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"Hernández-Lerma, O. (Onésimo)"
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Extreme Points of Sets of Randomized Strategies in Constrained Optimization and Control Problems
This paper concerns the existence and characterization of optimal randomized strategies for some constrained optimization and control problems. We first present a characterization of the extreme points of a set of randomized strategies that satisfy n moment-like constraints. Conditions are given under which those extreme points are randomizations of at most n+1 deterministic strategies. This result is then applied to obtain the existence and characterization of optimal strategies for a class of deterministic, allocation-like, optimization problems and their Young relaxations. Similar results are obtained for constrained Markov control processes in Borel spaces.
Journal Article
Nonzero-sum games for continuous-time Markov chains with unbounded discounted payoffs
In this paper, we study two-person nonzero-sum games for continuous-time Markov chains with discounted payoff criteria and Borel action spaces. The transition rates are possibly unbounded, and the payoff functions might have neither upper nor lower bounds. We give conditions that ensure the existence of Nash equilibria in stationary strategies. For the zero-sum case, we prove the existence of the value of the game, and also provide a recursive way to compute it, or at least to approximate it. Our results are applied to a controlled queueing system. We also show that if the transition rates are uniformly bounded, then a continuous-time game is equivalent, in a suitable sense, to a discrete-time Markov game.
Journal Article
Constrained Average Cost Markov Control Processes in Borel Spaces
This paper considers constrained Markov control processes in Borel spaces, with unbounded costs. The criterion to be minimized is a long-run expected average cost, and the constraints can be imposed on similar average costs, or on average rewards, or discounted costs or rewards. We give conditions under which the constrained problem (CP) is solvable and equivalent to an equality constrained (EC) linear program. Furthermore, we show that there is no duality gap between EC and the dual program EC* and that in fact the strong duality condition holds. Finally, we introduce an explicit procedure to solve CP in some cases which is illustrated with a detailed example.
Journal Article
Zero-Sum Stochastic Games in Borel Spaces: Average Payoff Criteria
This paper is concerned with two-person zero-sum dynamic stochastic games in Borel spaces, with possibly unbounded payoff function, and several average (or ergodic) payoff criteria. We give conditions under which the long-run expected average payoff criterion, the sample-path average criterion, the existence of solutions to the average payoff Shapley equations, and a certain \"martingale condition\" are all equivalent.
Journal Article
Bias and overtaking equilibria for zero-sum continuous-time Markov games
This paper deals with continuous-time zero-sum two-person Markov games with denumerable state space, general (Borel) action spaces and possibly unbounded transition and reward/cost rates. We analyze the bias optimality and the weakly overtaking optimality criteria. An example shows that, in contrast to control (or one-player) problems, these criteria are not equivalent for games. [PUBLICATION ABSTRACT]
Journal Article
Limiting Discounted-Cost Control of Partially Observable Stochastic Systems
This paper presents two main results on partially observable (PO) stochastic systems. In the first one, we consider a general PO system$$ x_{t+1}= F (x_t, a_t, \\xi_t), \\ \\ \\, y_t= G(x_t, \\eta_t) \\ \\ \\ (t=0,1,\\ldots) \\hspace{1in} \\ \\ \\ \\ (*) $$on Borel spaces, with possibly unbounded cost-per-stage functions, and we give conditions for the existence of$\\alpha$ -discount optimal control policies$(0 < \\alpha < 1).$In the second result we specialize (*) to additive-noise systems$$ x_{t+1}= F_n(x_t,a_t) + \\xi_t, \\ \\ \\, y_t= G_n(x_t) + \\eta_t \\ \\ \\ (t=0,1,\\ldots) $$in Euclidean spaces with Fn(x,a) and Gn(x) converging pointwise to${\\mbox{functions}}$$F_{\\infty}(x,a)$and$G_{\\infty}(x),$respectively, and we give conditions for the limiting PO model$$ x_{t+1}= F_{ınfty}(x_t, a_t) + \\xi_t, \\ \\ \\,\\, y_t=G_{ınfty}(x_t) + \\eta_t $$to have an$\\alpha$ -discount optimal policy.
Journal Article
Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates
This paper is a first study of two-person zero-sum games for denumerable continuous-time Markov chains determined by given transition rates, with an average payoff criterion. The transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. In the spirit of the ‘drift and monotonicity’ conditions for continuous-time Markov processes, we give conditions on the controlled system's primitive data under which the existence of the value of the game and a pair of strong optimal stationary strategies is ensured by using the Shapley equations. Also, we present a ‘martingale characterization’ of a pair of strong optimal stationary strategies. Our results are illustrated with a birth-and-death game.
Journal Article
The Scalarization Approach to Multiobjective Markov Control Problems: Why Does It Work?
This paper concerns discrete-time multiobjective Markov control processes on Borel spaces and unbounded costs. Under mild assumptions, it is shown that the usual \"scalarization approach\" to obtain Pareto policies for the multiobjective control problem is in fact equivalent to solving the dual of a certain multiobjective infinite-dimensional linear program. The latter program is obtained from a multiobjective measure problem which is also used to prove the existence of strong Pareto policies, that is, Pareto policies whose cost vector is the closest to the control problem's virtual minimum. [PUBLICATION ABSTRACT]
Journal Article
Strong duality of the Monge-Kantorovich mass transfer problem in metric spaces
This paper studies the Monge–Kantorovich mass transfer (MT) problem on metric spaces and with an unbounded cost function. Conditions are given under which the strong duality condition holds; that is, MT and its dual MT are both solvable and their optimal values coincide.
Journal Article