In a semi-infinite program only a countable subset of the constraints is essential (Q1079497)

The author considers the problem of minimizing a convex function f on a convex set described by convex inequality constraints \(g_ i(x)\leq 0\), \(i\in I\), I an infinite index set. It is shown that there exists a countable subset \(I_ 0\subset I\) such that the problem with \(I_ 0\) instead of I has the same feasible region and the same primal and dual value. By an example it is shown that not every subset \(I_ 1\subset I\) for which the feasible regions coincide leaves the dual value unchanged.