Premium
On a relaxation method for mathematical programs with vanishing constraints
Author(s) -
Achtziger Wolfgang,
Kanzow Christian,
Hoheisel Tim
Publication year - 2012
Publication title -
gamm‐mitteilungen
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.239
H-Index - 18
eISSN - 1522-2608
pISSN - 0936-7195
DOI - 10.1002/gamm.201210009
Subject(s) - karush–kuhn–tucker conditions , convergence (economics) , relaxation (psychology) , constraint (computer aided design) , stationary point , mathematical optimization , mathematics , field (mathematics) , class (philosophy) , point (geometry) , optimization problem , computer science , mathematical analysis , pure mathematics , geometry , psychology , social psychology , artificial intelligence , economics , economic growth
The 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)