters. The company said it needs to become leaner to accomplish ambitious and costly projects such as the Starship and Starlink, which would create a constellation of satellites to provide space-based broadband internet service.


A former newspaper editor who led four weekly titles during his career has died at the age of 80.
Geoff Cottrell, left, edited four titles in Worcester, the AdNews, Berrow's Worcester Journal, the Malvern Gazette and Ledbury Reporter, before retiring in 1998.
He started out in journalism following National Service at DC Thomson where he worked for The Sunday Post and met his wife Joyce there.
Geoff moved to County Cork in Ireland following his retirement and died there on Friday.
David Griffin, his former group managing editor in Worcestershire, said: "He was a definitely a journalist of the old school. He always put journalism and his editorial decisions before all other matters.
"He wasn't easily influenced by outside pressures and was completely committed to the newspapers he was editing and worked unstintingly on behalf of his readers, to present a comprehensive news service.
"He was well-liked and he was wonderfully supportive of his staff, especially his younger staff.
Geoff went into journalism following National Service in Royal Air Force Intelligence, after his commanding officer recommended it as a profession for him.
After working for DC Thomson, he went on to work as a sub-editor for the Worcester News, before becoming editor of the now-defunct AdNews series.
He went on to hold further editorships for its sister titles, which are now owned by Newsquest, and when he retired he was at the helm of the Malvern Gazette and the Ledbury Reporter.
While in Worcester, Geoff became a member of the Guild of British Newspaper Editors and was secretary of the West Midlands region.
He moved to County Cork in Ireland in 2000 to pursue his hobby of fishing but heart trouble later made that impossible.
Geoff is survived by his wife Joyce and son Roger.
I can only endorse what David Griffin has said about a great guy to work with and a true gentleman who truly cared about all the newspapers he worked on.


30 coaches are already signed on to help us beta test Instacoach. Feedback so far is great and they're looking forward to making money from the app when it goes mainstream.
Thanks guys, we owe you a sports drink!


text:
Simple New Experiment Reveals Which Face Masks Are Best at Blocking Droplets
Health10 August 2020
By Peter Dockrill
(Fischer et al., Science Advances, 2020)
We know that by wearing masks, we can help reduce the transmission of COVID-19, and we know that not every type of face mask is equally good at blocking viral droplets when we cough, sneeze, talk – or even simply breathe. But how can we really know for sure?
Aside from simply believing manufacturers' claims, if you wanted to somehow test different masks against one another to compare how much protection they offer in the real world, how would you go about it?
New research from scientists at Duke University shows you don't actually need all that much to devise a test. In a proof-of-concept study, they cobbled together a simple, low-cost laser device, and conducted an experiment comparing 14 different types of masks and face coverings.
The masks tested. (Fischer et al., Science Advances, 2020)
"The fundamental question is, how well does a specific mask type prevent droplets from spreading," lead researcher and molecular imaging specialist Martin Fischer says in a press Q&A.
The question is particularly relevant in the coronavirus pandemic, given many people have taken to buying fabric masks online, or making their own masks at home.
While the general consensus from experts is that all of these kinds of loose-fitting masks should help reduce COVID-19 transmission – which is why face coverings are mandated in many places around the world right now – most of the mask testing to date has been done on things like surgical masks and fitted N95 masks, and not on loose fabric masks or face coverings.
"Surgical masks are commonly worn by medical personnel and have received a fair amount of testing in clinical settings," Fischer says.
"But as far as we know, there was no quick, easy, and cost-effective way to demonstrate the effectiveness of such a wide variety of other mask types."
To fill that void, Fischer and his team devised an easy-to-make, inexpensive laser experiment, which can be used to test how different kinds of masks block tiny droplets that come out of people's mouths when they speak.
In the experimental setup, a lens turns an optical laser into a sheet of light. This light sheet, shone through a dark enclosure (made up of cardboard sheets and duct tape), reveals when droplets pass through it, with the results being filmed by a mobile phone camera.
In experiments, people would speak toward the sheet of light, saying the phrase "Stay healthy, people", while wearing 14 different kinds of face masks and coverings.
"We confirmed that when people speak, small droplets get expelled, so disease can be spread by talking, without coughing or sneezing," Fischer says.
"We could also see that some face coverings performed much better than others in blocking expelled particles."
Droplet count results for each mask. (Fischer et al., Science Advances, 2020)
The results showed that fitted N95 masks blocked the greatest amount of droplets released by the person speaking, followed by surgical masks, then masks made with polypropylene.
However, all sorts of other masks, including cotton masks and even knitted ones, showed an ability to block droplets, as did a valved version of the N95 mask, which didn't score as well as the fitted N95 mask due to its exhaust valve.
"These valves are closed when breathing in, but can open when speaking, hence letting out unfiltered air," Fischer says.
"In other words, they do a great job of protecting the wearer from the outside environment, but a bad job of protecting others from the wearer, and it is the second role that is the important one to reduce COVID-19 spread."
Most surprising, however, were the results at the bottom of the table. In terms of blocking droplets, bandanas were among the less effective, but worst of all is wearing a neck fleece, which the researchers found is actually worse than not wearing any kind of facial covering.
A schematic of the experimental setup. (Fischer et al., Science Advances, 2020)
That sounds counter-intuitive – and likely warrants further examination – but the researchers think the neck fleece actually makes droplets proliferate in the air.
"Common sense would dictate that wearing anything is better than wearing nothing – this was not the case here," Fischer says.
"We observed that the number of droplets increased when the speaker put on the neck fleece. We believe that the material of our fleece breaks down large droplets emitted during speaking into several smaller ones. This could make wearing such a mask counterproductive, since smaller droplets have an easier time being carried away by air currents and endangering nearby persons."
Of course, the researchers are eager to emphasise that the focus of the study is actually the low-cost testing method they developed, not their own test results of which masks are the most and least effective – as the same kind of testing could be conducted more robustly and more systematically than in the proof-of-concept study here.
That said, their own testing is certainly food for thought, highlighting again that not every mask is equal, and if you really want to protect other people and help to reduce the spread of coronavirus, you really need to think about what you're putting on your face.
The findings are reported in Science Advances.


text:
Q: Setting Selenium delay between testcases with in Test suite in Robotframework Have a requirement to set some delay between two test cases in a Robotframework Test suite to avoid cascading failure in one test case to another.

A: There is no built-in support to add a delay between test cases.
You can accomplish the same effect by calling the Sleep keyword as a test setup or test teardown.

A: I have used test specific actions in 'My Test Setup' keyword and added some test specific keywords to it to setup particular tests. Here is a simplified example:
*** Settings ***
Suite Setup        My Suite Setup     # Before the suite is executed (wait 5 secs)
Test Setup         My Test Setup      # Before the current test is executed (wait 1 sec)

*** Test Cases ***
# Tests goes here

*** Keywords ***
My Test Setup
    Sleep    5s

My Suite Setup
    Sleep    1s

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}
A wireless sensor network consists of a group of sensor nodes deployed in an area. The sensors sense and take measurements of the surrounding environment and transmit the obtained data through a wireless communication channel to a data aggregation center which performs state estimation. Thanks to their large-scale deployment, the wireless sensor networks have gained popularity in a wide range of applications such as environmental monitoring and target tracking~\cite{yick2008wireless}. However, resource constraints, including the amount of energy and the wireless communication channel bandwidth, pose challenges to system design and synthesis.

The sensor scheduling problem has been proposed to utilize the limited resources for enhancing the remote estimation quality. Energy consumption minimization along with sensor lifetime maximization have been studied  in~\cite{bitar2009energy,mo2011sensor,shi2011sensor}. In the mean time, another stream of research has been focusing on balancing the communication burden and a certain cost function related to estimation accuracy. Shi et al.~\cite{shi2011optimal} showed that the optimal schedule for a single sensor scheduling is periodic if packet drops are ignored. Chakravorty and Mahajan~\cite{chakravorty2017fundamental} obtained a general result which stated that the optimal policy is of threshold type. Similar threshold-type structural results are obtained for multiple sensors~\cite{molin2014price,gatsis2015opportunistic,leong2017sensor,ren2018attack}. The cost function in the above mentioned works is the summation of all the estimation errors. In some scenarios, one is not interested in optimizing a collective quantity of each individual, e.g., summation of all the estimation errors, but desires that each individual's performance is above certain threshold. Moreover, each individual has its own interests and may not be willing to reach a collective goal by sacrificing their own benefits. These demands lead to a requirement of fairness in scheduling policies.

In this work, we consider how to fairly allocate the limited communication resources to the sensors. To the best of our knowledge, fairness issue in the sensor scheduling problem has not been addressed. One major challenge is to propose a fairness metric for analysis. The fairness has been studied in operation research and communication community. The works in~\cite{moulin2002fair,correa2007fast,jiang2018towards} studied fairness issue in queuing systems, while the works in~\cite{wang2004distributed,karipidis2008quality,sadeghi2018max} addressed fairness issue for communication protocols. Lan et al.~\cite{lan2010axiomatic} summarized a variety of proposed fairness metrics and developed a set of axioms to identify one metric to be fair. Bertsimas et al.~\cite{bertsimas2011price} pointed out that \emph{max-min} fairness and \emph{proportional} fairness are two axiomatically justified and well accepted notations of fairness. The proportional fairness can be obtained by maximizing the summation of the $\log$ function of the utility of each user. Analysis and computation of solutions to proportional fairness can be done by using the same method as those for multiple sensor scheduling in the literature.

