text:
\section{Introduction}
More than 30 years ago, the European Muon Collaboration (EMC) discovered that the unpolarized structure function of a bound nucleon in a nucleus is different from that of a free nucleon, and different from one nucleus to another \cite{Aubert:1983xm}. This experimental fact triggered an innumerable series of analyses and their corresponding explanations (see \cite{Rith:2014tma} for an extensive list of references) and very interesting recent theoretical developments relating the EMC effect to Short Range Correlations \cite{Higinbotham:2010tb,Weinstein:2010rt,Piasetzky:2011zz,Arrington:2012ax,Hen:2012fm} ,\cite{Frankfurt:2009vv,Vanhalst:2012ur,GarciaCanal:2013dma}.

Not long ago there has been a proposal for an experimental study of the nuclear effect for the polarized structure function $g_1$, an experiment in which both projectile and the nuclear target are longitudinally polarized \cite{Joo}. This work has been motivated by some detailed model dependent calculations \cite{Cloet:2005rt,Cloet:2006bq,Smith:2005ra,Ganesamurthy:2011zza}, which have coined the development: {\it polarized EMC effect}. This problem was theoretically studied long ago   for the Deuteron by Frankfurt and Strikman \cite{Frankfurt:1981mk} and shortly thereafter for arbitrary spin by Jaffe and Manohar \cite{Jaffe:1988up} with a non-relativistic convolution model. The recent analyses are detailed model calculations which take into account a realistic low energy nuclear description, a model for hadron structure and include QCD evolution in their schemes. Their result shows a large effect in the nuclear to proton ratio and moreover,  the contribution of the quark could dramatically affect  it at small $x$.

Despite the fact that the proposed phenomenon is very interesting and might help to understand the behavior of the nucleons in nuclei,  the name, as pointed out by the PAC29 committee \cite{PAC29} might be misleading. In the experimental proposal the chosen nucleus has been  $^7 $Li, were the spin of the system, $3/2$, is naively mostly associated with a valence proton lying in a $p$-shell, while the remaining nucleons are coupled mostly in pairs to total angular momentum $0$. Thus the averaged medium behavior implied by the unpolarized EMC effect is not present in this polarized case. However, it is clear that the active valence proton is subject to the effect of its companions and therefore a medium effect must be present.

Our aim in this note is, by avoiding in as much as possible model dependence, to center our attention into physical ideas which might be at the origin of the discussed phenomenon. For that purpose we revisit the unpolarized EMC effect in terms of an $x$-rescaling description \cite{GarciaCanal:1984eh} and  a pion content approach  \cite{Epele:1994aq,deFlorian:1993hx} to fix the ideas and the parameters connected with the experimental data available at present. We then proceed to study the in medium polarized case.  We  have in mind in all our discussion the experimentally proposed $^7$ Li, which represents an ideal system to distinguish conceptually between the { \it polarized} and unpolarized EMC effects. We end by comparing the polarized and unpolarized phenomena with the intention of motivating further experimental research.

\section{Unpolarized DIS}

In this section we briefly summarize the analysis of DIS with nuclear targets in the two approaches mentioned before, the  $x$-rescaling approach \cite{GarciaCanal:1984eh},  and a qualitative approximation to the so called pion content approach \cite{Epele:1994aq,deFlorian:1993hx}.

This $x$-rescaling approach contains only one parameter, $\eta$,  and the EMC effect was described by suggesting that
the true scaling variable for deep-inelastic scattering off
nuclei should be taken to be $x^* = \eta \,x$ \cite{GarciaCanal:1984eh}. The main idea behind this approach  is that the  quark distributions in nuclei are shifted towards lower $x$ values as compared to those corresponding to free nucleons. Thus the name for the mechanism, $x$-rescaling.  This approach  was shown to be connected \cite{GarciaCanal:1986xe}  to the $Q^2$-rescaling approach \cite{Close:1984zn}.

 Consequently, the measured ratios of the nuclear to deuteron
structure functions can be written, for fixed $Q^2$  as,
\begin{equation}
R(A) = F_2^{A}(x^*)/F_2^{D}(x^*)
\label{f2A}
\end{equation}
where $F_2^{A}(x^*)$ is the nuclear structure function calculated
using a rescaled variable $x^*$ in the free proton and neutron
structure functions, and $F_2^{D}(x^*)$ is the Deuteron structure
function where the effects of rescaling are small.

The $x$-rescaling mechanism leads to fits of very good quality
for the EMC effect in the region $0.3< x < 0.75$ for all the nuclei experimentally analyzed \cite{GarciaCanal:1984eh,GarciaCanal:2013dma}. Recently, this approach was used to show an interplay between the quark-gluon and hadronic degrees of freedom in the unpolarized EMC \cite{GarciaCanal:2013dma}.

The second approach is a simplification of the so called pion content model \cite{Epele:1994aq,deFlorian:1993hx}.  In this  approach one introduces the pion presence into the nucleon structure function and makes a convolution model which takes this presence and the pion structure into account. The end result is twofold, on the one hand there is a $x$-rescaling effect associated to momentum conservation in the pion emission and on the other hand there is an explicit contribution of the sea associated with the pionic structure function.  This approach also leads to fits of very good quality for the EMC effect in the whole $x$-region for all the nuclei experimentally analyzed. The direct contribution of the sea is  important for low $x$ but not so much for intermediate $x$. However, and this is a peculiarity of this approach, it is necessary to consider also the pion sea effect  in the deuteron when one deals with the EMC ratio, if one wants to avoid the blowing up of the ratio at  low $x$ which  would destroy the agreement in the EMC ratio \cite{Epele:1994aq,deFlorian:1993hx}.

We  incorporate this twofold mechanism, in our phenomenological scheme, by means of two parameters. One, $\delta$, describes  the associated scaling mechanism into the free nucleon structure functions as before  $ x \rightarrow \delta x $;  analogously  to $\eta$, $\delta >1$.  Another,  $\sigma$, which magnifies the  contribution  from the sea associated to the $u$ and $d$ flavors in the intermediate $x$ region, concentrated initially for very  small $x$. For one nucleon in the medium the contribution to $F_2$ is
\begin{equation}
F_2^{N} (x) = F_2^{N} (\delta x) + F_2^{udsea_N} (\sigma x).
\end{equation}
This modified nucleon structure functions have to be incorporated into Eq. (\ref{f2A}) to perform the corresponding EMC average. We shall call this approach modified sea scheme (MSS).

\begin{figure}[htb]
\begin{center}
\includegraphics[scale= 1.]{f2xscalingf2sea.eps}
\caption{$R$ ratio as a function of $x$ in  the $x$-scaling approach (black), in the modified sea  approach with (dashed) and without (dotted) the added sea contribution. The parameter values: for $^7$Li: $\eta= 1.011\pm 0.002,\, \delta= 1.016\pm 0.001,\, \sigma= 1$; for $D$: $\eta= 1.0, \,\delta= 1.005,\, \sigma= 1$.}
\label{f2Li}
\end{center}
\end{figure}

Our study is concentrated on the experimentally wishful nucleus $^7$Li \cite{Joo}. We use the proton and neutron structure functions and sea distributions from the analysis of Ref. \cite{Martin:2009iq}  for fixed $Q^2 = 10 $GeV$^2$. The value of $\eta$ for $^7$Li is extracted from a linear extrapolation of a fit to the data of several nuclei. The value obtained for $^7$ Li is $\eta = 1.011 \pm 0.002$. The values of the parameters for the MSS scheme are obtained by fitting them to reproduce the Ratio, $R$, of the $x$-scaling description in the EMC region and are shown in the caption of Fig. \ref{f2Li}.

In Fig.\ref{f2Li} we show our prediction of both approaches and the size of the sea contribution in the MSS method. A good agreement between both approaches is obtained by simply doubling the sea contribution. Note, as it was already mentioned, that we had to incorporate an extra sea contribution for the Deuteron in order to avoid a dramatic increase at the origin, where it tends to dominate. Fig. \ref{f2noDsea} shows that effect.

\begin{figure}[htb]
\begin{center}
\includegraphics[scale= 1.]{f2noDsea.eps}
\caption{$R$ ratio as a function of $x$ in the modified sea approach  with  (black) and without (dashed) Deuteron sea. Parameter values: for $^7$ Li: $\delta= 1.016 \pm 0.001,\, \sigma= 1$, for $D$: $\eta=1.00$}
\label{f2noDsea}
\end{center}
\end{figure}

\section{Polarized DIS}

Several studies of the nuclear effects for the $g_1$ polarized structure functions in the typical EMC $x$ region have recently appeared. They are based on detailed dynamical nucleon structure models and models for nuclear matter. The analysis of Ref. \cite{Cloet:2005rt,Cloet:2006bq} predicts a strong increase of the ratio $R_{pol}= g_1^A/A g_1^p$ of the order of twice the size in the unpolarized case. This calculation is based upon a convolution where the nucleon is described as a bound state of a quark-diquark in the Nambu-Jona-Lasinio (NJL) model and the presence of the nuclear medium is taken into account through mean fields which act on the quarks in the nucleon. This is mainly a valence quark picture where the sea appears by evolution.  Another proposal \cite{Smith:2005ra} includes  sea quarks explicitly in order to be consistent  with both DIS and Drell-Yan results.  Consequently, the nuclear medium effects are present both in the valence and the  sea quark distributions. They are computed using the chiral quark-soliton model and the nuclear effects related to the valence quarks are similar to those obtained in Ref. \cite{Cloet:2005rt,Cloet:2006bq}, while the inclusion of sea quarks gives rise to an important increase for $x < 0.3$. Both description contain many parameters that have to be fitted.  A  third analysis \cite{Ganesamurthy:2011zza} is based on the phenomenological Thermodynamical Bag Model, a modification of the MIT bag model, and shows not much difference between the polarized and the unpolarized ratios.

