Multicriteria Control Problems for Stochastic Sequences

Piunovskiy, Alexey

doi:10.1016/s1474-6670(17)58273-7

Cited by 1 publication

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“…Note here that the team can be static or dynamic. For the special case of single-decision maker stochastic control problems with fully available state information at the controller (note that this can be viewed as a team problem with a classical information structure), convexity of the set of strategic measures has been established in [24] and [40]. We present a concise proof for the more general team problems with classical information structures.…”

Section: Convexity Properties Of Sets Of Strategic Measures and Redun...mentioning

confidence: 94%

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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Section: Convexity Properties Of Sets Of Strategic Measures and Redun...mentioning

confidence: 94%