The max-min fairness is taken as the notation of fairness in this work. We consider $n$ sensors measuring $n$ independent linear time-invariant dynamic processes. The sensors transmit their measurements to a remote state estimator through a shared wireless communication channel. Due to the constrained bandwidth, the allowable transmissions are limited. We want the maximal average estimation error covariance among the $n$ sensors to be as small as possible, which constitutes the max-min fairness problem. Interestingly, if the processes are all unstable, the max-min fairness formulation leads to an equal estimation performance (\textbf{Proposition~\ref{proposition: equal performance}}). Although the max-min fairness metric is clearly defined, the corresponding mathematical problem is to be formulated.

There are noticeable amount of works addressing max-min fair scheduling in wireless sensor networks. However, the problems they studied considered are different from ours. The works in~\cite{huang2001max,tassiulas2002maxmin} considered max-min fairness of data flows in the communication links. Their design goal is to maximize the bottleneck of the whole network. The works in~\cite{sridharan2004max,kim2017multi} considered how to schedule transmissions to maximize the lowest throughput over the MAC-layer. In summary, these works considered the fairness solely either from the data link or the routing perspective. None of the existing works addressed the max-min fairness by jointly considering the remote state estimation and bandwidth allocation.

The main results of this paper are summarized as follows.
\begin{enumerate}
	\item We consider the max-min fairness in terms of remote state estimation performance for sensor scheduling problem, which cannot be directly solved using existing numerical schemes. We formulate it as a general max-min fair resource allocation problem. We reformulate it as a two-player zero-sum game, which consists of two players, a ``judge" who adjusts the weights of each sensor and an ``allocator" who determines the resource allocation. We prove that the game possesses pure-strategy Nash Equilibriums (\textbf{Theorem~\ref{theorem: uniqueness of NE}}).
	\item For the general max-min fair allocation problem, we propose an algorithm to seek the equilibrium and show that the algorithm converges to the max-min fair allocation scheme (\textbf{Theorem~\ref{theorem: equilibrium and optimality}} and~\textbf{\ref{theorem: convergence to unique point}}). The max-min fair sensor scheduling problem does not satisfy some of the assumptions made for the general case. We adjust the algorithm as \textbf{Algorithm~\ref{alg:framework}} to seek a max-min fair allocation for sensor scheduling. This adjusted algorithm converges to the max-min allocation with a high accuracy. (\textbf{Theorem~\ref{theorem: fair policy unique and convergent}}).
\end{enumerate}

The remainder of this paper is organized as follows. In section~\Rmnum{2}, we develop a general framework for the max-min fair resource allocation problem. In section~\Rmnum{3}, we apply the theory developed to solve a fair sensor scheduling problem. The paper is summarized in section~\Rmnum{4}.

\emph{Notations}: The $n$-dimension Euclidean space is denoted by $\mathbb{R}^n$. The bold symbol letter stands for a vector which aggregates all its components, e.g., $\boldsymbol{x}=[x_1,\dots,x_n]^{\top}$. In particular, $\bm{1}=[1,\dots,1]^{\top}$. The inequality between a vector in the Euclidean space means that the inequality holds elementwise. For a matrix $X$, $X^{\top}$ and $\Tr(X)$ stand for the matrix transpose and the trace of the matrix. The operation $P_{\mathcal{X}}(\bm{x})$ denotes the projection of vector $\bm{x}$ into the constrained set $\mathcal{X}$, i.e.,
$P_{\mathcal{X}}(\bm{x}) = \argmin_{\bm{x'}\in\mathcal{X}} \|\bm{x}-\bm{x}'\|_2$,
where $\|\cdot\|_2$ stands for the Euclidean-norm. The probability and condition probability are denoted by $\mathtt{Pr}(\cdot)$ and $\mathtt{Pr}(\cdot | \cdot)$, respectively. The expectation of a random variable is $\mathbb{E}[\cdot]$.

\section{The Fair Resource Allocation Problem}

\subsection{Max-Min Fair Resource Allocation}
Consider a group of $n$ agents. Each agent $i=1,\dots,n$ aims to minimize a certain cost $J_i(r_i)$ by consuming certain resource $r_i$. The total resource is limited and characterized by $\sum_{i=1}^n r_i \leq R$. Moreover, the resource utilized by each agent has both an upper bound and a lower bound, i.e., $\underline{r_i} \leq r_i \leq \overline{r_i}$. We aim to find a centralized max-min fair resource utilization scheme $\bm{r}=\begin{bmatrix}r_1,\dots,r_n \end{bmatrix}^\top$ among the $n$ agents as follows.

\begin{problem}\label{problem: resource allocation fairness}
	\begin{align*}
	\min_{\bm{r}} \quad & \max_{i\in\mathcal{I}} J_i(r_i)\\
	\text{s.t.}\quad & \bm{1}^{\top}\bm{r} \leq R,\\
	&\underline{\bm{r}} \leq \bm{r} \leq \overline{\bm{r}},
	\end{align*}
where $\mathcal{I}=\{1,\dots,n\}$.
\end{problem}
This problem aims at optimizing the cost of the ``worst" agent. A central manager solves the above resource allocation problem and each individual uses the allocated resource to optimize its own performance.

\begin{remark}
Problem 1 involves a mixture of numerical (continuous) strategy $\bm{r}$ and categorical (discrete) strategy $i$. When the strategy is either purely continuous or discrete, efficient algorithms can be developed to solve the problem~\cite{nisan2007algorithmic}. If both sets of strategies are discrete, we can reformulate the problem as a linear program  by considering the notion of mixed strategy for discrete strategies. In the linear program setup, the strategy sets are continuous. In a continuous game, we can calculate the gradient of the game value with respect to the strategies, which is useful to seek the Nash equilibrium of the game. However, the mixture of continuous and discrete strategies prohibits direct calculation of the gradients.
\end{remark}

It is expected that more allocated resource leads to a smaller cost and the benefit of using resource should have a decreasing effect. Moreover, $J_i(r_i)$ should be continuous in $r_i$. We therefore made the following assumptions on $J_i(r_i)$.

\begin{assumption}[Continuity]\label{assumption: continuity}
For every $i$, the cost $J_i(r_i)$ is continuous in $r_i$.
\end{assumption}
\begin{assumption}[Strict Monotonicity]\label{assumption: monotonicity between objective and resource}
For every $i$, the cost $J_i(r_i)$ is strictly monotone decreasing in $r_i$, i.e.,
\begin{align*}
J_i(r'_i)>J_i(r_i)
\end{align*}
for any $r'_i<r_i$.
\end{assumption}
\begin{assumption}[Convexity]
For every $i$, the cost $J_i(r_i)$ is convex in $r_i$, i.e.,
\begin{align*}
t J_i(r'_i)+ (1-t)J_i(r_i) \geq J_i(t r'_i + (1-t)r_i)
\end{align*}
for any $r_i,r'_i$ and $t\in[0,1]$.
\end{assumption}
Moreover, we assume the costs are always strictly positive.
\begin{assumption}[Strict Positivity]\label{assumption: strict positivity of costs}
The cost of each agent $J_i(r_i)>0$ for $r_i\in[\underline{r_i},\overline{r_i}]$ and any $i\in\mathcal{I}$.
\end{assumption}

\begin{remark}
The strict monotonicity and the convexity assumption of the cost with respect to allocated resources can be interpreted as follows. Consider the following total cost minimization problem.
\begin{align*}
\min_x \quad &f_i(x)\\
\mathrm{s.t.} \quad &g_i(x)\leq r_i.
\end{align*}
The cost $f_i(x)$ and utilized resource $g_i(x)$ are negatively correlated, i.e., $f_i(x)>f_i(x')$ if and only if $g_i(x)<g_i(x')$ for any $x\neq x'$. Hence, minimization is attained only if $g_i(x)=r_i$. Let $J_i(r_i)=\min_{x:g_i(x)\leq r_i} f_i(x)$. The strict monotonicity of $J_i(r_i)$ then follows. We further assume that strong duality holds, i.e.,
\begin{align*}
\min_x \max_{\lambda\geq0} \; f_i(x)+\lambda (g_i(x)-r_i) = \max_{\lambda\geq0} \min_x \; f_i(x)+\lambda (g_i(x)-r_i).
\end{align*}
Let $x^\star$ and $\lambda^\star$ be the corresponding minimizer and maximizer.
Note that
\begin{align*}
J_i(r_i)=&\min_x f_i(x) + \lambda^\star (g_i(x)-r_i) \\
\leq&f_i(x) + \lambda^\star (g_i(x)-r_i)
\end{align*}
for all $x$. Let $x^{'\star}$ be the solution to the minimization with total cost $r'_i$. We can obtain
\begin{align*}
J_i(r_i) \leq &f_i(x^{'\star}) + \lambda^\star( g_i(x^{'\star}) -r_i )\\
=& J_i(r'_i) + \lambda^\star( r'_i -r_i ).
\end{align*}
In summary, for any $r_i$, there exists a $\lambda^\star$ such that
\begin{align}\label{ieq: convexity and monotonicity of each agent}
J_i(r_i) - \lambda^\star(r'_i-r_i) \leq J_i(r'_i)
\end{align}
for all $r'_i$. From the first order condition of a convex function, the cost $J_i(r_i)$ is convex in $r_i$.
\end{remark}

