Discretization in semi-infinite programming: The rate of approximation

The discretization approach for solving semi-infinite optimization problems is considered. We are interested in the rate of the approximation error between the solution of the semi-infinite problem and the solution of the discretized program depending on the discretization mesh-size $d$. It will be shown how this rate depends on whether the minimizer is strict of order one or two and on whether the discretization includes boundary points of the index set in a consistent way. This is done for common and for generalized semi-infinite problems.