A smoothing method for mathematical programs with equilibrium constraints.

*(English)*Zbl 0959.65079The authors propose an algorithm for solving optimization problems whose constraints include a strongly monotone variational inequality. Their idea is to reformulate the considered problem as a one-level nonsmoothly constrained optimization problem \((P)\) by using the Karush-Kuhn-Tucker conditions for the variational inequality.

Next, they introduce a sequence \((P_{\mu^k})\) of smooth, regular one-level problems which progressively approximate the nonsmooth problem \((P)\). It is proved that the sequence of solutions of the problems \((P_{\mu^k})\) is contained in a compact set and that each of its limit points is a solution of the original problem.

Furthermore, it is shown that the sequence of stationary points of the problems \((P_{\mu^k})\) is also contained in a compact set and that its limit points furnish strongly \(C\)-stationary points of the original problem.

Next, they introduce a sequence \((P_{\mu^k})\) of smooth, regular one-level problems which progressively approximate the nonsmooth problem \((P)\). It is proved that the sequence of solutions of the problems \((P_{\mu^k})\) is contained in a compact set and that each of its limit points is a solution of the original problem.

Furthermore, it is shown that the sequence of stationary points of the problems \((P_{\mu^k})\) is also contained in a compact set and that its limit points furnish strongly \(C\)-stationary points of the original problem.

Reviewer: Wolfgang W.Breckner (Cluj-Napoca)