To make the optimal value of Problem~\ref{problem: resource allocation fairness} bounded, we need further assumptions on the cost objective and the total resource. Define the set of feasible allocation schemes as $$\mathcal{R}\triangleq\{\bm{r}:~\bm{1}^{\top}\bm{r} \leq R,~\underline{\bm{r}} \leq \bm{r} \leq \overline{\bm{r}}\}.$$ We further have the following the assumption of the total resource $R$.
\begin{assumption}\label{assumption: feasiblity}
The lower bounds $\underline{r_i}$ satisfies $\sum_{i\in\mathcal{I}}\underline{r_i} < R$. In addition, there exists a feasible allocation scheme $\bm{r}\in\mathcal{R}$ such that $J_i(r_i)<\infty$ for all $i\in\mathcal{I}$.
\end{assumption}
If the first half of the assumption fails,  the feasible region does not contain an interior point. If the other half fails, the total cost under any allocation is unbounded, which leads to the nonexistence of any max-min fair allocation scheme.

\subsection{Game Reformulation}
The formulation in Problem~\ref{problem: resource allocation fairness} does not reveal any inner structures of the problem and cannot help determine whether there exists a solution to the resource allocation problem. In the following, we transform the original allocation problem as a two-player zero-sum game with continuous actions. With this reformulated problem, we can obtain existence of a solution and some properties of such a solution.

The max-min fair allocation problem can be transformed as a two-player zero-sum game. Let $\mathcal{W}\triangleq\{\bm{w}:\bm{1}^{\top}\bm{w}=1,~\bm{w}\geq\bm{0}\}$. We have two players with strategies $\bm{r}\in\mathcal{R}$ and $\bm{w}\in\mathcal{W}$, respectively. Each of the player is solving a coupled optimization problem
\begin{align}\label{eq: game formulation}
\min_{\bm{r}\in\mathcal{R}} g(\bm{w},\bm{r}) ~\text{and}~\max_{\bm{w}\in\mathcal{W}} g(\bm{w},\bm{r}),
\end{align}
where $g(\bm{w},\bm{r})=\sum_{i=1}^n w_iJ_i(r_i)$. This can be viewed as a zero-sum game between a ``judge" with action $\bm{w}$ and a resource allocator with action $\bm{r}$. The ``judge" wants to maximize it by adjusting $\bm{w}$, while the resource allocator wants to minimize it by allocating $\bm{r}$. If this game possesses at least one pure-strategy Nash equilibrium $(\bm{r}^\star,\bm{w}^\star)$, by the definition of a Nash equilibrium, $\bm{r}^\star$ is a solution of the original problem.

From the results in game theory, an equilibrium in the pure strategy exists for this game.
\begin{theorem}\label{theorem: uniqueness of NE}
The two-player zero-sum game in~\eqref{eq: game formulation} has the following properties:
\begin{enumerate}
\item there exists a pure-strategy Nash equilibrium;
\item the value of the game is unique;
\item there may exist multiple pure-strategy equilibrium pairs, but the weight $w$ at equilibriums is unique.
\end{enumerate}
\end{theorem}
\begin{IEEEproof}
The proof is done in three steps:
\begin{enumerate}
	\item there exists a pure strategy equilibrium;
	\item the value of the game is unique;
	\item $w$ at equilibriums is unique.
\end{enumerate}

We first verify the existence. Note that
\begin{itemize}
	\item both $\mathcal{W}$ and $\mathcal{R}$ is compact and convex;
	\item the utility of $\bm{w}$, $g(\bm{w},\bm{r})$, is continuous in $\bm{r}$, and the utility of $\bm{r}$, $-g(\bm{w},\bm{r})$, is continuous in $\bm{w}$;
	\item $g(\bm{w},\bm{r})$ is continuous and linear (thus also concave) in $\bm{w}$, and $-g(\bm{w},\bm{r})$ is continuous and concave in $\bm{r}$.
\end{itemize}
According to~\cite[Theorem 1.2]{fudenberg1991game}, this continuous game possesses at least one pure-strategy Nash equilibrium.

We next prove the uniqueness of the value of the game. Suppose both $(\bm{w}^\star,\bm{r}^\star)$ and $(\bm{w}^{\star\star},\bm{r}^{\star\star})$ are equilibrium strategies.

If $g(\bm{w}^\star,\bm{r}^\star) \leq g(\bm{w}^{\star\star},\bm{r}^{\star\star})$, we have
\begin{align*}
g(\bm{w}^{\star\star},\bm{r}^\star) \leq g(\bm{w}^\star,\bm{r}^\star)
\leq g(\bm{w}^{\star\star},\bm{r}^{\star\star}) \leq g(\bm{w}^{\star\star},\bm{r}^\star),
\end{align*}
which leads to
\begin{align*}
g(\bm{w}^\star,\bm{r}^\star) = g(\bm{w}^{\star\star},\bm{r}^{\star\star}).
\end{align*}

If $g(\bm{w}^\star,\bm{r}^\star) \geq g(\bm{w}^{\star\star},\bm{r}^{\star\star})$, we have
\begin{align*}
g(\bm{w}^\star,\bm{r}^{\star\star}) \geq g(\bm{w}^\star,\bm{r}^\star)
\geq g(\bm{w}^{\star\star},\bm{r}^{\star\star}) \geq g(\bm{w}^\star,\bm{r}^{\star\star}),
\end{align*}
which also leads to
\begin{align*}
g(\bm{w}^\star,\bm{r}^\star) = g(\bm{w}^{\star\star},\bm{r}^{\star\star}).
\end{align*}

We lastly prove the uniqueness of $w$ at equilibriums. Again assume both $(\bm{w}^\star,\bm{r}^\star)$ and $(\bm{w}^{\star\star},\bm{r}^{\star\star})$ are equilibrium strategies.
Note that
\begin{align*}
g(\bm{w}^{\star},\bm{r}^{\star}) \leq g(\bm{w}^{\star},\bm{r}^{\star\star}) \leq
g(\bm{w}^{\star\star},\bm{r}^{\star\star}) \\
g(\bm{w}^{\star\star},\bm{r}^{\star\star}) \leq g(\bm{w}^{\star\star},\bm{r}^{\star}) \leq g(\bm{w}^{\star},\bm{r}^{\star}),
\end{align*}
which leads to
\begin{align*}
g(\bm{w}^{\star},\bm{r}^{\star}) = g(\bm{w}^{\star},\bm{r}^{\star\star}) =  g(\bm{w}^{\star\star},\bm{r}^{\star}) = g(\bm{w}^{\star\star},\bm{r}^{\star\star}).
\end{align*}

Since $J_i(r_i)$ is strictly monotone in $r_i$, we have $J_i(r_i^\star)\neq J_i(r_i^{\star\star})$ for $r_i^{\star}\neq r_i^{\star\star}$. Suppose $\bm{w}^\star \neq \bm{w}^{\star\star}$. As $J_i(r_i)$ are strictly greater than zero, we have $g(\bm{w}^{\star},\bm{r}^{\star}) \neq g(\bm{w}^{\star\star},\bm{r}^{\star})$ and $g(\bm{w}^{\star},\bm{r}^{\star\star}) \neq g(\bm{w}^{\star\star},\bm{r}^{\star\star})$, which violates the uniqueness of the value of the game. Therefore, it holds that $\bm{w}^\star=\bm{w}^{\star\star}$.
\end{IEEEproof}

As the equilibrium in the pure strategy exists, we are able to obtain a solution of Problem~\ref{problem: resource allocation fairness}. In addition, we can observe some properties of a solution of Problem~\ref{problem: resource allocation fairness}. From the following example, we can see that the optimal value and $w$ at equilibriums of the game are unique, but there are multiple max-min allocation strategies.
\begin{example}
Let $J_1(t)=4-t$, $J_2(t)=J_3(t)=1.5-t$ and $\mathcal{R}=\{\bm{r}:\bm{1}^\top\bm{r}\leq1.5,~\bm{0}\leq\bm{r}\leq1 \}$. It is straightforward that any $\bm{r}$ in $\{\bm{r}: r_1=1, r_2+r_3\leq0.5 \}$ is a max-min fair allocation strategy. However, the corresponding weight $w^\star$ is unique as $\bm{w}^\star = \begin{bmatrix}1&0&0\end{bmatrix}^\top$.
\end{example}

The following corollary connects the max-min fairness with the equal performance. In this case, there exists a unique max-min fair allocation strategy.
\begin{corollary}\label{corollary: max achieve}
If $w^\star_i>0$ for each $i\in\mathcal{I}$, then there exists a unique max-min fair allocation strategy $\bm{r}^\star=[r^\star_1 \; \dots \; r^\star_n]^\top$ such that
\begin{align*}
J_i(r_i^\star) = \max_{j\in\mathcal{I}} J_j(r_j^\star).
\end{align*}
\end{corollary}

\begin{IEEEproof}
Let $w^\star$ be the the one at the equilibrium. As $w^\star$ is unique, the value of the game can be represented by
\begin{align*}
\max_{\bm{w}} \quad& \sum_{i=1}^n w_i J_i(r_i^\star)\\
\text{s.t.} \quad&\bm{1}^{\top}\bm{w}=1,~\bm{w}\geq\bm{0}.
\end{align*}
Noting that the constraints are linear, the constraint qualification conditions hold. The corresponding Karush-Kuhn-Tucker (KKT) conditions are
\begin{align*}
&\bm{J}(\bm{r}^\star)-\lambda^\star\bm{1}+\bm{\mu}^\star=0,&\text{(gradient condtion)}\\
&\bm{\mu}^\star\leq 0, &\text{(dual feasiblity)}\\
&\mu_i^\star w_i^\star=0,~i\in\mathcal{I}&\text{(complementary slackness)}
\end{align*}
where $\lambda^\star$ is the dual variable associated with the constraint $\bm{1}^\top\bm{w}^\star=1$ and $\bm{\mu}^\star$ is the dual variable associated with the constraint $\bm{w}^\star\geq0$.