For our analysis we use the fits to all polarized data of Ref. \cite{deFlorian:2008mr,deFlorian:2009vb} for $Q^2 = 10 $GeV$^2$.  We start by the rescaling approach and as in the case  of the unpolarized structure function we replace the scaling $x$ variable by a rescaled variable $x^*$ in the free nucleon structure functions.

In order to find reasonable values for this $\eta$ parameter, we follow two arguments. The first, which we already discussed previously, is the definition of the polarized in medium effect. Let us for clarity limit ourselves to $^7 Li$. In this case,  from a naive nuclear shell model point of view  we have one active proton in a $p$-shell and the remaining nucleons are coupled to total angular momentum zero in $s$ and $p$ shells. More sophisticated  calculations with Green Function Monte Carlo \cite{Pieper:2004qw} and a Cluster Model \cite{Walliser:1985zz}  give a mean proton polarization of $87\%$  and $3\% $ for the neutron. For simplicity we assume the shell model picture and we define the ratio

\begin{equation}
R_{pol}(x) = \frac{g_1^p(x^*)}{g_1^p (x)}.
\label{g1x}
\end{equation}
It is interesting to compare now with Eq. \ref{f2A} to realize the difference between the polarized and the unpolarized effects, namely the conventional EMC averages over nucleons, while not so the polarized in medium effect.

Accepting the previous naive definition, the second problem  we face is the relation of the $x$-scaling parameter $\eta$ with the dynamics. This parameter is related to the efective nucleon mass in the medium \cite{GarciaCanal:1984eh} which is an average property and therefore we expect that $\eta$ should not change. However, the fact that the nucleon is polarized might incorporate some spin interactions, that average out in the unpolarized case, which might modify it. The results of  Refs. \cite{Cloet:2005rt,Cloet:2006bq,Smith:2005ra}  seem to indicate that $\eta$ should increase. We plot in Fig. \ref{g1f2x} the value of the ratio Eq. (\ref{g1x}) in form of a band, whose smallest value corresponds to the unpolarized $\eta$ value and the largest to $\eta = 1.02$. The full curve shows the ratio $R$. We note that the $R_{pol}$ ratio is extremely sensitive to $\eta$ and therefore to the detailed dynamics of the polarized case.

\begin{figure}[htb]
\begin{center}
\includegraphics[scale= 1.]{g1f2x.eps}
\caption{$R_{pol}$ ratio for $^7$Li as a function of $x$ in the $x$-scaling approach. The dotted curves correspond to parameter values for $^7$Li: upper $\eta=1.01$, lower $\eta=1.02$. The black band corresponds to $R$ with $\eta = 1.011\pm 0.002$.}
\label{g1f2x}
\end{center}
\end{figure}

We next proceed to the MSS approach. As before the nucleus structure function will be defined in terms of modified nucleon structure functions,
\begin{equation}
g_1^N (x) = g_1^N (\delta x) + g_1^{udsea_N} (\sigma x).
\label{g1sea}
\end{equation}
For $^7 Li$, $N$ is a proton. In Fig.\ref{g1f2sea} we plot $ R_{pol}$ with the same parameters as for $R$, and following the same discussion as before we construct a band with a $2\%$ increase both in $\delta$ and  $\sigma$.

\begin{figure}[htb]
\begin{center}
\includegraphics[scale= 1.]{g1f2sea.eps}
\caption{$R_{pol}$ for $^7$Li as a function of $x$ in the MSS approach. The dotted curves correspond to parameter values for Li: upper $\delta=1.015, \,\sigma=1$, lower $\delta=1.035,\, \sigma=1.02$. The black band corresponds to $R$ with $\eta=1.011 \pm 0.002, \, \sigma = 1.0$.}
\label{g1f2sea}
\end{center}
\end{figure}

Finally in Fig.\ref{g1comparison} we compare our results with the two most extreme published analyses. Our result is intermediate between both. We could obtain a higher sea contribution close to the origin by increasing $\sigma$ in  line of Ref.\cite{Smith:2005ra}, but we can never get a result as low as that of Ref.\cite{Cloet:2005rt,Cloet:2006bq} for low $x$.

 We have only analyzed in here the low flavor sea contributions not worrying about heavy flavors and gluons, whose contribution are small in  the two approches.

\section{Conclusion}

The schemes presented in this note parametrize the change of parton distributions in medium in terms of one or two parameters. These parametrizations are motivated by QCD based dynamical pictures, $x$-scaling associated to a Renormalization Group Analysis, and the MSS,  a  chiral scheme convoluted into a partonic description. It is really surprising that in the unpolarized case,  with one or two parameters, one is able to fit the  EMC effect for all studied nuclei.  Thus our fits lead, based on QCD and chiral dynamics, to  parametrizations of the data with a minimal number of parameters.

What can a polarization experiment contribute to our undertanding of proton structure? As has been widely discussed this is not an EMC {``}average" type experimental analysis. The proposed nucleus  $^7$Li is, with percentage corrections, basically a polarized proton in a nuclear medium. In some sense the experiment reminds us of the  the proton spin  analysis \cite{Anselmino:1994gn}, but here with its parton distributions modified by nuclear dynamics.

\begin{figure}[htb]
\begin{center}
\includegraphics[scale= 1.]{g1comparison.eps}
\caption{$R_{pol}$ band for $^7$Li as a function of $x$ in the MSS approach as in the previous figure (dotted curves). The solid curve corresponds to the calculation in Ref. \cite{Smith:2005ra} with the sea contribution included, while the dashed curve the that of Ref. \cite{Cloet:2005rt,Cloet:2006bq}.}
\label{g1comparison}
\end{center}
\end{figure}

Our calculation has shown two important results. The most relevant one is that a dramatic change in the EMC region in $R_{pol}$ would imply a dramatic change in the value of the scaling parameters $\eta$ and $\delta$ and therefore an important influence of the spin-spin interactions in nuclei which lie dormant in the average procedure of the unpolarized EMC effect. The second result, which has been confirmed by the MSS approach is that the sea contribution is important for low $x$ if the ratio is performed against the free proton $g_1$ structure function. If one would perform the ratio with respect to the Deuteron, with its pionic sea included, the sea effect would be quantitatively diminished.
Therefore, our study strongly supports the realization of the proposed experiment \cite{Joo} in the whole range of the variable $x$  as the means of better understanding the structure of the nucleon parton distributions in nuclei and their relation to nuclear dynamics.

\section*{Acknowledgements}
We acknowledge Elliot Leader, who proposed the idea of the present analysis and for his careful reading of the manuscript.
We thank Rodolfo Sassot and  Robert Thorne for invaluable assistance in the use of their codes, and Arcadi Santamaria for
sharing his ample knowledge of Mathematica.   We have all been partially supported  ANPCyT Argentina. V.V. has been also supported by the Ministerio de Econom\'ia y Competitividad and  EU FEDER under contract FPA2010-21750-C02-01, by Generalitat Valenciana: Prometeo/2009/129 and by the EPLANET network under contract PIRSES-2009-GA-246806.

