Witsenhausen's counterexample: A view from optimal transport theory

Wu, Yihong; Verdú, S.

doi:10.1109/cdc.2011.6160829

Cited by 52 publications

(29 citation statements)

References 24 publications

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“…As k reaches 0.4-0.6 the step behavior of the output appears. This particular value of k where the system changes from being affine to have a more general shape is consistent with a result in [23], which states that the optimal cost is less than the optimal cost for an affine system if k < 0.564. Depending on the realization of the Monte-Carlo samples we get either a 3.5-step mapping or a 4-step mapping as shown in Fig.…”

Section: B Numerical Resultssupporting

confidence: 86%

Iterative source-channel coding approach to Witsenhausen's counterexample

Karlsson

Gattami

Oechtering

et al. 2011

Proceedings of the 2011 American Control Conference

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In 1968, Witsenhausen introduced his famous counterexample where he showed that even in the simple linear quadratic static team decision problem, complex nonlinear decisions could outperform any given linear decision. This problem has served as a benchmark problem for decades where researchers try to achieve the optimal solution. This paper introduces a systematic iterative source-channel coding approach to solve problems of the Witsenhausen Counterexample-character. The advantage of the presented approach is its simplicity. Also, no assumptions are made about the shape of the space of policies. The minimal cost obtained using the introduced method is 0.16692462, which is the lowest known to date. Index TermsOpen-loop control systems, decision making, iterative methods, discretization, quantizer design, stochastic control.

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Section: B Numerical Resultssupporting

confidence: 86%

Iterative source-channel coding approach to Witsenhausen's counterexample

Karlsson

Gattami

Oechtering

et al. 2011

Proceedings of the 2011 American Control Conference

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“…This is Witsenhausen's counterexample [4]. For this problem, Witsenhausen established that a solution exists (we note that Wu and Verdu provided an alternative proof using tools from Transportation theory [26]), and established that an optimal policy is non-linear.…”

Section: We Wish To Minimize E[(x −X)mentioning

confidence: 89%

“…When applied to the special case of spatially invariant systems, the controller still needs to be able to receive information faster than the plant can propagate its inputs over any given distance, analogous to (26), and the triangle inequality (25) discussed above becomes a condition that the support function imposed on the controller needs to be subadditive. This includes funnel causal systems, developed in the study of convexity for these problems [84].…”

Section: E Examplesmentioning

confidence: 99%

Information structures in optimal decentralized control

Mahajan

Martins

Rotkowitz

et al. 2012

2012 IEEE 51st IEEE Conference on Decision and Control (CDC)

136

153

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Abstract-This tutorial paper provides a comprehensive characterization of information structures in team decision problems and their impact on the tractability of team optimization. Solution methods for team decision problems are presented in various settings where the discussion is structured in two foci: The first is centered on solution methods for stochastic teams admitting state-space formulations. The second focus is on norm-optimal control for linear plants under information constraints.

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“…Existence of optimal policies for static and a class of sequential dynamic teams have been studied recently in [30]. More specific setups and non-existence results have been studied in [59], [55], [62] and [61]. Existence of optimal team policies has been established in [20] for a class of continuous-time decentralized stochastic control problems.…”

Section: 2mentioning

confidence: 99%

Convex Analysis in Decentralized Stochastic Control, Strategic Measures, and Optimal Solutions

Yüksel¹,

Saldi²

2017

SIAM J. Control Optim.

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Abstract. This paper is concerned with the properties of the sets of strategic measures induced by admissible team policies in decentralized stochastic control and the convexity properties in dynamic team problems. To facilitate a convex analytical approach, strategic measures for team problems are introduced. Properties such as convexity, compactness and Borel measurability under weak convergence topology are studied, and sufficient conditions for each of these properties are presented. These lead to existence of and structural results for optimal policies. It will be shown that the set of strategic measures for teams which are not classical is in general non-convex, but the extreme points of a relaxed set consist of deterministic team policies, which lead to their optimality for a given team problem under an expected cost criterion. Externally provided independent common randomness for static teams or private randomness for dynamic teams do not improve the team performance. The problem of when a sequential team problem is convex is studied and necessary and sufficient conditions for problems which include teams with a non-classical information structure are presented. Implications of this analysis in identifying probability and information structure dependent convexity properties are presented.Key words. Stochastic control, decentralized control, optimal control, convex analysis.AMS subject classifications. 93E03, 90B99, 49J551. Introduction. Team decision theory has its roots in control theory and economics. Marschak [37] was perhaps the first to introduce the basic elements of teams, and to provide the first steps toward the development of a team theory. Radner [42] provided foundational results for static teams, establishing connections between person-by-person optimality, stationarity, and team-optimality [38]. Contributions of Witsenhausen [56,57,58,54,53] on dynamic teams and characterization of information structures have been crucial in the progress of our understanding of dynamic teams. We refer the reader to Section 1.1, where Witsenhausen's intrinsic model, and characterization of information structures are discussed in detail. Further discussion on design of information structures in the context of team theory is available in [5,48,61].Convexity is a very important property for optimization problems. A property related to convex analysis that is relevant in team problems is the characterization of the sets of strategic measures; these are the probability measures induced on the exogenous variables, and measurement and action spaces by admissible control policies. In the context of single decision maker control problems, such measures have been studied extensively in [45,41,24,27]. A study of strategic measures for team problems has not been made to our knowledge, and it will be observed in this paper that many of the properties that are natural for fully-observed single-decision-maker stochastic control problems, such as convexity, do not generally extend to a large class of stochastic team problem...

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Witsenhausen's counterexample: A view from optimal transport theory

Cited by 52 publications

References 24 publications

Iterative source-channel coding approach to Witsenhausen's counterexample

Iterative source-channel coding approach to Witsenhausen's counterexample

Information structures in optimal decentralized control

Convex Analysis in Decentralized Stochastic Control, Strategic Measures, and Optimal Solutions

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