As $\bm{\mu}^\star \leq 0$, we always have $J_i(r_i^\star)\leq \lambda^\star$. Combining the gradient condition and the dual feasibility, we have $J_i(r_i^\star) = \lambda^\star$, if $\mu^\star_i=0$. From the complementary slackness, we know that
$\mu^\star_i=0$ if $w^\star_i>0$. Therefore if $w^\star_i>0$, we can obtain that $J_i(r^\star_i)=\lambda^\star$, which leads to that $J_i(r^\star_i)=\lambda^\star=\max_i J_i(r^\star_i)$. This also proves that $r^\star$ is unique.
\end{IEEEproof}

As a direct consequence of the corollary, for a general $\bm{w}^\star$, we can obtain $J_i(r^\star_i)=\max_{j\in\mathcal{I}} J_j(r^\star_j)$ if $w^\star_i>0$.

\begin{remark}
Note that $w_i>0$ is only sufficient for the $J_i(r_i^\star)=\max_i J_i(r^\star_i)$. However, this equality may still hold when $w_i=0$. Suppose $w^\star_i=0$ for some $i\in\mathcal{I}$, the consumed resource must be $r^\star_i=\underline{r_i}$ since $J_i(r_i)$ is strictly monotone decreasing. Otherwise, the resource available for other agents is smaller and $g(\bm{w},\bm{r})=\sum_{i:w_i\neq0} w_iJ_i(r_i)$ increases. Therefore, if $w_i^\star=0$, we have
\begin{align*}
\max_{r_i}J_i(r_i)=J_i(\underline{r_i})=J_i(r^\star_i) \leq \lambda^\star = \max_j J_j(r^\star_i),
\end{align*}
which still includes the possibility that $J_i(r^\star_i)= \max_j J_j(r^\star_i)$.
\end{remark}

\subsection{Cost-Based Solution Seeking}
The above theorem only ensures the existence of a pure-strategy equilibrium for the corresponding game, but does not provide a computation procedure to obtain such a equilibrium. One may expect to use a best response algorithm. Specifically, we alternate between solving a linear programming and a convex program. The solution of the linear program is the best response of $\bm{w}$ given $\bm{r}$, while the solution of the convex program is the best response of $\bm{r}$ given $\bm{w}$. A best response algorithm, however, does not converge in general as claimed by~\cite[Sec 1.4.3]{nisan2007algorithmic}. In this work, the best response also fails.

We propose a cost-based algorithm to obtain the max-min fair allocation scheme. Note that for a given allocation $\bm{r}$, if $J_i(r_i)$ is too large, we should increase $r_i$. Conversely, if $J_i(r_i)$ is small, it should donate its resource. Based on this observation, we propose the following algorithm:
\begin{align}\label{eq: cost based algorithm}
\bm{r}(t+1) = T(\bm{r}(t)) \triangleq P_{\mathcal{R}}\Big(\bm{r}_t + \varepsilon \bm{J}(\bm{r}(t))\Big).
\end{align}
The equilibrium of~\eqref{eq: cost based algorithm} is a solution of Problem~\ref{problem: resource allocation fairness}.
\begin{theorem}\label{theorem: equilibrium and optimality}
If $\bm{r}^\star$ is the equilibrium  of the discrete-time projected dynamics~\eqref{eq: cost based algorithm}, i.e., $\bm{r}^\star = T(\bm{r}^\star)$, then for any allocation $\bm{r}$
\begin{align*}
\max_{i\in\mathcal{I}} J_i(r^\star_i) \leq \max_{i\in\mathcal{I}} J_i(r_i).
	\end{align*}
\end{theorem}
The proof is technical and we left it in the appendix in the online version~\cite{wu2019max}.

The update inside the projection only requires local information of the cost function, i.e., the value of the cost $J_i(r_i(t))$ at $r_i(t)$. However, the projection still requires knowledge of every $r_i(t)$. We discuss a distributed extension of this algorithm in Appendix-B in the online version~\cite{wu2019max}. To guarantee the convergence of the algorithm in~\eqref{eq: cost based algorithm}, we impose the following assumption on $J_i(r_i)$.
\begin{assumption}\label{assumption: first order lipchitz condition}
The optimal value $J_i(r_i)$ satisfies, for any $r_i$ and $r_i'$ in $[\underline{r_i},\overline{r_i}]$,
\begin{enumerate}[(1)]
\item(strong monotonicity) there exists a positive real number  $\alpha_i$ such that
	\begin{align*}
	\alpha_i | r_i - r_i' | \leq | J_i(r_i) - J_i(r_i') |;
	\end{align*}
\item($\beta_i$-Lipschitz) there exists a positive real number $\beta_i$ such that
	\begin{align*}
	| J_i(r_i) - J_i(r_i') |  \leq  \beta_i | r_i - r_i' |.
	\end{align*}
\end{enumerate}
\end{assumption}

This assumption means that the variations of the costs with respect to allocated resources are mild.

\begin{theorem}\label{theorem: convergence to unique point}
There exists $\varepsilon \in (0,\frac{2\alpha}{\beta^2})$, where $\alpha=\min_i \alpha_i$ and $\beta = \max_i \beta_i$, such that the algorithm~\eqref{eq: cost based algorithm} converges with a linear convergence rate.
\end{theorem}
\begin{IEEEproof}
We prove the convergence to a unique point by showing that the update is contractive.
Because of the nonexpansive property of a projection to a convex set, we have
\begin{align*}
\| T(\bm{r}) - T(\bm{r}') \|^2
\leq \| \bm{r} + \varepsilon \bm{J}(\bm{r}) - \bm{r}' - \varepsilon \bm{J}(\bm{r}') \|^2
\end{align*}
Expanding the right hand side, we obtain
\begin{align*}
&\| T(\bm{r}) - T(\bm{r}') \|^2 \\
\leq & \| \bm{r} - \bm{r}' \|^2 + 2\varepsilon (\bm{r} - \bm{r}')^\top (\bm{J}(\bm{r}) - \bm{J}(\bm{r}')) \\
&+ \varepsilon^2\|\bm{J}(\bm{r}) - \bm{J}(\bm{r}')\|^2.
\end{align*}

Note that for each $i$, $(r_i-r'_i)(J_i(r_i)-J_i(r'_i))<0$ due to the monotonicity of $J_i(r_i)$. Therefore, we have
\begin{align*}
(\bm{r} - \bm{r}')^\top (\bm{J}(\bm{r}) - \bm{J}(\bm{r}'))
= - |\bm{r} - \bm{r}'|^\top|\bm{J}(\bm{r}) - \bm{J}(\bm{r}')|.
\end{align*}

From Assumption~\ref{assumption: first order lipchitz condition}, there exists a $\alpha_i>0$ such that
\begin{align*}
|J_i(r_i) - J_i(r'_i) | \geq \alpha_i |r_i - r'_i|.
\end{align*}
Let $\alpha=\min_i \alpha_i$, we have
\begin{align*}
|\bm{r} - \bm{r}'|^\top|\bm{J}(\bm{r}) - \bm{J}(\bm{r}')|
\geq  {\alpha}\| \bm{r} - \bm{r}' \|^2,
\end{align*}
which leads to
\begin{align*}
(\bm{r} - \bm{r}')^\top (\bm{J}(\bm{r}) - \bm{J}(\bm{r}')) \leq -{\alpha}\| \bm{r} - \bm{r}' \|^2.
\end{align*}
Consequently,
\begin{align*}
&\| T(\bm{r}) - T(\bm{r}') \|^2 \\
\leq & \| \bm{r} - \bm{r}' \|^2 - 2\varepsilon{\alpha} \|\bm{r} - \bm{r}'\|^2+ \varepsilon^2\|\bm{J}(\bm{r}) - \bm{J}(\bm{r}')\|^2.
\end{align*}

