Linear Semi-infinite Optimization: Recent Advances

Abstract

Linear semi-infinite optimization (LSIO) deals with linear optimization problems in which either the dimension of the decision space or the number of constraints (but not both) is infinite. This paper overviews the works on LSIO published after 2000 with the purpose of identifying the most active research fields, the main trends in applications, and the more challenging open problems. After a brief introduction to the basic concepts in LSIO, the paper surveys LSIO models arising in mathematical economics, game theory, probability and statistics. It also reviews outstanding real applications of LSIO in semidefinite programming, telecommunications and control problems, in which numerical experiments are reported. In almost all these applications, the LSIO problems have been solved by means of ad hoc numerical methods, and this suggests that either the standard LSIO numerical approaches are not well-known or they do not satisfy the users? requirements. From the theoretical point of view, the research during this period has been mainly focused on the stability analysis of different objects associated with the primal problem (only the feasible set in the case of the dual). Sensitivity analysis in LSIO remains an open problem.