Preface to the Special Issue on “Games, Dynamics and Optimization”

Radu Ioan Boţ; Mathias Staudigl

doi:10.1007/s00245-019-09636-4

Boţ, Radu Ioan; Staudigl, Mathias

2020-04-10 00:00:00

Applied Mathematics & Optimization (2020) 81:651–654 https://doi.org/10.1007/s00245-019-09636-4 Preface to the Special Issue on “Games, Dynamics and Optimization” 1 2 Radu Ioan Bot¸ · Mathias Staudigl Published online: 10 December 2019 © The Author(s) 2019 From March 13 to March 15, 2018, a 3-day interdisciplinary Workshop on mathe- matical game theory, continuous optimization and dynamical system was held at the University of Vienna. This conference was the ﬁrst edition of a a series of worhshops which is now already in its third edition, and it became a central element of the scientiﬁc work plan of the COST Action CA16228 “European Network for Game Theory”. The workshop was a jointly organized event together with the Vienna Center for Operations Research (VCOR) and the Department of Mathematics of the University of Vienna. The motivation for the creation of this workshop was that recently there are two inter- esting evolutions happening in disciplines which used to be quite close together but somehow have seen a diverging evolution: dynamical systems derived from games are a well-established research ﬁeld with many important applications in mathematical biology, physics, engineering and economics. Recently, dynamical systems have also witnessed important applications in optimization and machine learning. Here, the arse- nal of mathematical tools for studying continuous time systems have been successfully applied in shedding new light on the asymptotic properties of optimization algorithms, and to help practitioners and researchers in designing new algorithms. It is an interest- ing, and very notable, coincidence that both ﬁelds have one founding father for their evolution: the Lotka–Volterra system. In evolutionary game theory, Lotka–Volterra systems have been studied since the 1970s and led to the creation of the celebrated replicator dynamics. In optimization, the large class of Barrier-dynamics contains as special case Lotka–Volterra systems, and the connection between Riemannian gradient ﬂows, evolutionary game theory, and optimization is now well-documented. With all these in mind, the aim of the “Games, Dynamics and Optimization” (GDO) workshop series is to explore the analogies between dynamical systems in games and dynamical B Radu Ioan Bo¸t radu.bot@univie.ac.at Mathias Staudigl m.staudigl@maastrichtuniversity.nl Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria Department of Quantitative Economics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands 123 652 Applied Mathematics & Optimization (2020) 81:651–654 systems in optimization, to boost their evolution, popularize their use, and deepen and extend their theoretical foundations. With this purpose in mind, a special issue in the journal Applied Mathematics and Optimization has been announced, where rep- resentative papers on the theme of GDO are collected. This special issue consists of ten papers, all of which touching key topics in mathematical game theory, dynamical systems and nonlinear analysis. In what follows, we group the contributions according to these three ﬁelds. Papers on Operator Splitting – The work by Adly and Bourdin introduces a new decomposition formula for the resolvent operators of the sum of two monotone operators, a computational routine usually involved in the numerical treatment of monotone inclusions and PDEs via operator splitting algorithms. – Dynamical systems inspired by operator splitting techniques is a very active ﬁeld of research within optimization theory. The continuous time analysis often sheds new light on the asymptotic behavior of numerical algorithms, via a careful numerical discretization. In particular, continuous time analysis admits a direct analysis via Lyapunov techniques, and their structure gives us insights into the design of new algorithms. The most successful story in this area is the recent beautiful connection between Nesterov’s accelerated algorithm and second-order dynamical systems. Along these lines, the paper by Attouch and Cabot introduces a fairly broad fam- ily of iterative forward-backward schemes, which can be related to second-order dynamical systems by a careful explicit numerical discretization. The choice of the step parameters in the discretized dynamics