Optimality conditions for disjunctive programs based on generalized differentiation with application to mathematical programs with equilibrium constraints

摘要

We consider optimization problems with a disjunctive structure of theconstraints. Prominent examples of such problems are mathematical programs withequilibrium constraints or vanishing constraints. Based on the concepts ofdirectional subregularity and their characterization by means of objects fromgeneralized differentiation, we obtain the new stationarity concept of extendedM-stationarity, which turns out to be an equivalent dual characterization ofB-stationarity. These results are valid under a very weak constraintqualification of Guignard type which is usually very difficult to verify. Wealso state a new constraint qualification which is a little bit stronger butverifiable. Further we present second-order optimality conditions, bothnecessary and sufficient. Finally we apply these results to the special case ofmathematical programs with equilibrium constraints and compute explicitly allthe objects from generalized differentiation. For this type of problems we alsointroduce the concept of strong M-stationarity which builds a bridge betweenS-stationarity and M-stationarity.