On a relaxation method for mathematical programs with vanishing constraints

Abstract

AbstractThe paper considers a numerical approach for the solution of mathematical problems with vanishing constraints (MPVC). It is well known that direct numerical approaches for the treatment of MPVCs typically fail because standard constraint qualifications usually are not satisfied at a local minimizer. This parallels the situation in the related class of mathematical programs with equilibrium constraints (MPEC). For MPVC several concepts of stationarity have been proposed in the last few years generalizing the usual KKT‐conditions. The paper considers a direct relaxation approach where the original problem is replaced by a perturbed problem. This approach has also been investigated by Izmailov and Solodov [1]. Under standard regularity conditions, we prove the convergence of KKT‐points of the perturbed problem to suitable stationary points of the MPVC. Moreover, conditions are provided showing convergence to a strongly stationary point (i.e., a KKT‐point) of MPVC. The paper closes with a numerical test of this approach for a problem arising in the field of topology optimization of mechanical structures with vanishing stress constraints (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)