Recent advances in multiobjective convex semi-infinite optimization

This paper reviews the existing literature on multiobjective (or vector) semi-infinite optimization problems, which are defined by finitely many convex objective functions of finitely many variables whose feasible sets are described by infinitely many convex constraints. The paper shows several applications of this type of optimization problems and presents a state-of-the-art review of its methods and theoretical developments (in particular, optimality, duality, and stability)

Abide With Me, Tis Eventide (4): Unison Voices

4 pages

A comparative note on the relaxation algorithms for the linear semi-infinite feasibility problem

The problem (LFP) of finding a feasible solution to a given linear semi-infinite system arises in different contexts. This paper provides an empirical comparative study of relaxation algorithms for (LFP). In this study we consider, together with the classical algorithm, implemented with different values of the fixed parameter (the step size), a new relaxation algorithm with random parameter which outperforms the classical one in most test problems whatever fixed parameter is taken. This new algorithm converges geometrically to a feasible solution under mild conditions. The relaxation algorithms under comparison have been implemented using the extended cutting angle method for solving the global optimization subproblems.This research was partially supported by MICINN of Spain, Grant MTM2014-59179-C2-1-P and Sistema Nacional de Investigadores, Mexico

Relaxation methods for solving linear inequality systems: Converging results

The problem of finding a feasible solution to a linear inequality system arises in numerous contexts. In [12] an algorithm, called extended relaxation method, that solves the feasibility problem, has been proposed by the authors. Convergence of the algorithm has been proven. In this paper, we consider a class of extended relaxation methods depending on a parameter and prove their convergence. Numerical experiments have been provided, as well

Stability and Well-Posedness in Linear Semi-Infinite Programming

This paper presents an approach to the stability and the Hadamard well-posedness of the linear semi-infinite programming problem (LSIP). No standard hypothesis is required in relation to the set indexing of the constraints and, consequently, the functional dependence between the linear constraints and their associated indices has no special property. We consider, as parameter space, the set of all LSIP problems whose constraint systems have the same index set, and we define in it an extended metric to measure the size of the perturbations. Throughout the paper the behavior of the optimal value function and of the optimal set mapping are analyzed. Moreover, a certain type of Hadamard well-posedness, which does not require the boundedness of the optimal set, is characterized. The main results provided in the paper allow us to point out that the lower semicontinuity of the feasible set mapping entails high stability of the whole problem, mainly when this property occurs simultaneously with the boundedness of the optimal set. In this case all the stability properties hold, with the only exception being the lower semicontinuity of the optimal set mapping.This research was partially supported by grants PB95-0687 and SAB 95-0311 from DGES and by grants GV-2219/94 and GV-C-CN-10-067-96 from Generalitat Valenciana

Generic primal-dual solvability in continuous linear semi-infinite programming

In this article, we consider the space of all the linear semi-infinite programming (LSIP) problems with a given infinite compact Hausdorff index set, a given number of variables and continuous coefficients, endowed with the topology of the uniform convergence. These problems are classified as inconsistent, solvable with bounded optimal set, bounded (i.e. finite valued), but either unsolvable or having an unbounded optimal set, and unbounded (i.e. with infinite optimal value), giving rise to the so-called refined primal partition of the space of problems. The mentioned LSIP problems can be also classified with a similar criterion applied to the corresponding Haar's dual problems, which provides the refined dual partition of the space of problems. We characterize the interior of the elements of the refined primal and dual partitions as well as the interior of the intersections of the elements of both partitions (the so-called refined primal-dual partition). These characterizations allow to prove that most (primal or dual) bounded problems have simultaneously primal and dual non-empty bounded optimal set. Consequently, most bounded continuous LSIP problems are primal and dual solvable.Research supported by DGES and FEDER, Grant MTM2005-08572-C03-01. Research partially supported by CONACyT of MX. Grant 55681

Ill-posedness in continuous linear optimization via partitions of the space of parameters

In this paper we consider the parameter space of continuous linear optimization problems with a given decision space and a given index set. We consider different partitions of this space, on the basis of the primal, the dual, and the primal-dual status of each parameter. We define ill-posedness and relative ill-posedness w.r.t. a given set and absolute ill-posedness w.r.t. a given family of sets. These concepts are characterized for the elements of the partitions considered in this paper

On Motzkin decomposable sets and functions

A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. The main result in this paper establishes that a closed convex set is Motzkin decomposable if and only if the set of extreme points of its intersection with the linear subspace orthogonal to its lineality is bounded. The paper characterizes the class of the extended functions whose epigraphs are Motzkin decomposable sets showing, in particular, that these functions attain their global minima when they are bounded from below. Calculus of Motzkin decomposable sets and functions is provided.This work has been supported by MICINN of Spain, Grants MTM2008-06695-C03-01/03, by Generalitat Valenciana, by Generalitat de Catalunya, by the Barcelona GSE Research Network, and by CONACyT of Mexico, Grant 55681

Motzkin decomposition of closed convex sets via truncation

A nonempty set F is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set C with a closed convex cone D. In that case, the sets C and D are called compact and conic components of F. This paper provides new characterizations of the Motzkin decomposable sets involving truncations of F (i.e., intersections of FF with closed halfspaces), when F contains no lines, and truncations of the intersection F̂ of F with the orthogonal complement of the lineality of F, otherwise. In particular, it is shown that a nonempty closed convex set F is Motzkin decomposable if and only if there exists a hyperplane H parallel to the lineality of F such that one of the truncations of F̂ induced by H is compact whereas the other one is a union of closed halflines emanating from H. Thus, any Motzkin decomposable set F can be expressed as F=C+D, where the compact component C is a truncation of F̂. These Motzkin decompositions are said to be of type T when F contains no lines, i.e., when C is a truncation of F. The minimality of this type of decompositions is also discussed.This work has been supported by the MICINN of Spain, Grants MTM2011-29064-C03-01&02, by the Barcelona Graduate School of Economics, by the Government of Catalonia, by CONACyT of Mexico, Grant 55681, and by CNPq of Brazil, Grant 301280/86. The first author is Partner Investigator in the Australian Research Council Discovery Project DP120100467. The third author is affiliated toMOVE(Markets, Organizations and Votes in Economics)

On implicit active constraints in linear semi-infinite programs with unbounded coefficients

The concept of implicit active constraints at a given point provides useful local information about the solution set of linear semi-infinite systems and about the optimal set in linear semi-infinite programming provided the set of gradient vectors of the constraints is bounded, commonly under the additional assumption that there exists some strong Slater point. This paper shows that the mentioned global boundedness condition can be replaced by a weaker local condition (LUB) based on locally active constraints (active in a ball of small radius whose center is some nominal point), providing geometric information about the solution set and Karush-Kuhn-Tucker type conditions for the optimal solution to be strongly unique. The maintaining of the latter property under sufficiently small perturbations of all the data is also analyzed, giving a characterization of its stability with respect to these perturbations in terms of the strong Slater condition, the so-called Extended-Nürnberger condition, and the LUB condition.MICINN of Spain, Grant MTM2008-06695-C03-01, CONACyT of MX.Grant 55681 and SECYT-UNCuyo of Argentina, Grant Res. 1094/09-R