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

Abstract

The problem (LF P ) 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 (LF P ). 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 (ECAM) for solving the global optimization subproblems.

Recent contributions to linear semi-infinite optimization: an update

“…This relaxation scheme admits di¤erent implementations, e.g., if k = 1 for all k; then x k+1 is an approximate projection of x k onto H k ; and, if k = 2 for all k; then x k+1 is an approximate symmetric of x k with respect to H k : The relaxation parameter k is maintained …xed in [130] and [135] (with k = 1 for all k), as well as in [117] and [118] (with k = for all k; for some 2 (0; 2]), whereas k is taken at random in some subinterval of (0; 2] in [79], where all this variants of the relaxation algorithm are compared from the computational e¢ ciency point of view.…”

Recent contributions to linear semi-infinite optimization

“…This relaxation scheme admits di¤erent implementations, e.g., if k = 1 for all k; then x k+1 is an approximate projection of x k onto H k ; and, if k = 2 for all k; then x k+1 is an approximate symmetric of x k with respect to H k : The relaxation parameter k is maintained …xed in [113] and [118] (with k = 1 for all k), as well as in [102] and [103] (with k = for all k; for some 2 (0; 2]), whereas k is taken at random in some subinterval of (0; 2] in [68], where all this variants of the relaxation algorithm are compared from the computational e¢ ciency point of view.…”

Selected Applications of Linear Semi-Infinite Systems Theory

“…Regarding numerical methods for LSIS, it is worth mentioning the linear semi-in…nite feasibility problem, which consists in …nding a solution of a given LSIS (see [34] and references therein). Such kind of problem arises in practice in statistics [4], when one tries to solve a given linear or quadratic semi-in…nite optimization problem by means of some feasible direction method (see, e.g., [47] and [26], respectively), in variational inequalities [33] and in image recovery problems [23].…”