# Volume 09 (2005) No. 1

## Volume 09 (2005) No. 1

### Approximation of general optimization problems

Jorge Álvarez-Mena and Onésimo Hernández-Lerma

**Abstract:**

This paper concerns the approximation of a general optimization problem (OP) for which the cost function and the constraints are defined on a Hausdorff topological space. This degree of generality allows us to consider OPs for which other approximation approaches are not applicable. First we obtain convergence results for a general OP, and then we present two applications of these results. The first application is to *approximation schemes* for infinite-dimensional linear programs. The second is on the approximation of the optimal value and the optimal solutions for the so-called *general capacity* problem in metric spaces.

### Linear programming relaxations of the mixed postman problem

Francisco Javier Zaragoza Martínez

**Abstract:**

The mixed postman problem consists of finding a minimum cost tour of a connected mixed graph traversing all its vertices, edges, and arcs at least once. We prove in two different ways that the linear programming relaxations of two well-known integer programming formulations of this problem are equivalent. We also give some properties of the extreme points of the polyhedra defined by one of these relaxations and its linear programming dual.

### A nonmeasurable set as a union of a family of increasing well-ordered measurable sets

Juán González-Hernández and César E. Villarreal

**Abstract:**

Given a measurable space $(\mathcal{X}, \mathcal{A})$in which every singleton is measurable and which contains a nonmeasurable subset, we prove the existence of a nonmeasurable set which is the union of a well-ordered increasing family of measurable sets.

### Noncooperative continuous-time Markov games

Héctor Jasso-Fuentes

**Abstract:**

This work concerns noncooperative continuous-time Markov games with Polish state and action spaces. We consider finite-horizon and infinite-horizon discounted payoff criteria. Our aim is to give a unified presentation of optimality conditions for general Markov games. Our results include zero-sum and nonzero-sum games.