Meanwhile, from Assumption~\ref{assumption: first order lipchitz condition}, we also know that
\begin{align*}
\|\bm{J}(\bm{r}) - \bm{J}(\bm{r}')\|^2 \leq \beta\|\bm{r} - \bm{r}' \|^2,
\end{align*}
where $\beta= \max_i \beta_i$. We can obtain
\begin{align*}
\| T(\bm{r}) - T(\bm{r}') \|^2 \leq  \Big[1+(\varepsilon^2\beta^2-2\varepsilon\alpha)\Big]\| \bm{r} - \bm{r}' \|^2.
\end{align*}
Therefore, as long as $0<\varepsilon<\frac{2\alpha}{\beta^2}$, there exists a real number $0<c<1$ and a point $\bm{r}^\star=T(\bm{r}^\star)$ such that, for any $\bm{r}\in\mathcal{R}$,
\begin{align*}
\|T(\bm{r})-\bm{r}^\star\|^2 = \|T(\bm{r})-T(\bm{r}^\star)\|^2 \leq c \|\bm{r}-\bm{r}^\star\|^2.
\end{align*}
Denoting the error of $\bm{r}(k)$ as $e(k)=\|\bm{r}(k)-\bm{r}^\star\|^2$, we can obtain $e(k+1) \leq c e(k)$, which proves the linear convergence rate.
\end{IEEEproof}

\section{Application in Fair Sensor Scheduling}

\subsection{Problem Setup}
Consider $n$ LTI processes, each measured by one sensor. The dynamics are as follows:
\begin{align}
x^{(i)}(k+1) = A_ix^{(i)}(k) + w^{(i)}(k),\label{eq: process dynamics}\\
y^{(i)}(k) = H_i{x}^{(i)}(k) + {v}^{(i)}(k),\label{eq: measurement}
\end{align}
where $i \in N \triangleq \{1,\ldots,n\}$, ${x}^{(i)}(k)\in\mathbb{R}^{n_i}$ is the state of the $i$-th system at time $k$ and ${y}^{(i)}(k)\in\mathbb{R}^{m_i}$ is the noisy measurement taken by the sensors. For all processes and $k\geq0$, the state disturbance noise ${w}^{(i)}(k)$'s, the measurement noise ${v}^{(i)}(k)$'s and the initial state ${x}^{(i)}(0)$'s are mutually independent Gaussian random variables, which follow Gaussian distributions as ${w}^{(i)}(k)\sim\mathcal{N}({0},{Q}_i)$, ${v}^{(i)}(k)\sim\mathcal{N}({0},{R}_i)$ and ${x}^{(i)}(0)\sim\mathcal{N}({0}, {\Pi}_i)$. The covariance matrices ${Q}_i$ and ${\Pi}_i$ are positive semidefinite, and ${R}_i$ is positive definite. We assume that, for every $i \in N$, the pair $({A}_i,{H}_i)$ is observable and the pair $({A}_i,\sqrt{{Q}}_i)$ is controllable.

The sensors measure the process states and compute local state estimates using a Kalman filter. After that, the sensors transmit the estimate data through a communication network under resource constraints to a remote state estimator. The remote state estimator will either synchronize the remote state estimates with the local state estimate if the updated data is received, or use process dynamics to predict the state estimates if no data is received. For sensor $i$ at time $k$, we use $a_i(k)=1$ to denote transmission and $a_i(k)=0$ otherwise. The estimation error covariance of the remote estimator for each process $P^{(i)}$ at time $k+1$ can be computed as follows:
\begin{align*}
P^{(i)}(k+1)=
\begin{cases}
\overline{P}_i, &\text{if~}a_i(k)=1,\\
A_iP^{(i)}(k)A_i^{\top}+Q_i, &\text{if}~a_i(k)=0,
\end{cases}
\end{align*}
where $\overline{P}_i$ is the steady state of the state estimation error covariance of the local Kalman filter.

In this work, we assume that the information available to transmission decision is
\begin{align*}
\tau_i(k) = \min\{t\geq0:a_i(k-t)=1\},
\end{align*}
which is the time elapsed since the last successful transmission. Transmission decisions of the senors are defined by
\begin{align*}
f_i(\tau_i(k)) := \mathtt{Pr}(a_i(k)|\tau_i(k)),
\end{align*}
where $a_i(k)=1$ denotes transmission and $a_i(k)=0$ denotes no transmission. We are interested in allocating the average communication rates of the senors to minimize the largest average remote state estimation error. This can be formulated as
\begin{align}\label{problem: sensor fair problem}
\min_{\bm{r}\in\mathcal{R}} \max_{i\in\mathcal{I}} J_i(r_i),
\end{align}
where the constraint set is
\begin{align*}
\mathcal{R} := \{\bm{r}: \bm{1}^\top\bm{r}\leq R,~\bm{0}\leq\bm{r}\leq 1 \},
\end{align*}
and the performance metric is
\begin{align*}
J_i(r_i):=\limsup_{T\to\infty}\frac{1}{T}\sum_{k=0}^{T-1} \Tr(P^{(i)}(k))).
\end{align*}
To transform the problem as the standard setting in the last section, we need to study properties of $J_i(r_i)$. This requires analysis of optimal scheduling polices for a single sensor.

\begin{remark}
	Some papers studied the state estimation problem from a zero-sum game perspective~\cite{banavar1992game,shen1997game}. Their problems are different from ours. They studied the worst-case estimation error estimator in face of model uncertainty. The related game is between the state estimator and the nature who exerts uncertain disturbance to the system. The estimator wants to minimize the estimation error while the nature wants to maximize the estimation error.
\end{remark}

\subsection{Single Sensor Scheduling}
\begin{figure}[t]
	\centering
	\includegraphics[width=0.3\textwidth]{single_constrained.eps}
	\caption{The optimal average estimation error of the single sensor scheduling under constrained communication is piecewise linear, convex and nonincreasing w.r.t. communication rate.}
	\label{fig:single_constrained}
\end{figure}
Given an average communication rate $r_i$ for sensor $i$, the following single sensor scheduling problem is well solved
\begin{align*}
\min_{f_i} \quad& J_i(f_i)\\
\text{s.t.} \quad& C_i(f_i) \leq r_i,
\end{align*}
where $f_i$ is the scheduling policy for sensor $i$ and $C_i(f_i)=\lim_{T\to\infty} \frac{1}{T+1}\mathbb{E}[\sum_{k=0}^{T} a_i(k)]$. The settings in~\cite{chakravorty2017fundamental} are similar to this problem. By using a similar approach, we can obtain the same results for the single sensor scheduling problem as those in~\cite{chakravorty2017fundamental}. We recap essential results of this single sensor scheduling problem.

Recall that $\tau_i(k)$ stands for the time elapsed since the last successful transmission. A threshold policy with threshold $\xi_i$ is defined as follows. Transmit if $\tau_i(k)\geq\xi_i$ and do not transmit otherwise. An optimal policy of the constrained single sensor scheduling is a randomized threshold strategy, which randomizes between two thresholds $\xi_i$ and $\xi_i+1$, i.e.,
\begin{align*}
a_i(k) =
\begin{cases}
0,&\text{ if }\tau_i(k)<\xi_i,\\
0,&with~probability~1-{b_i}, \text{ if }\tau_i(k)=\xi_i,\\
1,&with~probability~{b_i}, \text{ if }\tau_i(k)=\xi_i,\\
1,&\text{if }\tau_i(k)>\xi_i.
\end{cases}
\end{align*}

The communicate rate of the randomized threshold optimal policy is exactly $r_i$. Therefore, given the allocated communication rate $r_i$, we can calculate the corresponding threshold $\xi_i$ and randomization parameter $b_i$ as
\begin{align*}
\xi_i = \lfloor \frac{1}{r_i} - 1 \rfloor,~
b_i =\xi_i+1+\frac{r_i-1}{r_i},
\end{align*}
where $\lfloor\cdot\rfloor$ denotes round downwards.

Given the parameters of the optimal randomized policy, we can compute the corresponding optimal estimation error $J_i(r_i)$. For the optimal single sensor scheduling under constrained communication, the relation of the communication rate and the average estimation error is shown in Fig. \ref{fig:single_constrained}. It can be proven that the optimal estimation error $J_i(r_i)$ is piecewise linear, convex, continuous and strictly decreasing with respect to $r_i$~\cite{chakravorty2017fundamental}.

