×

Exact penalty results for mathematical programs with vanishing constraints. (English) Zbl 1185.90187

Summary: A mathematical program with vanishing constraints (MPVC) is a constrained optimization problem arising in certain engineering applications. The feasible set has a complicated structure so that the most familiar constraint qualifications are usually violated. This, in turn, implies that standard penalty functions are typically non-exact for MPVCs. We therefore develop a new MPVC-tailored penalty function which is shown to be exact under reasonable assumptions. This new penalty function can then be used to derive (or recover) suitable optimality conditions for MPVCs.

MSC:

90C30 Nonlinear programming
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Luo, Z.-Q.; Pang, J.-S.; Ralph, D., Mathematical programs with equilibrium constraints, (1996), Cambridge University Press Cambridge, UK
[2] Outrata, J.V.; Kočvara, M.; Zowe, J., Nonsmooth approach to optimization problems with equilibrium constraints, (1998), Kluwer Academic Publishers Dordrecht, The Netherlands · Zbl 0947.90093
[3] Achtziger, W.; Kanzow, C., Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications, Mathematical programming, 114, 69-99, (2008) · Zbl 1151.90046
[4] Hoheisel, T.; Kanzow, C., On the abadie and guignard constraint qualification for mathematical progammes with vanishing constraints, Optimization, 58, 431-448, (2009) · Zbl 1162.90560
[5] Hoheisel, T.; Kanzow, C., Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications, Journal of mathematical analysis and applications, 337, 292-310, (2008) · Zbl 1141.90572
[6] Hoheisel, T.; Kanzow, C., First- and second-order optimality conditions for mathematical programs with vanishing constraints, Applications of mathematics, 52, 495-514, (2007), Special issue dedicated to J.V. Outrata’s 60th birthday · Zbl 1164.90407
[7] W. Achtziger, T. Hoheisel, C. Kanzow, A smoothing-regularization approach to mathematical programs with vanishing constraints, Preprint 284, Institute of Mathematics, University of Würzburg, Würzburg, November 2008 · Zbl 1291.90234
[8] Izmailov, A.F.; Solodov, M.V., Mathematical programs with vanishing constraints: optimality conditions, sensitivity and a relaxation method, Journal of optimization theory and applications, 142, 501-532, (2009) · Zbl 1180.90312
[9] Peterson, D.W., A review of constraint qualifications in finite-dimensional spaces, SIAM review, 15, 639-654, (1973)
[10] Rockafellar, R.T.; Wets, R.J.-B., ()
[11] Mordukhovich, B.S., ()
[12] Mordukhovich, B.S., ()
[13] Rockafellar, R.T., Convex analysis, (1970), Princeton University Press Princeton, NJ · Zbl 0229.90020
[14] Flegel, M.L.; Kanzow, C.; Outrata, J.V., Optimality conditions for disjunctive programs with application to mathematical programs with equilibrium constraints, Set-valued analysis, 15, 139-162, (2007) · Zbl 1149.90143
[15] Burke, J.V., Calmness and exact penalization, SIAM journal on control and optimization, 29, 493-497, (1991) · Zbl 0734.90090
[16] Burke, J.V., An exact penalization viewpoint of constrained optimization, SIAM journal on control and optimization, 29, 968-998, (1991) · Zbl 0737.90060
[17] Clarke, F.H., Optimization and nonsmooth analysis, (1983), John Wiley & Sons New York, Reprinted by SIAM, Philadelphia, PA, 1990 · Zbl 0727.90045
[18] Pang, J.-S., Error bounds in mathematical programming, Mathematical programming, 79, 299-332, (1997) · Zbl 0887.90165
[19] Henrion, R.; Outrata, J.V., Calmness of constraint systems with applications, Mathematical programming, 104, 437-464, (2005) · Zbl 1093.90058
[20] Mangasarian, O.L., Nonlinear programming, (1969), McGraw Hill New York, Reprinted by SIAM, Philadelphia, PA, 1994 · Zbl 0194.20201
[21] Borwein, J.M.; Lewis, A.S., Convex analysis and nonlinear optimization, () · Zbl 0808.46022
[22] Han, S.-P.; Mangasarian, O.L., Exact penalty functions in nonlinear programming, Mathematical programming, 17, 251-269, (1979) · Zbl 0424.90057
[23] Ioffe, A.D.; Outrata, J.V., On metric and calmness qualification conditions in subdifferential calculus, Set-valued analysis, 16, 199-227, (2008) · Zbl 1156.49013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.