redpajama_set_name: RedPajamaArXiv


text:
COVID-19 Has Not Surged In Cities With Big Protests, But It Has In States That Reopened Early. Here Are Possible Reasons
coronaviruscovidcovid_19lockdownprotestsreopeningspread
By Marie McCullough The Philadelphia Inquirer
WWR Article Summary (tl;dr) Marie McCullough takes a look at the latest patterns of COVID_19 spread.
The United States may be seeing the most compelling evidence yet that the best way to stop the coronavirus is also the most disruptive and difficult: Stay home and avoid other people.
At first glance, the evidence seems conflicting.
States in the South and West that reopened their economies early and with few precautions are now grappling with huge surges in daily case counts, hospitalizations, and deaths.
Yet Philadelphia, Chicago and other cities across the country have not seen the sharp, sustained spikes that were expected after hundreds of thousands of people gathered for protests against police brutality. Daily demonstrations, some capped by riots, began about a week after George Floyd was killed by Minneapolis police on May 25. The maximum incubation period for COVID-19, the time between getting infected and showing symptoms, is believed to be two weeks, so any fallout should be clear by now.
The leading theory to explain this paradox is that being outdoors is safer than indoors. Numerous studies support this. Japanese researchers, for example, found the odds of indoor transmission of the coronavirus were about 19 times greater than in the open air.
But a new study funded by the National Bureau of Economic Research offers another explanation: The protests prompted an overall increase in people staying home, which mostly offset any impact of transmission among protesters.
"This is consistent with avoidance behavior," said Andrew Friedson, a health economist at the University of Colorado, Denver and co-author of the paper. "These protests are large gatherings which are likely to spread the disease. But if there's an offsetting behavior, an increase in social distancing, then that may have caused the overall spread to stay flat or even go down a bit."
The researchers used cell-phone tracking data from Safegraph to see what percentage of the population did not leave home in big cities with protests, compared with cities that didn't have protests.
The results clearly linked an increase in stay-at-home behavior to the civil unrest. Curfews played a part in this behavior but did not completely account for it.
"It's entirely possible that protesters have gotten COVID-19" while demonstrating, Friedson said. "That's a question for contact tracers. But other things, going to bars, seeing friends, not wearing masks, also increase the spread."
Those "other things" are the main driver of the renewed outbreaks in states including Arizona, Nevada, Oklahoma, Texas, Mississippi and Florida, states that had largely reopened. While expanded testing accounts for some of the soaring case numbers, the proportion of positive tests suggests increased transmission. In Arizona, 23% of tests over the last week came back positive. Several governors have suspended further reopenings or reimposed some restrictions.
There are no data on infections among protesters. But circumstantial evidence, anecdotes, and basic biology suggest some protesters got sick and spread the coronavirus even if overall case numbers didn't spike.
An increasing number of new COVID-19 cases across the country are among people in their 20s, 30s, and 40s, the New York Times reported. Protesters were predominantly in these younger age groups.
Philadelphia health officials urged protesters to get a diagnostic test for the virus, even if they wore masks. Vybe Urgent Care Centers and Sayre Health Clinic, which don't restrict testing to people with symptoms or a doctor's order, reported an uptick in testing as some people followed the city's advice.
Another reason to expect transmission during protests: Philadelphia police used teargas and pepper spray that left peaceful protesters reeling, coughing, and retching. If any of them had the virus, and many people who are infected have no symptoms, they would have spewed infectious droplets into the air.
Philadelphia has seen an increase in cases among people ages 20 to 34, who now make up about a quarter of all cases. But the timing of the increase does not point to the protests, said Thomas Farley, the city's health commissioner.
On Friday, Farley warned that an uptick in cases in recent days, coinciding with the gradual reopening of businesses, is worrisome. He said he would reconsider whether the city can move into the "green" phase next week. "Cases in the community are no longer decreasing," Farley said as he announced 143 new cases Friday. ___ Distributed by Tribune Content Agency, LLC.
Nonprofit LAUNCH Chefs Learn While Helping Others
Local Restaurant, Food Truck Business Provides...
Headhunter Tells Millennials How To Land Jobs And Keep Them
How NYC Is Dealing With The Growing Number Of Serial Subway Perverts
New Twist On Downtown Hartford Apartments: 'Co-Living' For Startup Entrepreneurs
Airbnb Fights Unfriendly Regulations With Wave Of Lawsuits against San Francisco, other cities

redpajama_set_name: RedPajamaCommonCrawl


text:
Mr. Li Ning & Dwyane Wade Formally Announce WADE Brand
D-Wade and Li-Ning's company founder announce an all-new brand, WADE. Find out more from the press event and the "Way of Wade" model ahead.
words // Nick DePaula
images // Bob Metelus
Across the globe in China, Li-Ning Company founder Mr. Li Ning and Miami Heat star guard Dwyane Wade formally announced their new partnership moments ago at a press event in Beijing. With the Heat in town as part of the NBA's "China Games," the timing couldn't have worked out better for the newly inked endorsement deal, as Wade and the defending champs will take on the LA Clippers tomorrow in Beijing.
Together, Mr. Li Ning and Dwyane officially launched a new brand simply named WADE. Dwyane will personally oversee all products as the Chief Brand Officer, taking on more responsibility earlier in the product creation process for his own brand that is slated to produce statement level footwear and apparel for both adults and kids.
"I am so proud to welcome Dwyane Wade to the Li-Ning family," said founder Mr. Li Ning. "Dwyane is a superstar for his athletic achievements, and also for the way he carries himself on and off the NBA court. I have come to understand what motivates him, what he cares about, and the personal and professional path he wants to take in the coming years. There are remarkable similarities between Dwyane and myself in this regard, and I could not be more excited about developing a brand with him that speaks to those shared views."
Not only will Wade be earning a hefty annual endorsment tag as the company's new lead athlete for the duration of his NBA career, but he's also being given an equity stake in the company. Last year, Li-Ning reported $1.4 Billion in revenues. It's quite the landmark deal for a Chinese footwear brand to sign a superstar athlete still in his prime and slated to compete for multiple NBA championships in the coming seasons.
"I first visited China in 2008, when I competed in the Beijing Olympics, and was deeply moved by the country's genuine passion for basketball," said Wade. "I have been blessed with an opportunity to make an impact on the game in the United States, and I look forward to now being active with the sport on a more global level. I am honored to join forces with Li-Ning and grow the game, beginning in China."
Dwyane's very first signature model, named the "Way of Wade," can be seen below in both home and away colorways. Expected to be priced in the $125 zone with an early 2013 launch date, he'll be breaking out both versions on-court this week. He's slated to wear the Away pair against the Clippers tomorrow in Beijing, and we'll see the Home pair worn on Sunday in Shanghai's game. Both pairs will be auctioned after each game for charity.
Stay tuned for more on the monumental signing for Li-NIng and the formation of WADE. We'll be speaking with Dwyane shortly for his thoughts on his new endorsement deal, as well as Vice General Manager of Basketball Brian Cupps to discuss where the partnership is headed.


Gone to Philadelphia — for a while. – Tranter's Journal: Marcel Proust's favourite orchid: the cattleya.
Gone to Philadelphia — for a while.
I shall continue this Journal, indefinitely, in the long run, but there will be a hiatus of a few months, while I write a series of regular entries for a Journal (titled "Commentaries") for the Jacket2 people at the University of Pennsylvania at Philadelphia, from October 2012 to about January 2013. Check it out here.
Lots of other writers have Commentary streams on the Jacket2 site. Mine is titled "Under the Horizon", as that's where Sydney is, if you are looking south-west from Philadelphia into the sunset. It will have a slightly different focus to my personal Journal here, as it relates mainly to the history of Jacket magazine, and is published by the people running Jacket2. That new incarnation of Jacket magazine is part of the busy literary doings at the University of Pennsylvania, linked the Kelly Writers House and their many events, and to the PennSound audio archive of literary readings. These are the people I gave Jacket magazine to in 2010, and I feel honoured that they asked me to contribute.
Glad to hear this, John. Your online presence via Jacket has been missed. I'll be tuning in regularly.
Thanks, Larry. It's good to be back.


Since our inception, we have provided medical and non-medical care to over 60,000 individuals in the Valley including the elderly, developmentally disabled, chronically ill and children with disabilities. Last year alone, we provided care to over 4,000 patients. Our source of patients has come through referrals from physicians, insurance companies, state and county agencies, schools, case managers and private parties. We receive periodic audits of our operations from many of these referral sources as well as by the Arizona Department of Health Services. We continue to earn their trust through our high standards and excellent track record of quality care, business practices and ethics.
Debbie oversees the clinical and administrative operations of both MD Home Health and MD Home Assist. She is a graduate of Indiana University School of Nursing and has been a Registered Nurse for 33 years and the Administrator of MD since 2005. She sits on the Professional Advisory Committee and is an active member of the Arizona Home Care Association where she is currently a member of its Board. Debbie has developed several community programs and continues to make home visits to make sure she stays connected with the community needs.
Alicia is responsible for the day to day operations of MD Home Assist. She hires, trains and manages the paraprofessional staff who perform housekeeping, custodial and personal care. Alicia is a Certified Direct Care Worker (DCW) and Article 9 Instructor and has been with the Company for 10 years.
Lorre implements, evaluates and maintains a company wide data-driven Quality Assessment and Performance Improvement program (QAPI). This program monitors the adequacy, appropriateness, and effectiveness of selected indicators related to client outcomes, client safety, and quality of care. She has been with the Company 5 years.
Michael oversees all Billing, Collections, Payroll, A/P and A/R processes. He earned his BS in Accounting and Computer Information Systems at Arizona State University. Michael started with MD in 2001.
The Clinical Manager's function is to guide, instruct, and supervise the clinical personnel in the provision of home health care in accordance with the policies and procedures of the Agency.
MD Home Health is a State Licensed/Medicare Certified Home Health Agency.
Additional information on home care can be obtained from many of the following companies and agencies. Just click on the link to be connected to their web site.