\subsection{Max-Min Fair Allocation Approach}
\begin{algorithm}
	\caption{Solving the average constraint problem}
	\label{alg:framework}
	\begin{algorithmic}[1]
		\State Initialize $\bm{r}(1)$, $0<\eta<1$, $\varepsilon_r>0$, $\varepsilon>0$, $t=1$
		\State Set
		\begin{align*}
		\underline{r_i} \leftarrow
		\begin{cases}
		0,&\text{if process~}i\text{~is stable},\\
		\eta, &\text{if process~}i\text{~is unstable}.
		\end{cases}
		\end{align*}
		\State \textbf{repeat}
		\State \quad  $\mathcal{R} \leftarrow  \{\bm{r}:~\bm{1}^{\top}\bm{r} \leq R,~\underline{\bm{r}} \leq \bm{r} \leq \bm{1}\}$
		\State \quad \textbf{repeat}
		\State	\qquad	\begin{align*}
		&r_i(t+1) \leftarrow r_i(t) + \varepsilon J_i(r_i(t))\\
		&\bm{r}(t+1) \leftarrow \argmin_{\bm{r}\in\mathcal{R}} \|\bm{r}(t+1) - \bm{r}\|_2^2 \\
		&t \leftarrow t+1\\
		& {\varepsilon} \leftarrow  \frac{1}{\frac{1}{\varepsilon}+1}
		\end{align*}
		\State \quad \textbf{until} $\|\bm{r}(t) - \bm{r}(t-1) \| \leq\varepsilon_r$
		\State \quad $\tilde{\bm{r}}\leftarrow \bm{r}(t)$
		\State \quad \textbf{if} for all unstable processes
		\State	\qquad $\tilde{r}_i>\underline{r_i}$
		\State \qquad \textbf{break}
		\State \quad \textbf{else} for all unstable processes
		\State \qquad  $\underline{r_i} \leftarrow  \eta \cdot \underline{r_i}$
		\State \quad \textbf{end if}
		\State \textbf{until} $\tilde{r}_i>\underline{r_i}$ for all unstable processes
	\end{algorithmic}
\end{algorithm}

An optimal allocation for~\eqref{problem: sensor fair problem} can be found by solving the following zero-sum game
$
\max_{\bm{w}\in\mathcal{W}} \min_{\bm{r}\in\mathcal{R}} \sum_{i=1}^n w_i J_i(r_i).
$
Note that Assumption~\ref{assumption: monotonicity between objective and resource}-\ref{assumption: strict positivity of costs} are satisfied from the results in the single sensor scheduling problem. Moreover, the total bandwidth $R$ should be strictly greater than $0$. Then an allocation scheme $r_i=\frac{R}{n}$ suffices to verify Assumption~\ref{assumption: feasiblity}. By applying the existence, we see that there exists at least one fair allocation policy $\bm{r}^\star$.

In addition, this formulation reveals the following result which bridges the max-min fairness and the identical estimation error, which is an extension of Corollary~\ref{corollary: max achieve}.
\begin{proposition}\label{proposition: equal performance}
	The remote estimation errors of the sensors are equal for unstable processes.
\end{proposition}
\begin{IEEEproof}
Note that we must have $w^\star_i>0$ for unstable processes. Otherwise, if $w^\star_i=0$, we have $J_i(\underline{r_i})$ being unbounded, which certainly does not meet the max-min fair requirement. According to Corollary~\ref{corollary: max achieve}, the result follows.
\end{IEEEproof}

\begin{remark}
	According to the above proposition, the min-max criterion leads to fairness in the sense of equal remote estimation performance if the processes are unstable. Nevertheless, if there exist stable processes, it could be infeasible to find a scheduling policy such that the remote estimation performances of each process are equal. This occurs if the estimation error of the state prediction for the $i$-th processes is less than the steady state estimation error of the Kalman filter for the $j$-th process, i.e.,
$\Tr(P_i) < \Tr(P_j)$,
	where $P_i=f_p^{(i)}(P_i)$ and $P_j=f_c^{(j)}(f_p^{(j)}(P_j))$ with $f_p^{j}(X)=A_jXA_j^\top+Q_j$ and $f_c^{j}(X)=X-XC_j^\top(C_jXC_j^\top+R_j)^{-1}C_jX$. In this case, we never have $J_i(r_i) = J_j(r_j)$ for any allocation scheme because $\Tr(P_i)$ is the largest estimation error of sensor $i$ while $\Tr(P_j)$ is the lowest estimation error of sensor $j$ .
\end{remark}

To compute the max-min fair allocation scheme using the algorithm in~\eqref{eq: cost based algorithm}, the validity of Assumption~\ref{assumption: first order lipchitz condition} should also be verified. Note that the optimal average estimation performance can be unbounded on $r_i\in(0,1]$, which makes the Lipschitz condition in Assumption~\ref{assumption: first order lipchitz condition} invalid. We overcome this issue by imposing a very small lower bound of the minimum communication rate $r_i\in[\underline{r_i},1]$ for those unstable processes. On the compact interval $[\underline{r_i},1]$ with $\underline{r_i}>0$, the cost objective $J_i(r_i)$ satisfies Assumption~\ref{assumption: first order lipchitz condition}. For any lower bound $\underline{r_i}$, if $r^\star_i=\underline{r_i}$, we set $\underline{r_i}$ to be a smaller value, e.g., $\eta\underline{r_i}$ with $0<\eta<1$, and then run the algorithm~\eqref{eq: cost based algorithm} again. This iterative scheme stops when $r_i^\star>\underline{r_i}$. In practice, however, if the initial $\underline{\bm{r}}$ is small enough, only one iteration for $\underline{\bm{r}}$ is needed. This revised fair allocation scheme seeking algorithm is summarized in Algorithm~\ref{alg:framework}. Essentially, the inner loop corresponds to the original algorithm in~\eqref{eq: cost based algorithm}, and the outer loop seeks the appropriate $\underline{r_i}$.

Note that we use diminishing step sizes in the algorithm. This is proposed to ensure that the step size meets the convergence condition that $\varepsilon\in(0,\frac{2\alpha}{\beta^2})$ in finite steps as mentioned in Theorem~\ref{theorem: convergence to unique point}. This condition requires knowledge of $\alpha$ and $\beta$ before implementing the algorithm. By using the diminishing step sizes, such a requirement is removed. Moreover, the diminishing step sizes are always greater than zero, which means that they also meet the positivity condition of the step sizes in Theorem~\ref{theorem: convergence to unique point}.
The convergence of the revised algorithm is guaranteed in the following theorem.
\begin{theorem}\label{theorem: fair policy unique and convergent}
Algorithm~\ref{alg:framework} converges to $\tilde{\bm{r}}^\star$, which lies in an $\frac{c}{1-c}\varepsilon_r$-neighborhood of a max-min fair allocation policy of~\eqref{problem: sensor fair problem}, i.e., $\|\bm{r}^\star-\tilde{\bm{r}}^\star\|<\frac{c}{1-c}\varepsilon_r$ with $\bm{r}^\star$ being a solution of the max-min fair allocation for~\eqref{problem: sensor fair problem} and $c$ being the contraction constant for the mapping $T(\cdot)$. Moreover, $c=1+(\varepsilon^2\beta^2-2\varepsilon\alpha)$, where $\varepsilon$ is the stepsize when the algorithm terminates, and $\alpha$ and $\beta$ are the same as those in Theorem~\ref{theorem: convergence to unique point}. In addition, for each $\underline{\bm{r}}$, the convergence rate is linear in time.
\end{theorem}
\begin{IEEEproof}
For each $\mathcal{R}$, the iteration~\eqref{eq: cost based algorithm} satisfies the Assumptions required for convergence. The positive decreasing step sizes $\varepsilon$'s satisfy the requirement for the iteration to be a contraction mapping in a finite time. By using the contraction property developed in Theorem~\ref{theorem: convergence to unique point}, the iteration~\eqref{eq: cost based algorithm} converges for each $\mathcal{R}$ and the convergence rate is linear. If $\tilde{r}_i=\underline{r_i}\neq 0$ for any unstable process, there exists a smaller $\underline{r_i}$, which decreases $\tilde{r}_i$ and $\max_j J_j(\tilde{r}_i)$ simultaneously. If $\tilde{r}_i>\underline{r_i}$ for all unstable processes, then $\max_j J_j(\tilde{r}_i)$ cannot decrease by choosing smaller $\underline{r_i}$'s. As $\underline{r}_i$ is lower bounded by zero, the outer loop terminates in a finite time. Finally, let $\bm{r}$ be such that $T(\bm{r})=\tilde{\bm{r}}^\star$. We can observe that
$
\|\bm{r} - \bm{r}^\star\| = \|\bm{r} - T(\bm{r}) + T(\bm{r})- \bm{r}^\star\|
\leq  \varepsilon_r + c \|\bm{r} - \bm{r}^\star \|,
$
which is equivalent to
$
\|\bm{r} - \bm{r}^\star\| \leq \frac{\varepsilon_r}{1-c}.
$
As $\|\tilde{\bm{r}}^\star- \bm{r}^\star\| \leq c\|\bm{r} - \bm{r}^\star \|$, we can obtain $\|\tilde{\bm{r}}^\star- \bm{r}^\star\| \leq \frac{c}{1-c}\varepsilon_r$. This completes the proof.
\end{IEEEproof}