is a key issue to guarantee a good performance of the resulting scheme. This paper derives explicit modeling recom- mendations on step-sizes, proximal parameters and inertial parameters, to ensure stability of the discrete-time dynamical system. An application to zero-sum games sheds light on the connection between operator splitting and game theory. – The paper by Csetnek, Malitsky and Tam proposes a new algorithm for ﬁnding a zero of the sum of two maximally monotone operators, where one is assumed to be single-valued and Lipschitz continuous, which arises from a non-standard discretization of a continuous dynamical system associated with the celebrated Douglas–Rachford algorithm. This is a truly novel results in this ﬁeld, and interest- ing connections between discrete and continuous time dynamical systems become apparent. Papers on Partial Differential Equations – Differential games describe a competitive optimal control problem between play- ers who jointly control an ordinary differential equation in order to steer the system towards certain states in which each player is guaranteed a certain payoff. In the setting of non-anticipative strategies determining the value of such a game when players’ action spaces are constrained is a delicate issue. An even more challeng- ing setting is studied in the paper by Bettiol, Quincampoix and Vinter, in which 123 Applied Mathematics & Optimization (2020) 81:651–654 653 players controls are even affected by coupled constraints. Such Generalized Nash games are classical in economics and operations research, and play an increas- ingly important role in PDE constrained optimization as well. Under a viability condition, the authors prove the existence of the game value in such a complex strategic environment. – Continuing the line of challenging optimal control problems, the paper by Breiten, Kunisch and Pfeiffer, investigates stabilization of the 2-dimensional Navier–Stokes equation by classical numerical optimal control tools, namely the value function approximation. Key to this results is an interesting discovery related to the local continuity properties of the value function around the steady state. This allows the authors to perform a delicate linearization, which they use to produce an approxi- mate feedback control. – The limit behavior of singular perturbations problems of biLaplacian type is inves- tigated in the paper by Dipierro, Karakhanyan and Valdinoci. This study provides new solutions to free boundary problems, and contains an interesting search game in continuous time and space. – Gradient ﬂows are the paradigm for dissipative evolution equations, which are also of key importance in games and optimization. One standard numerical dis- cretization scheme for gradient ﬂows is the classical Euler scheme. To preserve the dissipation dynamics along a time discretization, the paper by Jüngel, Stefanelli and Trussardi investigate two modiﬁcations of the discretized Euler scheme, and demonstrates important improvements with respect to the latter. – The paper by Rodrigues and Santos aims to extend classical results on varia- tional and quasi-variational inequalities over convex sets deﬁned through gradient constraints, by replacing the gradient with a fractional gradient. The continuous dependence of the solution on the data is proved and generalized Lagrange multi- pliers rules are derived. Papers on Optimization – Finite sum models are the canonical model in machine learning and empirical risk minimization. In the paper by Dinh, Goberna and Volle, a ﬁnite sum model is investigated in an abstract setting, and new optimality conditions are derived. These abstract results are then used to derive interesting new results for the best- approximation problem. – In large scale applications, the use of randomized algorithms is a standard tool to improve the iteration complexity. The paper by Nedic and Necoara deals with the minimization of a nonsmooth convex function with respect to a feasible set, which is represented as large intersection of functional constraints. In this chal- lenging setting, a parallel and sequential stochastic subgradient algorithm based on minibatch estimators is studied. 123 654 Applied Mathematics & Optimization (2020) 81:651–654 Acknowledgements Open access funding provided by University of Vienna. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna- tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations.