Special interests were promoting a federal tobacco bill in the name of protecting the children. But the claim about 3,000 kids a day starting smoking was exposed as completely phony. Another claim – that 53,000 Americans were dying every year from secondhand smoke—was also demolished. It turned out that the exact figure was 3,000. And this much-publicized figure, which came from the Environmental Protection Agency, the EPA, has also been refuted.
In fact, a federal judge recently ruled that the 1993 EPA report including that figure was seriously flawed. U.S. District Judge William Osteen Sr. overturned the report, saying that it had failed to disclose important findings and reasoning; had jumped to conclusions without evidence; and had adjusted its procedures to fit the agency's conclusions.
This is not to say that secondhand smoke is not bothersome, harmful or offensive, only that it is not a dangerous killer. But the flawed EPA study has already had a major impact. The EPA's 1993 report caused many state and local governments, as well as private employers, to institute bans on smoking inside offices, stadiums and restaurants. The EPA's report was also used by trial lawyers trying to shake down money from the tobacco industry.
This is a classic case of junk science. In the words of the Washington Post, the EPA has been found to have "manipulated scientific studies and ignored scientific and statistical practices." The judge said "there is evidence in the record supporting the accusation that EPA 'cherry-picked' its data" to reach a desired conclusion. The Post insisted in a follow-up story that there are many other studies supporting the EPA conclusion on secondhand smoke, but it admitted that proving a link to cancer remains problematic. The paper added that "there's solid evidence linking environmental smoke to serious respiratory illnesses, especially in children," and that these concerns may be enough to warrant policies curbing smoking in public places.
That may be the case, but it misses the point. If the government wants to curb secondhand smoke because of professed concern for children, that is one thing. But when it claims a scientific link to cancer deaths, that is quite another. Yet that is what the EPA did in this case. Also troubling is the reaction of the American Cancer Society, which issued a news release after the judge's ruling saying that, "If a new [EPA] report is issued, it will say the same thing—secondhand smoke is a killer." Thus, the American Cancer Society has once again prejudged the evidence and has come to an unwarranted conclusion. The group has done a lot of good work in the past alerting the public to cancer dangers, but in this case it is compromising its own credibility with dubious claims.
This wouldn't be the first time the government has exaggerated a cancer danger. New research publicized in the New England Journal of Medicine indicates that the EPA has greatly overestimated the risk of lung cancer from asbestos. That EPA finding led to billions of dollars being spent to remove asbestos from public buildings. It seems as if the EPA is quickly developing a reputation for lying to the American people.


text:
\section{Introduction}
The primal-dual gradient dynamics (PDGD) (or saddle--point dynamics) were first introduced in \cite{arrow1958studies,kose1956solutions} and have been widely used in obtaining the primal-dual solutions of a convex (or concave) optimization problem. The partial primal-dual gradient dynamics (P-PDGD) is similar to PDGD, which is first proposed in \cite{Li:Connecting} to solve specific types of optimization problems with separable decision variables, i.e., $ \min f({x})+h(y), \ \text{s.t.}\  A{x}+By=C$.
Typical applications of PDGD and P-PDGD include power systems \cite{Changhong:Design, Li:Connecting, mallada2017optimal, wang2019unified}, wireless communication \cite{chiang2007layering}, distributed optimization \cite{Yi:Distributed} and seeking the Nash Equilibrium in game \cite{gharesifard2013distributed}.

Despite its wide applications, general theoretical studies of PDGD and P-PDGD are focused on its asymptotic stability analysis \cite{Feijer:Stability,Cherukuri:Asymptotic,cherukuri2018role,holding2014convergence} and exponential stability analysis \cite{cortes2019distributed,niederlander2016exponentially,dhingra2018proximal,qu2018exponential, tang2019semi,chen2019exponential, liang2019exponential,bansode2019exponential}.
In the existing literature, the global asymptotic stability of the PDGD under different settings is investigated. In \cite{Feijer:Stability}, the projection is proposed to address the inequality constraints and the PDGD is modeled as a hybrid automaton. This will result in the discontinuity of the right-hand side of primal-dual dynamics and bring difficulties in the convergence proof. Then, \cite{Cherukuri:Asymptotic} improves the convergence proof by using the invariance principle for discontinuous Caratheodory systems, which are further extended in \cite{cherukuri2018role} by using a weaker assumption to show global asymptotic stability for locally strong convex-concave Lagrangian. Extensions are given to subgradient methods in \cite{holding2014convergence}, which constrain the dynamics to a convex domain.

Exponential stability is a desirable property of a dynamic system both theoretically and in practice. In many practical systems like the power grid, it is better to have stronger stability guarantees. In addition, a discrete-time iterative algorithm can be obtained from an exponentially stable dynamics through explicit Euler discretization, which achieves linear convergence for sufficiently small step sizes \cite{stuart1994numerical,stetter1973analysis}.
For PDGD to solve constrained convex optimization, its locally exponential stability can be obtained by investigating spectral bounds of saddle matrices \cite{benzi2005numerical}. Regarding the global exponential stability, in \cite{qu2018exponential}, the augmented PDGD is introduced to solve convex optimization with affine inequality constraints, where the exponential stability is proved. The method is further extended in \cite{tang2019semi} for convex optimization problems with convex inequality constraints, and in \cite{chen2019exponential} for convex optimization problems with partially nonstrongly convex objective functions. In \cite{liang2019exponential}, PDGD is used to solve the distributed optimization problem with nonstrongly convex objectives, where the metrically subregular condition is adopted to prove the exponential stability. In \cite{bansode2019exponential}, a projected dynamics is proposed to solve a convex optimization problem with set and linear inequality constraints, where the exponential stability is proved on a Riemannian manifold.
To solve nonsmooth convex optimization problems, Clark generalized gradients are adopted to replace the regular gradients. In \cite{niederlander2016exponentially,cortes2019distributed}, saddle-point-like dynamics is proposed and proved to be globally exponentially stable with equality constraints. It is further improved in \cite{dhingra2018proximal} to consider affine inequality constraints.

This work studies the exponential stability of P-PDGD to solve optimization problems with strongly convex and non-smooth objective functions. Affine equality and set constraints are included. Compared with existing literature \cite{cortes2019distributed,niederlander2016exponentially,dhingra2018proximal,qu2018exponential, tang2019semi,chen2019exponential, liang2019exponential}, the main difference is that set constraints are considered. This is very important in practice. In many cases, set constraints are hard limits and even need to be satisfied during transient process. To this end, the projection onto the tangent cone of the set is adopted. This is also different from \cite{bansode2019exponential}, where the projection onto the set itself is used. However, the method in \cite{bansode2019exponential} does not apply to optimization problems with nonsmooth objectives. As pointed out in \cite{zeng2018distributed}, the projection of a subdifferential set cannot guarantee the existence of trajectories. This paper is also partially motivated by \cite{li2020exponentially}, which designs an exponentially convergent algorithm for the consensus problem. In this work, the optimization problem is more general and dual variables are explicitly included.

\textit{Notations}: In this paper, use $\mathbb{R}^n$ to denote the $n$-dimensional  Euclidean space. For a column vector $ x\in \mathbb{R}^n$ (matrix $A\in \mathbb{R}^{m\times n}$), $ x^{\mathrm{T}}$($A^{\mathrm{T}}$) denotes its transpose. For vectors $ x, y\in \mathbb{R}^n$, $ x^{\mathrm{T}} y=\left\langle  x, y \right\rangle$ denotes the inner product of $ x, y$. $\left\| x \right\|=\sqrt{ x^{\mathrm{T}} x}$ denotes the Euclidean norm of $ x$. For any real symmetric matrices $P$ and $Q$, $P \succeq Q$ and $Q \preceq P$ mean that $P-Q$ is positive semidefinite.

\section{Preliminaries}\label{Preliminaries}
\subsection{Convex analysis}
Let $\Omega \subset \mathbb{R}^{n}$ be a convex set. Let $ f(x):\Omega\rightarrow\mathbb{R} $ be a locally Lipschitz continuous function and denote its Clarke generalized gradient by $\partial f( x)$ \cite[Page 27]{clarke:optimization}. For a $\mu$-strongly convex function $ f( x):\mathbb{R}^n\rightarrow\mathbb{R} $, we have $ (g_{ x}-g_{ y})^{\mathrm{T}}( x- y)\ge \mu\|x- y\|^{2},\ \forall  x,  y \in \Omega$, for all $g_{ x}\in \partial f( x)$ and $g_{ y}\in \partial f( y)$.

