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Title: First- and second-order optimality conditions for mathematical programs with vanishing constraints (English)
Author: Hoheisel, Tim
Author: Kanzow, Christian
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 52
Issue: 6
Year: 2007
Pages: 495-514
Summary lang: English
Category: math
Summary: We consider a special class of optimization problems that we call Mathematical Programs with Vanishing Constraints, MPVC for short, which serves as a unified framework for several applications in structural and topology optimization. Since an MPVC most often violates stronger standard constraint qualification, first-order necessary optimality conditions, weaker than the standard KKT-conditions, were recently investigated in depth. This paper enlarges the set of optimality criteria by stating first-order sufficient and second-order necessary and sufficient optimality conditions for MPVCs. (English)
Keyword: mathematical programs with vanishing constraints
Keyword: mathematical programs with equilibrium constraints
Keyword: first-order optimality conditions
Keyword: second-order optimality conditions
MSC: 90C30
MSC: 90C33
idZBL: Zbl 1164.90407
idMR: MR2357577
DOI: 10.1007/s10492-007-0029-y
Date available: 2009-09-22T18:31:34Z
Last updated: 2020-07-02
Stable URL:
Reference: [1] W.  Achtziger, C.  Kanzow: Mathematical programs with vanishing constraints: Optimality conditions and constraint qualifications.Math. Program (to appear). MR 2386163
Reference: [2] M. S.  Bazaraa, H. D.  Sherali, and C. M.  Shetty: Nonlinear Programming. Theory and Algorithms. 2nd edition.John Wiley & Sons, Hoboken, 1993. MR 2218478
Reference: [3] M. L.  Flegel, C.  Kanzow: A direct proof for $M$-stationarity under MPEC-ACQ for mathematical programs with equilibrium constraints.In: Optimization with Multivalued Mappings: Theory, Applications and Algorithms, S. Dempe, V. Kalashnikov (eds.), Springer-Verlag, New York, 2006, pp. 111–122. MR 2243539
Reference: [4] C.  Geiger, C.  Kanzow: Theorie und Numerik restringierter Optimierungsaufgaben.Springer-Verlag, Berlin, 2002. (German)
Reference: [5] T.  Hoheisel, C.  Kanzow: On the Abadie and Guignard constraint qualification for mathematical programs with vanishing constraints.Optimization (to appear). MR 2561810
Reference: [6] T.  Hoheisel, C.  Kanzow: Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications.J.  Math. Anal. Appl. 337 (2008), 292–310. MR 2356071, 10.1016/j.jmaa.2007.03.087
Reference: [7] Z.-Q.  Luo, J.-S.  Pang, and D.  Ralph: Mathematical Programs with Equilibrium Constraints.Cambridge University Press, Cambridge, 1997. MR 1419501
Reference: [8] O. L.  Mangasarian: Nonlinear Programming.McGraw-Hill Book Company, New York, 1969. Zbl 0194.20201, MR 0252038
Reference: [9] J. V.  Outrata: Optimality conditions for a class of mathematical programs with equilibrium constraints.Math. Oper. Res. 24 (1999), 627–644. Zbl 1039.90088, MR 1854246, 10.1287/moor.24.3.627
Reference: [10] J. V.  Outrata, M. Kočvara, and J. Zowe: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Nonconvex Optimization and its Applications.Kluwer, Dordrecht, 1998.
Reference: [11] H.  Scheel, S.  Scholtes: Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity.Math. Oper. Res. 25 (2000), 1–22. MR 1854317, 10.1287/moor.
Reference: [12] S. Scholtes: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints.SIAM J. Optim. 11 (2001), 918–936. Zbl 1010.90086, MR 1855214, 10.1137/S1052623499361233
Reference: [13] S.  Scholtes: Nonconvex structures in nonlinear programming.Oper. Res. 52 (2004), 368–383. Zbl 1165.90597, MR 2066033, 10.1287/opre.1030.0102
Reference: [14] J. J.  Ye: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints.J.  Math. Anal. Appl. 307 (2005), 350–369. Zbl 1112.90062, MR 2138995, 10.1016/j.jmaa.2004.10.032


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