On duality in semi-infinite programming and existence theorems for linear inequalities

Linear semi-infinite programming deals with the optimization of linear functionals on finite-dimensional spaces under infinitely many linear constraints. For such kind of programs, a positive duality gap can occur between them and their corresponding dual problems, which are linear programs posed on infinite-dimensional spaces. This paper exploits some recent existence theorems for systems of linear inequalities in order to obtain a complete classification of linear semi-infinite programming problems from the point of view of the duality gap and the viability of the discretization numerical approach. The elimination of the duality gap is also discussed.