For $x \in \Omega,$ the tangent cone to $\Omega$ at $x$ is defined as
$$
\mathcal{T}_{\Omega}(x) \triangleq\left\{\lim _{k \rightarrow+\infty} \frac{x_{k}-x}{\tau_{k}} | x_{k} \in \Omega, x_{k} \rightarrow x, \tau_{k}>0, \tau_{k} \rightarrow 0\right\}
$$

The normal cone to $\Omega$ at $x$ is defined by
$$\mathcal{N}_\Omega(x)=\{ v|\left\langle  v,  y- x\right\rangle\le 0, \forall  y\in \Omega\}$$

By \cite{brogliato2006equivalence}, the tangent cone is the polar of the normal cone, which implies
$$ T_\Omega(x) = \{y\in R^n | \langle s,y\rangle  \le 0, \forall s \in N_\Omega (x) \} $$

Define the projection of $x$ onto $\Omega$ as
\begin{eqnarray}
\label{def_projection}
\mathcal{P}_{\Omega}( x)=\arg \min\nolimits_{ y\in \Omega}\left\| x- y \right\|
\end{eqnarray}
We have
\begin{align}
	x-\mathcal{P}_{\Omega}(x) \in \mathcal{N}_{\Omega}(x)
\end{align}

By \cite{brogliato2006equivalence}, the projection onto $ \mathcal{T}_{\Omega}(x) $ is computed by
\begin{align}\label{Compute_tangent}
	P_{T_{\Omega}(x)}(y)&=\lim _{\delta \rightarrow 0} \frac{P_{\Omega}(x+\delta y)-x}{\delta}\nonumber\\
	&=y-\varpi z^{*}
\end{align}
where $\varpi=\max \left\{0, \langle y,z^{*}\rangle\right\},$ and $z^{*}=\arg\max\nolimits_{\|z\|=1, z \in \mathcal{N}_{\Omega}(x)}\langle y, z\rangle$.

\subsection{Differential inclusion}
Following \cite{aubin2012differential}, a differential inclusion is given by
\begin{align}\label{Differential_inclusion}
	\dot{x}(t) \in \mathcal{F}(x(t)), x(0)=x_{0}, t \geq 0
\end{align}
where $\mathcal{F}$ is a set-valued map from points in $\mathbb{R}^{n}$ to a nonempty, compact, convex subset of $\mathbb{R}^{n}$. For each $x \in \mathbb{R}^{n},$ system \eqref{Differential_inclusion} specifies a set of possible evolutions instead of a single one. A solution of \eqref{Differential_inclusion} defined on $[0, \tau] \subset[0, \infty)$ is an absolutely continuous function $x:[0, \tau] \rightarrow \mathbb{R}^{n}$ such that \eqref{Differential_inclusion} holds for almost all $t \in[0, \tau]$ for $\tau>0$. The solution $t \mapsto x(t)$ to \eqref{Differential_inclusion} is a right maximal solution if it cannot be extended in time. Suppose that all the right maximal solutions to \eqref{Differential_inclusion} exists on $[0,+\infty)$. If $0_{m} \in \mathcal{F}\left(x^*\right)$, then $x^*$ is an equilibrium point of \eqref{Differential_inclusion}.

By \cite[Proposition 2 in p. 266, and Theorem in p. 267]{aubin2012differential}, we have
\begin{lemma}\label{Equivalence}
	Let $\Omega$ be a closed convex subset of $\mathbb{R}^{n}$, and $\mathcal{F}$ be a map with non-empty compact value from $\Omega$ to $\mathbb{R}^{n}$. Consider two differential inclusions given by
	\begin{subequations}
		\setlength{\abovedisplayskip}{4pt}
		\setlength{\belowdisplayskip}{4pt}
		\begin{align}\label{Normal_cone_pro}
		&\dot{x}(t) \in \mathcal{F}(x(t))-\mathcal{N}_{\Omega}(x(t)), x(0)=x_{0}\\
		\label{Tangent_cone_pro}
		&\dot{x}(t) \in P_{\mathcal{T}_{\Omega}}[\mathcal{F}(x(t))], \quad x(0)=x_{0}
		\end{align}
	The trajectory x(t) is a solution of \eqref{Normal_cone_pro} if and only if it is a solution of \eqref{Tangent_cone_pro}. Moreover, if $\mathcal{F}$ is also upper semi-continuous and bounded, there exists a solution to dynamics \eqref{Normal_cone_pro}.
	\end{subequations}
\end{lemma}

Let $V: \mathbb{R}^{n} \rightarrow \mathbb{R}$ be a locally Lipschitz continuous function and $\partial V$ be the Clarke generalized gradient of $V(x)$ at $x$. The set-valued Lie derivative $\mathcal{L}_{\mathcal{F}} V$ of $V$ with respect to \eqref{Differential_inclusion} is defined as $\mathcal{L}_{\mathcal{F}} V(x) \triangleq\left\{a \in \mathbb{R}: a=p^{\mathrm T} v, p \in \partial V(x), v \in \mathcal{F}(x)\right\}$.

\section{Problem Formulation and Algorithm Design}\label{Problem_Formulation}
\subsection{Optimization problem}
The optimization problem is
\begin{subequations}
	\setlength{\abovedisplayskip}{4pt}
	\setlength{\belowdisplayskip}{4pt}
	\label{OLC}
	\begin{align}
	\min\limits_{x\in X,y\in\Omega}\quad& f({x})+h(y)
	\label{OLC_1}
	\\
	\text{s.t.} \quad	&A{x}+By=C
	\end{align}
\end{subequations}
where $X\subset\mathbb{R}^n$, $ \Omega\subset\mathbb{R}^m $ are compact convex sets. $A\in\mathbb{R}^{p\times n}, B\in \mathbb{R}^{p\times m}, C\in\mathbb {R}^{p}$ are constant matrices.
We make following assumptions.
\begin{assumption}\label{Assumption_Slater}
	The Slater's condition  \cite[Chapter 5.2.3]{boyd2004convex} of \eqref{OLC} holds.
\end{assumption}
\begin{assumption}\label{Assumption_f}
	For some $ \alpha >0  $, $ f(x) $ is $\alpha$-strongly convex and twice differentiable, i.e., $ \nabla^{2} f(x) \succeq \alpha I$.
\end{assumption}
\begin{assumption}\label{Assumption_g}
	The function $ h(y) $ is Lipschitz continuous, and $\beta$-strongly convex on $\Omega$ for some $ \beta>0 $, that is,
	$
	\langle y_1-y_2, g_{h}(y_1)-g_{h}(y_2)\rangle \geq \beta\|y_1-y_2\|^{2}, \ \forall y_1, y_2 \in \Omega
	$,
	where $g_{h}(y_1) \in \partial h(y_1)$ and $g_{h}(y_2) \in \partial h(y_2)$.
\end{assumption}
It should be noted that $h(y)$ could be nonsmooth.
\begin{assumption}\label{Assumption_A}
	The  matrix $A$ has full row rank and $\kappa_{1}I \preceq A A^\mathrm{T}$ for some $\kappa_{1}>0$.
\end{assumption}

\begin{remark}
	Many practical problems have the same type as problem \eqref{OLC}. For example,  distributed frequency control problem in power systems  \cite{Li:Connecting,Distributed_I:Wang,Distributed_II:Wang}, and distributed voltage control in distribution networks \cite{liu2018hybrid, wang2019asynchronous} can be generalized as \eqref{OLC}. Thus, the result in this paper can be applied to many problems in practice.
\end{remark}

\subsection{Algorithm Design}
The Lagrangian of \eqref{OLC} is
\begin{align}\label{Lagrangian}
	\mathop{L(x, y, \lambda)}\limits_{x\in X, y\in\Omega} & = f({x})+h(y)+\lambda^\mathrm{T}(A x+B y-C)
\end{align}
where $ \lambda\in\mathbb{R}^p $ is the Lagrangian multiplier vector.
The partial primal-dual gradient algorithm is
\begin{subequations}
	\setlength{\abovedisplayskip}{4pt}
	\setlength{\belowdisplayskip}{4pt}
	\label{Controller}
	\begin{align}
	\label{Controller1}
	x&=\arg \min _{x\in X}\left\{f(x)+\lambda^\mathrm{T} A x\right\}  \\
	\label{Controller2}
	\dot y &\in P_{\mathcal{T}_{\Omega}\left(y \right)}\left(-\partial h\left(y \right)-B^\mathrm{T}\lambda\right)\\
	\dot\lambda&=Ax+By-C
	\end{align}
\end{subequations}

Define functions
\begin{align}
	\varphi(\lambda) & \triangleq \min _{x\in X}\left\{f(x)+\lambda^\mathrm{T} A x\right\} \\
	\label{Partial_Lagrangian}
	\mathop	{\hat{L}(y, \lambda)}\limits_{y\in\Omega} & \triangleq \varphi(\lambda)+h(y)+\lambda^\mathrm{T}(B y-C)
\end{align}

