Juan González-Hernández

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.

Constrained Markov control processes in Borel spaces: the discounted case

Abstract. We consider constrained discounted-cost Markov control processes in Borel spaces, with unbounded costs. Conditions are given for the constrained problem to be solvable, and also equivalent to an equality-constrained (EC) linear program. In addition, it is shown that there is no duality gap between EC and its dual program EC*, and that, under additional assumptions, also EC* is solvable, so that in fact the strong duality condition holds. Finally, a Farkas-like theorem is included, which gives necessary and sufficient conditions for the primal program EC to be consistent.

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.

Markov control processes with randomized discounted cost

In this paper we consider Markov Decision Processes with discounted cost and a random rate in Borel spaces. We establish the dynamic programming algorithm in finite and infinity horizon cases. We provide conditions for the existence of measurable selectors. And we show an example of consumption-investment problem.

On the consistency of the mass transfer problem

Conditions are given under which the Monge-Kantorovich mass transfer problem on general metric spaces and with unbounded cost function has a feasible solution.

An approximation scheme for the Kantorovich-Rubinstein problem on compact spaces

Abstract This paper presents an approximation scheme for the Kantorovich–Rubinstein mass transshipment (KR) problem on compact spaces. A sequence of finite-dimensional linear programs, minimal cost network flow problems with bounds, are introduced and it is proven that the limit of the sequence of the optimal values of these problems is the optimal value of the KR problem. Numerical results are presented approximating the Kantorovich metric between distributions on [0, 1].

Measurable order and simulation

In this article is proposed a method for simulating random objects in a sample space Ω provided with a probability measure, a total order p and general conditions, such as the fact that for all ω0 ∈ Ω the set {ω ∈ Ω: ω p ω0} is measurable, the order topology relative to p is first countable, and a completion of Ω relative to the order is also first countable.