\section{Numerical Examples}
In this section, we consider five LTI processes. The process parameters in~\eqref{eq: process dynamics} and~\eqref{eq: measurement} are as follows:
\begin{align*}
&{A}_1=\begin{bmatrix}1.2 &0 \\ 0 &0\end{bmatrix}, &{Q}_1&=\begin{bmatrix}4 &0 \\ 0 &1\end{bmatrix};\\
&{A}_2=\begin{bmatrix}1.1 &1 \\ 0 &1\end{bmatrix},  &{Q}_2&=\begin{bmatrix}1 &0 \\ 0 &4\end{bmatrix};\\
&{A}_3=\begin{bmatrix}1.2 &1 \\ 0 &0.8\end{bmatrix}, &{Q}_3&=\begin{bmatrix}1 &0 \\ 0 &4\end{bmatrix}\\
&{A}_4=\begin{bmatrix}0.8 &0.6 \\ 0 &0.9\end{bmatrix}, &{Q}_4&=\begin{bmatrix}16 &0 \\ 0 &1\end{bmatrix};\\
&{A}_5=\begin{bmatrix}0.3 &1 \\ 0 &0.1\end{bmatrix}, &{Q}_5&=\begin{bmatrix}0.3 &0 \\ 0 &1.2\end{bmatrix},
\end{align*}
and $C_i$ and $R_i$ are two-dimensional identity matrix for all $i=1,2,3,4,5$.

\begin{figure}[t]
	\centering
	\includegraphics[width=0.35\textwidth]{2.eps}
	\caption{The bandwidth allocation of each processes.}
	\label{fig:2}
	\includegraphics[width=0.35\textwidth]{2_cost.eps}
	\caption{The remote estimation error of each processes.}
	\label{fig:2_cost}
\end{figure}

By running Algorithm~\ref{alg:framework}, we obtain the max-min fair allocation scheme and corresponding average estimation error for each processes. The results of the bandwidth resource allocation for $r_{tot}=2$ are shown in the following Fig.~\ref{fig:2}. It can be seen that the resource allocation converges. Moreover, the average remote estimation errors of all processes, except process~$5$, converge to the same value as shown in Fig.~\ref{fig:2_cost}.

According to Proposition~\ref{proposition: equal performance}, the estimation errors of the unstable processes converge to the same value as expected. Interestingly, although process~$4$ is stable, its estimation error also converges to the same value as those of the unstable processes. Moreover, the communication rate of process~$4$ is the highest among all processes. Meanwhile, process~$5$ as a stable processes, does not occupy any communication bandwidth. Intuitively, process~$4$ is most unpredictable among the five processes in terms of the process noise covariance $Q_i$ while process~$5$ is predictable in the sense that its spectral radius is small and the process noise covariance is very small. Although process 5 is allocated with zero communication budget, its time-averaged estimation error is smaller than other processes with nonzero communication resources.

\begin{figure}[t]
	\centering
	\includegraphics[width=0.35\textwidth]{rate.eps}
	\caption{The error of $\bm{r}(k)$ in each iteration.}
	\label{fig:rate}
\end{figure}

As $\underline{r_i}$ is initialized to be small ($0.001$ in the example), the outer loop of the algorithm terminates after one run. Fig.~\ref{fig:rate} shows that the log of the error decays linearly in each iteration in the inner loop as claimed in Theorem~\ref{theorem: convergence to unique point}.

\section{CONCLUSION}
In this work, we considered the max-min fairness criterion for resource allocation. We specified the conditions which guarantee the convexity and monotonicity structure of the cost objective with respect to its allocated resources. We used a Lagrangian multiplier method to transform the original problem into a zero-sum game. We developed an algorithm to compute the policy at the equilibrium and proved its convergence. The developed method was applied to a fair sensor scheduling problem. We showed that our formulation includes ``equal" performance of the sensors if it is feasible to have an ``equal-performance" scheduling policy. We utilized the results from the single sensor scheduling problem to efficiently compute the update equations of the subproblems for the decomposed problem.

The coupling between each individual is imposed in terms of the average consumption. In a more strict situation, where the total resource consumption constraint need to be satisfied in each time step, a Markov game setup should be considered. This is out of the framework developed in this work, which is left as future work.

\section*{Appendix}
\subsection{Proof of Theorem~\ref{theorem: equilibrium and optimality}}
We first characterize key properties of an optimal solution of Problem~\ref{problem: resource allocation fairness} and then the equilibrium of the iterative algorithm. Lastly, we combine the two to prove their equivalence. The proof relies on the notation of the normal cone of a set at its boundary. We give its definition here for completeness.
\begin{definition}~\cite{nagurney2012projected}
	Given a set $\mathcal{R}\subset\mathbb{R}^n$ and a point $\bm{r}\in\mathcal{R}$, the normal cone of $\mathcal{R}$ at $\bm{r}$ is
	\begin{align*}
	N_\mathcal{R}(\bm{r}) := \{ \, \bm{x} :  \bm{x}^\top (\bm{r}'-\bm{r}) \leq 0, \,\forall \bm{r}'\in\mathcal{R} \, \}.
	\end{align*}
\end{definition}

\subsubsection{Characterization of an optimal solution}
The objective function is a pointwise maximum function and is non-differentiable. However, its  subdifferentials exist.
\begin{lemma}\cite{boyd2018subgradients}\label{lemma: subgradient of the pointwise max}
	If the functions $J_i(r_i)$ are subdifferentiable, the subdifferential of $\max_{i\in\mathcal{I}} J_i(r_i)$ is the convex hull of the union the subdifferentials of every $J_j(r_j)$ satisfying $J_j(r_j) = \max_{i\in\mathcal{I}} J_i(r_i)$, i.e.,
	\begin{align*}
	\partial_{\bm{r}} \max_{i\in\mathcal{I}} J_i(r_i) = \textbf{Co}\cup\{ \, \partial_{\bm{r}} J_j(r_j) : \, J_j(r_j) = \max_{i\in\mathcal{I}} J_i(r_i) \, \},
	\end{align*}
	where $\textbf{Co}$ is the convex hull of a set.
\end{lemma}

For subdifferentials, there exists a generalized KKT condition, which involves the normal cone at an optimal solution. The relation between the normal cone and the subdifferential of an optimal solution of Problem~\ref{problem: resource allocation fairness} is as follows.
\begin{lemma}\cite[Theorem 3.33]{ruszczynski2006nonlinear}
	A point $\bm{r}^\star\in\mathcal{R}$ is an optimal solution to Problem~\ref{problem: resource allocation fairness} if and only if
	\begin{align*}
	\bm{0} \in \partial_{\bm{r}} \max_{i\in\mathcal{I}} J_i(r^\star_i) + N_{\mathcal{R}}(\bm{r}^\star),
	\end{align*}
	where $N_{\mathcal{R}}(r)$ is the normal cone of $\mathcal{R}$ at $\bm{r}$ and the plus sign stands for addition in terms of set, i.e., $X+Y=\{\,x+y|: \, x\in X,y\in Y\,\}$ for two sets $X$ and $Y$.
\end{lemma}

Denote the active set of an optimal solution as
\begin{align*}
\mathcal{I}^\star = \{ \, i: \, J_i(r_i^\star) = \max_{j\in\mathcal{I}} J_j(r_j^\star) \, \}.
\end{align*}
There exists one element in the subdifferentials of $\max_{i\in\mathcal{I}} J_i(r^\star_i)$ lying in the space spanned by $\{ \bm{e}_i \}_{i\in\mathcal{I}^\star}$
\begin{lemma}
	There exists $\alpha>0$ such that $\sum_{i\in\mathcal{I}^\star} -\alpha \bm{e}_i\in \partial_{\bm{r}} \max_{i\in\mathcal{I}} J_i(r^\star_i)$, where $\bm{e}_i$ is the unit vector with the $i$-th component being one.
\end{lemma}
\begin{IEEEproof}
	As $\partial_{\bm{r}} \max_{i\in\mathcal{I}} J_i(r^\star_i)$ is the convex hull of $\cup\{ \partial_{\bm{r}} J_i(r^\star_i): \, i\in\mathcal{I}^\star \}$ and the subdifferentials of $J_i(r_i)$ are all negative, the result follows directly.
\end{IEEEproof}

\subsubsection{Characterization of the equilibrium}
For $\bm{r}\in\mathcal{R}$ and $v\in N_\mathcal{R}(\bm{r})$, we have $P_{\mathcal{R}} (\bm{r} + \bm{v}) = \bm{r}$. The equilibrium can thus be characterized with its normal cone.
\begin{lemma}\cite[pp. 12]{nagurney2012projected}
	If $\bm{r}^\star$ is the equilibrium of the discrete-time projected dynamics~\eqref{eq: cost based algorithm}, then
	\begin{align*}
	\bm{J}(\bm{r}^\star) \in N_\mathcal{R}(\bm{r}^\star).
	\end{align*}
\end{lemma}