Let $ H:=\nabla^{2} f(x) $, and its maximal eigenvalue be $ \alpha_{m} $.	Its inverse $ H^{-1} $ is also positive definite, and the minimal eigenvalue is $ \frac{1}{\alpha_{m}} $.
Then, we have the following result demonstrating properties of $ \varphi(\lambda) $ and $ \hat{L}(y, \lambda) $.
\begin{lemma}\label{Strongly_convex}
	Suppose Assumptions \ref{Assumption_f}, \ref{Assumption_A} hold. The function $\varphi(\lambda)$ is continuously differentiable and	$ \frac{\kappa_1}{\alpha_{m}} $-strongly concave. As a consequence, $ \hat{L}(y, \lambda) $ is strongly concave on $\lambda$.
\end{lemma}
\begin{proof}
	Because $f(x)$ is differentiable, $\varphi(\lambda)$ is continuously differentiable by \cite[Proposition 6.1.1]{Bertsekas2008Nonlinear}. The gradient of $\varphi(\lambda)$ is $ \nabla \varphi(\lambda)=Ax $.
	From \cite[Equation (6.9)]{Bertsekas2008Nonlinear}, the Hessian of $\varphi(\lambda)$ is
	\begin{align}\label{Hessian}
		\nabla^{2} \varphi(\lambda)=-A H^{-1} A^\mathrm{T}
	\end{align}
	For any $z\in\mathbb{R}^p$, we have
	\begin{align}
		-z^\mathrm{T}\nabla^{2} \varphi(\lambda)z&=z^\mathrm{T}A H^{-1} A^\mathrm{T}z\nonumber\\
		&\ge \frac{1}{\alpha_{m}}z^\mathrm{T}A A^\mathrm{T}z \nonumber\\
		&\ge \frac{\kappa_1}{\alpha_{m}}z^\mathrm{T} z
	\end{align}
	where the second inequality is due to Assumption \ref{Assumption_A}. This implies that $ -\varphi(\lambda) $ is $ \frac{\kappa_1}{\alpha_{m}} $-strongly convex, i.e., $ \varphi(\lambda) $ is $ \frac{\kappa_1}{\alpha_{m}} $-strongly concave. This completes the proof.
\end{proof}
From Lemma \ref{Strongly_convex}, taking any $ \lambda_1, \lambda_2 \in \mathbb{R}^p $, we have
\begin{align}\label{Strongly_convex_q}
	\left\langle \lambda_1-\lambda_2, \nabla \varphi(\lambda_1)-\nabla \varphi(\lambda_2)\right\rangle \leq -\frac{\kappa_1}{\alpha_{m}}\|\lambda_1-\lambda_2\|^{2}
\end{align}
Then, the algorithm \eqref{Controller} is rewritten as

\noindent \textbf{P-PDGD:}
\begin{subequations}
	\setlength{\abovedisplayskip}{4pt}
	\setlength{\belowdisplayskip}{4pt}
	\label{Controller_PPD}
	\begin{align}
	\label{Controller_PPD1}
	x&=\arg \min _{x\in X}\left\{f(x)+\lambda^\mathrm{T} A x\right\}  \\
	\label{Controller_PPD2}
	\dot y &\in P_{\mathcal{T}_{\Omega}\left(y \right)}\left(-\partial h\left(y \right)-B^\mathrm{T}\lambda\right)\\
	\dot\lambda&=\nabla \varphi(\lambda)+By-C
	\end{align}
\end{subequations}
In the rest of the paper, we will study the properties of the algorithm \eqref{Controller_PPD}.
\begin{remark}
	The algorithm \eqref{Controller_PPD} has some significant improvement from the initial version in \cite{Li:Connecting}. First, it considers the nonsmooth objective functions, where the Clark generalized gradient is utilized. Second, the set constraint is included, where the projection onto the tangent cone is adopted. Thus, the proposed algorithm is more general.

	In \eqref{Controller_PPD2}, the projection onto $ \mathcal{T}_{\Omega}(y) $ is applied. The reason is that the projection from	$\partial h\left(y\right)$ onto $\Omega$ may be a nonconvex differential inclusion. As a result, the existence of trajectories of \eqref{Controller_PPD} is not guaranteed, which sets a difficult barrier for convergence analysis.
\end{remark}

\section{Optimality and Exponential Convergence}
In this section, we first investigate the optimality of the equilibrium point of \eqref{Controller_PPD}. Then, we prove that the trajectory of dynamics \eqref{Controller_PPD} converges to the equilibrium point exponentially.
\subsection{Optimality}
Before proving the optimality, we introduce the existence of solutions to \eqref{Controller_PPD}.
\begin{lemma}\label{bounded}
	Suppose Assumptions \ref{Assumption_f}, \ref{Assumption_g}, and \ref{Assumption_A} hold. Consider dynamics \eqref{Controller_PPD}.
	\begin{enumerate}
		\item If ${y}(0) \in \Omega$, then $y(t) \in \Omega$ for all $t \geq 0$.
		\item If $\lambda(0)$ is bounded, then $\lambda(t) $ is bounded for all $t \geq 0$.
		\item There exists a solution to \eqref{Controller_PPD}.
	\end{enumerate}
\end{lemma}
\begin{proof}
	For 1), define a function
	\begin{align}
	E\left(y(t)\right)=\frac{1}{2}\left\|y(t)-P_{\Omega}\left(y(t)\right)\right\|^{2}
	\end{align}
	Its gradient is \cite[Theorem 1.5.5]{facchinei2003finite}
	\begin{align}
	\nabla E\left(y(t)\right)=y(t)-P_{\Omega}\left(y(t)\right)
	\end{align}
	The time derivative of $ E\left(y(t)\right) $ is
	\begin{align}
	\dot E\left(y(t)\right)=\left\langle\nabla E\left(y(t)\right),\dot{y}(t)\right\rangle=\left\langle y(t)-P_{\Omega}\left(y(t)\right),\dot{y}(t)\right\rangle
	\end{align}
	Because $ y(t)-P_{\Omega}(y(t)) \in \mathcal{N}_{\Omega}(y(t)) $ and $ \dot{y}(t) \in \mathcal{T}_{\Omega}\left(y(t)\right) $, we have $ \dot E\left(y(t)\right)\le0 $. This implies that $ E\left(y(t)\right) $ is non-increasing. From $ E\left(y(t)\right)\ge0,\ \forall t\ge0 $ and $ E\left(y(0)\right)=0 $, we have $ E\left(y(t)\right)=0,\ \forall t\ge0  $, i.e., $ y(t)=P_{\Omega}\left(y(t)\right), \ \forall t\ge0 $.

	For 2), define a function
	\begin{align}
		\tilde E\left(\lambda(t)\right)=\frac{1}{2}\left\|\lambda(t)\right\|^{2}
	\end{align}
	The time derivative of $\tilde E\left(\lambda(t)\right)$ along \eqref{Controller_PPD} is
	\begin{align}
		\dot{\tilde E}(\lambda) &=\lambda^{\rm T}(\nabla \varphi(\lambda)+By-C)\nonumber\\
		&=\lambda^{\rm T}\nabla \varphi(\lambda)+\lambda^{\rm T}(By-C)\nonumber\\
		&\le -\frac{\kappa_1}{\alpha_{m}}\|\lambda\|^{2} + \lambda^{\rm T}\nabla \varphi(0)+\lambda^{\rm T}(By-C)\nonumber\\
		&\le -\frac{\kappa_1}{\alpha_{m}}\|\lambda\|^{2} + a_\lambda\|\lambda\| \nonumber\\
		&= -\frac{2\kappa_1}{\alpha_{m}} \tilde E\left(\lambda\right)+a_\lambda\sqrt{2\tilde E\left(\lambda\right)}
	\end{align}
	where $ a_\lambda=\max_{y\in\Omega}\left(\|\nabla \varphi(0)\|+\|By-C\|\right) $, and the first inequality is due to $ \left\langle \lambda, \nabla \varphi(\lambda)\right\rangle \leq  -\frac{\kappa_1}{\alpha_{m}}\|\lambda\|^{2} + \lambda^{\rm T}\nabla \varphi(0)$. The second inequality is due to the boundedness of $y(t)$.
	Then, we have $ \|\tilde{E}(\lambda(t))\|\le \max\left\{\tilde{E}(\lambda(0)),\frac{{{\alpha ^2_m}{a^2_\lambda }}}{{2{\kappa^2 _1}}}\right\} $. Thus, $\tilde{E}(\lambda(t)), t \geq 0$ is bounded, so is $\lambda(t), t \geq 0$.

	Because $h(y)$ is Lipschitz, $ \partial h\left(y \right) $ is nonempty, compact, convex, and upper semicontinuous \cite[Proposition 6]{cortes2008discontinuous}.
	From 1) and 2), we have $(y(t),\lambda(t))$ is bounded. Then, by Lemma \ref{Equivalence}, we can prove 3).

