Relaxation-based bounds for semi-infinite programs

Finite formulations are presented for the calculation of lower and upper bounds on the optimal solution value of semi-infinite programs (SIPs) involving smooth, potentially nonconvex objective function and constraints. The lower bounding problem is obtained by a formulation that combines the first- and second-order KKT necessary conditions of the lower-level problem with a discretization of the parameter set. The resulting mathematical program with equilibrium constraints (MPEC) is a relaxation of the original SIP and furnishes valid lower bounds. If the parameter set is subdivided, the optimal solution value of the lower bounding problem converges to the optimal solution value of the SIP. The upper bounding problem is based on convex and linear relaxations of the lower-level problem resulting in a restriction of the SIP. If the parameter set is subdivided, the constructed relaxations converge to the original lower-level program. The existence of SIP Slater points ensures convergence of the upper bounding problems to the optimal solution value of the SIP. Several alternatives for the upper bounding problem are presented and discussed. Numerical results are presented for a number of test problems from the literature.