\subsubsection{Proof of Theorem~\ref{theorem: equilibrium and optimality}}
The direction of $\bm{r}(k)$ in~\eqref{eq: cost based algorithm} forms an acute angle with one of the subdifferentials of the pointwise maximum function as
\begin{align*}
\bm{J}(\bm{r})^\top\Big(-\partial_{\bm{r}} \max_{i\in\mathcal{I}} J_i(r_i) \Big) > 0.
\end{align*}
It is expected that both vectors lie in the normal cone of the equilibrium. The acute angle relation can be further exploited at an optimal solution as follows.
\begin{lemma}\label{lemma:max achieved for active agent}
	If $i\in\mathcal{I}^\star$ and there exists $i'$ such that $J_{i'}(r^\star_{i'})<J_i(r^\star_{i})$, then $r_i^\star = \min\{R,1\}$, which is the maximum allowable resource.
\end{lemma}
\begin{IEEEproof}
	We prove by contradiction. Suppose $i\in\mathcal{I}^\star$ and $r_i^\star < \min\{R,1\}$. Let $r_i = \min\{R,1\}$. In addition, fix one $i'\in\mathcal{I}\backslash\mathcal{I}^\star$, and let $r_{i'} = r_{i'}^\star - (\min\{R,1\} - r^\star_i)$. For other $j\in\mathcal{I}\backslash\{i,i'\}$, let $r_j=r^\star_j$. We can obtain
	\begin{align*}
	&\bm{J}(\bm{r}^\star)^\top (\bm{r} - \bm{r}^\star) \\
	= &J_i(r_i^\star)(\min\{R,1\} - r_i^\star) + J_{i'}(r_{i'}^\star)( r_{i}^\star - \min\{R,1\})\\
	= & ( J_i(r_i^\star) - J_{i'}(r_{i'}^\star) ) (\min\{R,1\} - r_i^\star) > 0.
	\end{align*}
	This contradicts that $\bm{J}(\bm{r}^\star) \in N_\mathcal{R}(\bm{r}^\star)$.
\end{IEEEproof}

Now we are ready to prove Theorem~\ref{theorem: equilibrium and optimality}. We separate the discussion for two cases.
Firstly, if $J_i(r_i^\star) = \max_{j\in\mathcal{I}} J_j(r_j^\star)$ for every $i\in\mathcal{I}$, then
\begin{align}\label{eq: centralized seeking algorithm}
\sum_{i\in\mathcal{I}^\star} J_i(r_i^\star) ( r_i - r_i^\star )
= \sum_{i\in\mathcal{I}} J_i(r_i^\star) ( r_i - r_i^\star ) \leq 0.
\end{align}

Secondly, if there exists $i'$ such that $J_{i'}(r^\star_{i'})<J_i(r^\star_{i})$, then according to Lemma~\ref{lemma:max achieved for active agent}, we have
\begin{align*}
\sum_{i\in\mathcal{I}^\star} J_i(r_i^\star) ( r_i - r_i^\star )
= \sum_{i\in\mathcal{I}^\star} J_i(r_i^\star) ( r_i - \min\{R,1\} ) \leq 0.
\end{align*}
By the definition of $N_{\mathcal{R}}(\bm{r}^\star)$, we have $\sum_{i\in\mathcal{I}^\star} \bm{e}_i\in N_{\mathcal{R}}(\bm{r}^\star)$, where $\bm{e}_i$ is the unit vector with the $i$-th component being one. Since there exists a constant $\alpha>0$ such that
\begin{align*}
\sum_{i\in\mathcal{I}^\star} -\alpha\bm{e}_i\in&\partial_{\bm{r}} \max_{i\in\mathcal{I}} J_i(r_i) \\
= &\textbf{Co}\cup\{\, \partial_{\bm{r}} J_j(r_j) : \, J_j(r_j) = \max_{i\in\mathcal{I}} J_i(r_i) \, \}.
\end{align*}Therefore,
\begin{align*}
\bm{0} \in \partial_{\bm{r}} \max_{i\in\mathcal{I}} J_i(r_i) +N_{\mathcal{R}}(\bm{r}^\star),
\end{align*}
which completes the proof.

\subsection{Distributed Algorithm}
The algorithm in~\eqref{eq: cost based algorithm} is centralized because of the projection operation. We can develop a distributed version of it by distributing the projection operation. To accomplish such a goal, we take a step backward. Algorithm~\eqref{eq: cost based algorithm} can be perceived as a projected gradient descent algorithm for the following optimization problem
\begin{align}
\min_{\bm{r}} \quad &\sum_{i\in\mathcal{I}} \tilde{J}_i(r_i)\nonumber\\
\text{s.t.} \quad &\sum_{i\in\mathcal{I}} r_i\leq R,~r_i\in[\underline{r_i},\overline{r_i}],\label{eq: equivalent problem}
\end{align}
where
\begin{align*}
\tilde{J}_i(r_i) := \int_{0}^{r_i} -J_i(t)\mathrm{d}t.
\end{align*}
This formulation can be included in a distributed optimization framework as both the objective function and the constraint is written in a separable summation form. We propose one intuitive implementation\footnote{Other implementations such as~\cite{yi2016initialization,nedic2018improved} are also available.}. The dual problem of~\eqref{eq: equivalent problem} can be written as
\begin{align*}
\max_{\lambda\geq0} q(\lambda),
\end{align*}
where $q(\lambda) = \min_{\bm{\underline{r}}\leq\bm{r}\leq\bm{\overline{r}}} \sum_{i\in\mathcal{I}} \Big( \tilde{J}_i(r_i)+\lambda (r_i - R/n) \Big)$. By making $n$ copies of the dual variable $\lambda$, this dual problem can be reformulated as
\begin{align*}
\max_{\bm{\lambda}\geq \bm{0}} \quad &\sum_{i\in\mathcal{I}} q_i(\lambda_i)\\
\text{s.t.} \quad & \lambda_i=\lambda_j \quad \forall i,j\in\mathcal{I},
\end{align*}
where $q_i(\lambda_i)=\min_{\underline{r_i}\leq r_i \leq \overline{r_i}} \tilde{J}_i(r_i)+\lambda_i (r_i - R/n)$. The above constraint can also be relaxed over an undirected graph. Each sensor $i$ can acquire $\lambda_j$ from a subset of $\mathcal{I}$ and the constraint can be compactly written as
\begin{align*}
L\bm{\lambda} = \bm{0},
\end{align*}
where $L$ is a $n\times n$ matrix with its element in the $i$-th row $j$-column being
\begin{align*}
L_{[i,j]} = \begin{cases}
0,& ~\text{if $i$ and $j $ are not connected},\\
-\frac{1}{N_i},&~\text{if $j$ is connceted to $i$},\\
1,&~\text{if}~i=j,
\end{cases}
\end{align*}
where $N_i$ is the number of nodes connecting to node $i$. Consider the following augmented Lagrangian
\begin{align*}
\mathcal{L}(\bm{r},\bm{\lambda}) = \sum_{i\in\mathcal{I}} \Big(\tilde{J}_i(r_i) + \lambda_i(r_i-R/n) \Big) - \bm{\lambda}^\top L \bm{\lambda},
\end{align*}
which is convex in $\bm{r}$ as $J_i(r_i)$ are decreasing and continuous, and concave in $\bm{\lambda}$. The perturbed primal-dual subgradient update can be written as
\begin{align*}
\bm{r}(k+1) &= P_{[\bm{\underline{r}},\bm{\overline{r}}]} \Big(\bm{r}(k) - \epsilon(k)\nabla_{\bm{r}}\mathcal{L}(\bm{r}(k),\hat{\bm{\lambda}}(k+1))\Big)\\
\bm{\lambda}(k+1) &= P_{[\bm{0},+\infty]} \Big(\bm{\lambda}(k) + \epsilon(k)\nabla_{\bm{\lambda}}\mathcal{L}(\hat{\bm{r}}(k+1),\bm{\lambda}(k))\Big),
\end{align*}
where $[\bm{\underline{r}},\bm{\overline{r}}]$ denotes the box constraint for $\bm{r}$ and $[\bm{0},+\infty]$ denotes the positivity constraint for $\bm{\lambda}$.
Note that in the update terms, $\bm{\lambda}(k)$ and $\bm{r}(k)$ has been replaced by two perturbation terms $\hat{\bm{\lambda}}(k+1)$ and $\hat{\bm{r}}(k+1)$. This is adopted to ensure convergence for general convex-concave Lagrangian functions. These perturbation terms $\hat{\bm{\lambda}}(k)$ and $\hat{\bm{r}}(k))$ can be calculated by
\begin{align*}
\hat{\bm{r}}(k+1) &= P_{[\bm{\underline{r}},\bm{\overline{r}}]} \Big(\bm{r}(k) - \hat{\epsilon}(k)\nabla_{\bm{r}}\mathcal{L}(\bm{r}(k),\bm{\lambda}(k))\Big)\\
\hat{\bm{\lambda}}(k+1) &= P_{[\bm{0},+\infty]} \Big(\bm{\lambda}(k) + \hat{\epsilon}(k)\nabla_{\bm{\lambda}}\mathcal{L}(\bm{r}(k),\bm{\lambda}(k))\Big).
\end{align*}
The selection of the step sizes $\epsilon(k)$ and $\hat{\epsilon}(k)$ and the convergence of this perturbed primal-dual subgradient algorithm have been shown in~\cite{kallio1999large}. Since the updating terms
\begin{align*}
\nabla_{\bm{r}}\mathcal{L}(\bm{r}(k),\bm{\lambda}(k)) =& \bm{J}(\bm{r}(k)) + \bm{\lambda}(k)\\
\nabla_{\bm{\lambda}}\mathcal{L}(\bm{r}(k),\bm{\lambda}(k)) =& \bm{r}(k) - R/n - L\bm{\lambda}(k)
\end{align*}
does not require knowledge of $J_i(r_i)$ and $r_i(k)$ of other nodes and the projections can be done for each component, this algorithm is distributed.

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