	This completes the proof.
\end{proof}

\begin{remark}
	Lemma \ref{bounded} shows that the trajectory of $y(t)$ will stay in $\Omega$ as long as $y(0)\in \Omega$. This is very important in practice besides paving the way for convergence proof. Many domain constraints are hard limits, which cannot be violated even in the transient process \cite{trip2019optimal}. For example, the power generation limits of generators and capacity limits of inverters cannot be violated physically \cite{Distributed_I:Wang,Distributed_II:Wang}. The voltage limits should not be violated, otherwise, it is dangerous for system operators \cite{wang2019unified}. Thus, the results in the paper can be applied to many practical problems.
\end{remark}

Let $ (x^*, y^*, \lambda^* ) $ be an equilibrium of \eqref{Controller_PPD}. Then
\begin{subequations}\label{Equilibrium}
	\begin{align}
	\label{Equilibrium1}
	x^*&=\arg \min _{x\in X}\left\{f(x)+(\lambda^*)^\mathrm{T} A x\right\}  \\
	\label{Equilibrium2}
	0 &\in P_{\mathcal{T}_{\Omega}\left(y^* \right)}\left(-\partial h\left(y^* \right)-B^\mathrm{T}\lambda^*\right)\\
	\label{Equilibrium3}
	0&=Ax^*+By^*-C
	\end{align}
\end{subequations}

\begin{theorem}\label{Optimality}
	Suppose Assumptions \ref{Assumption_Slater}, \ref{Assumption_f}, \ref{Assumption_g} hold. The point $ (x^*, y^*, \lambda^* ) $ satisfies \eqref{Equilibrium}, if and only if it is the primal-dual optimal solution to \eqref{OLC} and its dual problem.
\end{theorem}
\begin{proof}
	According to the Karush-Kuhn-Tucker (KKT) optimal conditions \cite[Theorem 3.25]{ruszczynski2006nonlinear}, the primal-dual optimal solution should satisfy
	\begin{subequations}\label{KKT}
		\begin{align}
		\label{KKT1}
		0&\in\nabla f(x^*)+A^\mathrm{T}\lambda^*+\mathcal{N}_{X}(x^*) \\
		\label{KKT2}
		0 &\in \partial h\left(y^* \right)+B^\mathrm{T}\lambda^*+\mathcal{N}_{\Omega}(y^*)\\
		\label{KKT3}
		0&=Ax^*+By^*-C
		\end{align}
	\end{subequations}
	Compare \eqref{KKT} with \eqref{Equilibrium}, and we know \eqref{KKT1} is equivalent to \eqref{Equilibrium1}. By Lemma \ref{Equivalence}, we have \eqref{KKT2} is equivalent to \eqref{Equilibrium2}. Thus, \eqref{KKT} is equivalent to \eqref{Equilibrium}. Because the optimization problem \eqref{OLC} is convex and with strongly convex objective functions, $ (x^*, y^*, \lambda^* ) $ is the primal-dual optimal solution to \eqref{OLC} and its dual problem. This completes the proof.
\end{proof}

\subsection{Exponential convergence}

In this subsection, we analyze the convergence rate for algorithm dynamics \eqref{Controller_PPD}.

\begin{theorem}\label{Convergence}
	Suppose Assumptions \ref{Assumption_Slater}, \ref{Assumption_f}, \ref{Assumption_g} hold.  The solution algorithm dynamics \eqref{Controller_PPD} converges to its equilibrium point $ (x^*, y^*, \lambda^* ) $ exponentially.
\end{theorem}

\begin{proof}
Define the Lyapunov function candidate
\begin{align}
	V(y,\lambda)=\frac{1}{2}\|y-y^{*}\|^2+ \frac{1}{2}\|\lambda-\lambda^{*}\|^2
\end{align}

The time derivative of $ V(y,\lambda) $ is
\begin{align}
	&\mathcal{L}_{\mathcal{F}} V(y, \lambda)=\left\{a \in \mathbb{R}: \right.\nonumber\\
	&\qquad  a=\nabla_{y} V(y, \lambda)^{\mathrm{T}} P_{\mathcal{T}_{\Omega}\left(y(t)\right)}\left(-\partial h\left(y(t)\right)-B^\mathrm{T}\lambda\right) \nonumber\\
	&\qquad \left. +\nabla_{\lambda} V(y, \lambda)^{\mathrm{T}}(\nabla \varphi(\lambda)+By-C)\right\}
\end{align}

Suppose $a \in \mathcal{L}_{\mathcal{F}} V(y, \lambda)$. There is $\eta(y) \in \partial h(y)$ such that
\begin{align}
	a=&\left(y-y^{*}\right)^\mathrm{T}P_{\mathcal{T}_{\Omega}\left(y(t)\right)}\left(-\eta(y)-B^\mathrm{T}\lambda\right)\nonumber \\
	&+\left(\lambda-\lambda^{*}\right)^\mathrm{T}(\nabla \varphi(\lambda)+By-C)
\end{align}

From \eqref{Normal_cone_pro}, we have
\begin{align}
	-\eta(y)-B^\mathrm{T}\lambda-\dot{y} \in \mathcal{N}_{\Omega}(y)
\end{align}
From the definition of $ \mathcal{N}_{\Omega}(y(t)) $ and the fact that $ y^* \in \Omega $, we have
\begin{align}
	&\left\langle y^{*}-y,-\eta(y)-B^\mathrm{T}\lambda-\dot{y}\right\rangle \le 0\nonumber\\
	&\left\langle y-y^{*},\dot{y}\right\rangle \le \left\langle y-y^{*},-\eta(y)-B^\mathrm{T}\lambda\right\rangle
\end{align}

Then
\begin{align}\label{Inequality3}
	a\le&\left(y-y^{*}\right)^\mathrm{T}\left(-\eta(y)-B^\mathrm{T}\lambda\right)\nonumber \\
	&+\left(\lambda-\lambda^{*}\right)^{{T}}(\nabla \varphi(\lambda)+By-C)
\end{align}
From \eqref{KKT}, the definition of $ \mathcal{N}_\Omega(x) $, and the fact that $y(t)\in\Omega, \forall t$, at the equilibrium, we have
\begin{subequations}
	\setlength{\abovedisplayskip}{4pt}
	\setlength{\belowdisplayskip}{4pt}
	\label{Eqlibr}
	\begin{align}
	\label{Eqlibr1}
	&\left\langle-\eta\left(y^*\right)-B^\mathrm{T}\lambda^*,y-y^* \right\rangle\le 0\\
	\label{Eqlibr2}
	&0=\nabla \varphi(\lambda^*)+By^*-C
	\end{align}
\end{subequations}

Combine \eqref{Eqlibr} with \eqref{Inequality3}, and we have
\begin{align}
	a&\le\left(y-y^{*}\right)^\mathrm{T}\left(-(\eta(y)-\eta\left(y^*\right))-B^\mathrm{T}(\lambda-\lambda^*)\right)\nonumber \\
	&\quad+\left(\lambda-\lambda^{*}\right)^{{T}}(\nabla \varphi(\lambda)-\nabla \varphi(\lambda^*)+By-By^*)\nonumber \\
	&\le -\left(y-y^{*}\right)^\mathrm{T}\left( \eta(y)-\eta\left(y^*\right )\right)\nonumber\\
	&\quad+\left(\lambda-\lambda^{*}\right)^{{T}}(\nabla \varphi(\lambda)-\nabla \varphi(\lambda^*)) \nonumber\\
	&\le -\beta\|y-y^{*}\|^{2}-\frac{\kappa_1}{\alpha_{m}}\|\lambda-\lambda^{*}\|^{2}\nonumber\\
	&\le -\gamma V
\end{align}
where $\gamma=\min\left\{2\beta,\ 2\frac{\kappa_1}{\alpha_{m}}\right\}$. As a result, $V(t) \leq V(0) e^{-\gamma t}$, and we have $ \left\| \left({y\left( t \right) - {y^*}}, {\lambda \left( t \right) - {\lambda ^*}}\right) \right\|\le \sqrt{2V(0)} e^{-\frac{\gamma}{2} t} $ converges to $(y^*,\lambda^*)$ exponentially, and convergence rate is no less than $ \frac{\gamma}{2} $. This completes the proof.
\end{proof}
\begin{remark}[Decaying rate]\label{Decaying_rate}
	Give some $\tau>0$, the P-PDGD can be written as
	\begin{subequations}
		\setlength{\abovedisplayskip}{4pt}
		\setlength{\belowdisplayskip}{4pt}
		\label{Controller_PPD_tau}
		\begin{align}
		\label{Controller_PPD1_tau}
		x&=\arg \min _{x\in X}\left\{f(x)+\lambda^\mathrm{T} A x\right\}  \\
		\label{Controller_PPD2_tau}
		\dot y &\in \tau P_{\mathcal{T}_{\Omega}\left(y \right)}\left(-\partial h\left(y \right)-B^\mathrm{T}\lambda\right)\\
		\dot\lambda&=\tau(\nabla \varphi(\lambda)+By-C)
		\end{align}
	\end{subequations}
	Then, the Lyapunov function is
	\begin{align}
		V_2(y,\lambda)=\frac{1}{2\tau}\|y-y^{*}\|^2+ \frac{1}{2\tau}\|\lambda-\lambda^{*}\|^2
	\end{align}