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Preface to the Special Issue on “Games, Dynamics and Optimization”

$Applied Mathematics & Optimization (2020) 81:651–654 https://doi.org/10.1007/s00245-019-09636-4 Preface to the Special Issue on “Games, Dynamics and Optimization” 1 2 Radu Ioan Bot¸ · Mathias Staudigl Published online: 10 December 2019 © The Author(s) 2019 From March 13 to March 15, 2018, a 3-day interdisciplinary Workshop on mathe- matical game theory, continuous optimization and dynamical system was held at the University of Vienna. This conference was the ﬁrst edition of a a series of worhshops which is now already in its third edition, and it became a central element of the scientiﬁc work plan of the COST Action CA16228 “European Network for Game Theory”. The workshop was a jointly organized event together with the Vienna Center for Operations Research (VCOR) and the Department of Mathematics of the University of Vienna. The motivation for the creation of this workshop was that recently there are two inter- esting evolutions happening in disciplines which used to be quite close together but somehow have seen a diverging evolution: dynamical systems derived from games are a well-established research ﬁeld with many important applications in mathematical biology, physics, engineering and economics. Recently, dynamical systems have also witnessed important applications in optimization and machine learning. Here, the arse- nal of mathematical tools for studying continuous time systems have been successfully applied in shedding new light on the asymptotic properties of optimization algorithms, and to help practitioners and researchers in designing new algorithms. It is an interest- ing, and very notable, coincidence that both ﬁelds have one founding father for their evolution: the Lotka–Volterra system. In evolutionary game theory, Lotka–Volterra systems have been studied since the 1970s and led to the creation of the celebrated replicator dynamics. In optimization, the large class of Barrier-dynamics contains as special case Lotka–Volterra systems, and the connection between Riemannian gradient ﬂows, evolutionary game theory, and optimization is now well-documented. With all these in mind, the aim of the “Games, Dynamics and Optimization” (GDO) workshop series is to explore the analogies between dynamical systems in games and dynamical B Radu Ioan Bo¸t radu.bot@univie.ac.at Mathias Staudigl m.staudigl@maastrichtuniversity.nl Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria Department of Quantitative Economics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands 123 652 Applied Mathematics & Optimization (2020) 81:651–654 systems in optimization, to boost their evolution, popularize their use, and deepen and extend their theoretical foundations. With this purpose in mind, a special issue in the journal Applied Mathematics and Optimization has been announced, where rep- resentative papers on the theme of GDO are collected. This special issue consists of ten papers, all of which touching key topics in mathematical game theory, dynamical systems and nonlinear analysis. In what follows, we group the contributions according to these three ﬁelds. Papers on Operator Splitting – The work by Adly and Bourdin introduces a new decomposition formula for the resolvent operators of the sum of two monotone operators, a computational routine usually involved in the numerical treatment of monotone inclusions and PDEs via operator splitting algorithms. – Dynamical systems inspired by operator splitting techniques is a very active ﬁeld of research within optimization theory. The continuous time analysis often sheds new light on the asymptotic behavior of numerical algorithms, via a careful numerical discretization. In particular, continuous time analysis admits a direct analysis via Lyapunov techniques, and their structure gives us insights into the design of new algorithms. The most successful story in this area is the recent beautiful connection between Nesterov’s accelerated algorithm and second-order dynamical systems. Along these lines, the paper by Attouch and Cabot introduces a fairly broad fam- ily of iterative forward-backward schemes, which can be related to second-order dynamical systems by a careful explicit numerical discretization. The choice of the step parameters in the discretized dynamics is a key issue to guarantee a good performance of the resulting scheme. This paper derives explicit modeling recom- mendations on step-sizes, proximal parameters and inertial parameters, to ensure stability of the discrete-time dynamical system. An application to zero-sum games sheds light on the connection between operator splitting and game theory. – The paper by Csetnek, Malitsky and Tam proposes a new algorithm for ﬁnding a zero of the sum of two maximally monotone operators, where one is assumed to be single-valued and Lipschitz continuous, which arises from a non-standard discretization of a continuous dynamical system associated with the celebrated Douglas–Rachford algorithm. This is a truly novel results in this ﬁeld, and interest- ing connections between discrete and continuous time dynamical systems become apparent. Papers on Partial Differential Equations – Differential games describe a competitive optimal control problem between play- ers who jointly control an ordinary differential equation in order to steer the system towards certain states in which each player is guaranteed a certain payoff. In the setting of non-anticipative strategies determining the value of such a game when players’ action spaces are constrained is a delicate issue. An even more challeng- ing setting is studied in the paper by Bettiol, Quincampoix and Vinter, in which 123 Applied Mathematics & Optimization (2020) 81:651–654 653 players controls are even affected by coupled constraints. Such Generalized Nash games are classical in economics and operations research, and play an increas- ingly important role in PDE constrained optimization as well. Under a viability condition, the authors prove the existence of the game value in such a complex strategic environment. – Continuing the line of challenging optimal control problems, the paper by Breiten, Kunisch and Pfeiffer, investigates stabilization of the 2-dimensional Navier–Stokes equation by classical numerical optimal control tools, namely the value function approximation. Key to this results is an interesting discovery related to the local continuity properties of the value function around the steady state. This allows the authors to perform a delicate linearization, which they use to produce an approxi- mate feedback control. – The limit behavior of singular perturbations problems of biLaplacian type is inves- tigated in the paper by Dipierro, Karakhanyan and Valdinoci. This study provides new solutions to free boundary problems, and contains an interesting search game in continuous time and space. – Gradient ﬂows are the paradigm for dissipative evolution equations, which are also of key importance in games and optimization. One standard numerical dis- cretization scheme for gradient ﬂows is the classical Euler scheme. To preserve the dissipation dynamics along a time discretization, the paper by Jüngel, Stefanelli and Trussardi investigate two modiﬁcations of the discretized Euler scheme, and demonstrates important improvements with respect to the latter. – The paper by Rodrigues and Santos aims to extend classical results on varia- tional and quasi-variational inequalities over convex sets deﬁned through gradient constraints, by replacing the gradient with a fractional gradient. The continuous dependence of the solution on the data is proved and generalized Lagrange multi- pliers rules are derived. Papers on Optimization – Finite sum models are the canonical model in machine learning and empirical risk minimization. In the paper by Dinh, Goberna and Volle, a ﬁnite sum model is investigated in an abstract setting, and new optimality conditions are derived. These abstract results are then used to derive interesting new results for the best- approximation problem. – In large scale applications, the use of randomized algorithms is a standard tool to improve the iteration complexity. The paper by Nedic and Necoara deals with the minimization of a nonsmooth convex function with respect to a feasible set, which is represented as large intersection of functional constraints. In this chal- lenging setting, a parallel and sequential stochastic subgradient algorithm based on minibatch estimators is studied. 123 654 Applied Mathematics & Optimization (2020) 81:651–654 Acknowledgements Open access funding provided by University of Vienna. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna- tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations.$

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Preface to the Special Issue on “Games, Dynamics and Optimization”

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Preface to the Special Issue on “Games, Dynamics and Optimization”

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