	Follow the similar analysis in Theorem \ref{Convergence}, and we have
	\begin{align}
	a&\le -\frac{1}{2}\gamma \left\| \left({y\left( t \right) - {y^*}}, {\lambda \left( t \right) - {\lambda ^*}}\right) \right\|^2 \nonumber\\
	&\le -\gamma\tau V_2(y,\lambda)
	\end{align}
	As a result, $V_2(t) \leq V_2(0) e^{-\gamma\tau t}$, and we have $ \left\| \left({y\left( t \right) - {y^*}}, {\lambda \left( t \right) - {\lambda ^*}}\right) \right\|\le \sqrt{2\tau V_2(0)} e^{-\frac{\gamma\tau}{2}t} $. Thus, the decaying rate bound can be regulated by $\tau$. If $\tau=1$, it will be same as that in Theorem \ref{Convergence}.
\end{remark}

\section{Illustrative Examples}
We consider a convex problem that models the optimal voltage control in distribution networks. The problem is formulated as
\begin{subequations}\label{eq_opt2}
	\setlength{\abovedisplayskip}{4pt}
	\setlength{\belowdisplayskip}{4pt}
	\begin{align}
	\min\limits_{{U}, {q}\in \mathbb{R}^n} &\quad   f=\frac{a}{2}\|{U}-\textbf{1}\|^2 + \sum\nolimits_{j=1}^n h_j(q_j)
	\label{eq_opt2a}
	\\
	\text{s.t.}
	&  \quad	{B}{U}={q}+C
	\label{eq_opt2b} \\
	&  \quad	\underline{q}\le {q}\le \overline{q} \label{eq_opt2f}
	\end{align}
\end{subequations}
where $ U $ is the voltage, $q$ is the reactive power. $ {B}{U}={q}+C $ is the linear model derived from the DistFlow equations \cite{baran1989optimal}. $ \underline{q}, \overline{q} $ are lower and upper bound of $q$. $a$ is a real positive constant. The first part of objective function, $ \frac{a}{2}\|{U}-\textbf{1}\|^2 $, is the voltage difference, where $ \textbf{1} $ is the nominal voltage. The second part, $ h_j(q_j) $, is the regulation cost of reactive power, which is strongly convex and nonsmooth, defined by
\begin{equation}
	\setlength{\abovedisplayskip}{4pt}
	\setlength{\belowdisplayskip}{4pt}
	\begin{aligned}
	h_j(q_j)=\left\{ \begin{array}{l}
	q_j^2-0.02,\quad\ \ q_j\le -0.2\\
	\frac{1}{2}q_j^2 ,\quad \quad \qquad -0.2<q_j\le  0.2\\
	q_j^2-0.02,\quad\ \ 0.2<q_j
	\end{array} \right.
	\end{aligned}
\end{equation}
An 8-bus feeder is utilized as the test system, the detailed description is given in \cite{wang2019asynchronous}. The bus 0 is the substation with voltage as $1$ and has no generator. Other have reactive power regulation capability with  $\overline{{q}}=-\underline{{q}}=[80, 80, 88, 80, 104, 80, 96]$kVar. We set $a=8$, $ C=(1.011,-0.009,-0.1,0.14,-0.26,-0.019,-0.06)^{\rm T} $. The minimal eigenvalue of $B$ is $0.1165$. Moreover, we have $ \beta = 1, \kappa_1 = 0.1165^2 = 0.01357, \alpha_m = a = 8 $. Therefore $ \gamma = \min\{2\beta, 2\kappa_1/\alpha_m\} = \min\{2, 2\times0.01357/8\} = 0.00339 $.

Define $ \Omega\triangleq \{q:\underline{q}\le {q}\le \overline{q}\} $, and then the P-PDGD for \eqref{eq_opt2} is
\begin{subequations}
	\setlength{\abovedisplayskip}{4pt}
	\setlength{\belowdisplayskip}{4pt}
	\label{Controller_V}
	\begin{align}
	\label{Controller_V1}
	U&=\arg \min _{U}\left\{\frac{a}{2}\|U-\textbf{1}\|^2+\lambda^\mathrm{T}BU\right\}=-\frac{1}{a}B^\mathrm{T}\lambda+\textbf{1}  \\
	\label{Controller_V2}
	\dot q &\in P_{\mathcal{T}_{\Omega}\left(q \right)}\left(-\partial h\left(q \right)+\lambda\right)\\
	\dot\lambda&=-\frac{1}{a}B B^{\rm T}\lambda-q+B\times\textbf{1}-C
	\end{align}
\end{subequations}
The method in \eqref{Compute_tangent} is utilized to compute the projection onto $ {\mathcal{T}_{\Omega}\left(q \right)} $. The simulations are implemented on Matlab R2013b, and the function ``ode23tb" is adopted to solve \eqref{Controller_V}.
The equilibrium point of \eqref{Controller_V} is
$$ q^*=(0.031, 0.031, 0.042, 0.075, 0.104, 0.035, 0.044)^{\rm T}, $$
$$U^*=(0.992, 1.003, 1.004, 1.031, 0.959, 1.004, 0.997)^{\rm T}.$$

The dynamic performance of \eqref{Controller_V} are given in Fig.\ref{Reactive_power} and Fig.\ref{Exponential_figure}. In Fig.\ref{Reactive_power}, the dynamics of reactive power are illustrated, which shows that $q_i(t), \forall i$ is within its lower and upper limits. This validates Lemma \ref{bounded}.
\begin{figure}[t]
	\setlength{\abovecaptionskip}{0pt}
	\centering
	\includegraphics[width=0.45\textwidth]{Reactive_power}
	\caption{Dynamics of reactive power $ q $ at each bus}
	\label{Reactive_power}
\end{figure}

In Fig.\ref{Exponential_figure}, the convergence of $ \left\| \left({q\left( t \right) - {q^*}}, {\lambda \left( t \right) - {\lambda ^*}}\right) \right\| $ is illustrated. The dotted line is the trajectory of $ \sqrt {{{\left\| {q\left( 0 \right) - {q^*}} \right\|}^2} + {{\left\| {\lambda \left( 0 \right) - {\lambda ^*}} \right\|}^2}} e^ { - \gamma t/2}   $, which is the upper bound of exponential convergence obtained from Theorem \ref{Convergence}. The blue line is the trajectory of $ (q(t),\lambda(t)) $. It is shown that $ (q(t),\lambda(t)) $ converges to $ (q^*,\lambda^*) $ rapidly. The convergence speed is much faster than computed upper bound.
\begin{figure}[t]
	\setlength{\abovecaptionskip}{0pt}
	\centering
	\includegraphics[width=0.45\textwidth]{Exponential_figure}
	\caption{Illustration of the relative distances to $ (q^*, \lambda^*) $}
	\label{Exponential_figure}
\end{figure}

We further investigate the decaying rates under different $\tau$, which is illustrated in Fig.\ref{Exponential_different_tau}. With the increasing of $\tau$, the decaying rates also increase, and the exponential convergence always holds. The result is consistent with the analysis in Remark \ref{Decaying_rate}.
\begin{figure}[t]
	\setlength{\abovecaptionskip}{0pt}
	\centering
	\includegraphics[width=0.45\textwidth]{Different_tau}
	\caption{Illustration of the decaying rate with different $\tau$. The dotted lines are trajectories of $ \sqrt{2\tau V_2(0)} e^{-\frac{\gamma\tau}{2}t} $ with different $\tau$. The solid lines are the corresponding trajectories of $ \left\| \left({y\left( t \right) - {y^*}}, {\lambda \left( t \right) - {\lambda ^*}}\right) \right\| $. }
	\label{Exponential_different_tau}
\end{figure}

\section{Conclusion}\label{Conclusion}
This paper investigates the P-PDGD for solving convex optimization with strongly convex and non-smooth objectives. Affine equality and set constraints are considered. We prove the exponential stability of P-PDGD, where bounds on decaying rates are also provided. It is also validated that the algorithm is almost initialization free as long as the initial point satisfies the set constraints.

Our results are promising in many practical problems, such as the frequency and voltage control in power systems, which can provide a stronger stability guarantee. However, there are still some limitations on the problem form. In the future, we will investigate exponentially convergent algorithms for more general optimization problems.


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