zbMATH — the first resource for mathematics

Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications. (English) Zbl 1151.90046
In this very interesting paper, a new class (MPVC) of optimization problems is considered where a part of the constraints needs to be satisfied only if the variables are contained in some specified set and are absent otherwise. Besides interesting applications, the relations to mathematical programs with equilibrium constraints (MPEC) are explained. Regularity conditions like LICQ or MFCQ are violated at any feasible point of an MPEC, for MPVC the situation is better. The mentioned as well as the Abadie constraint qualification can be satisfied under certain assumptions which is investigated in detail in the paper. If the Abadie constraint qualification (which is weaker than MFCQ) is satisfied, then the Karush-Kuhn-Tucker necessary optimality conditions are satisfied at a local optimum of MPVC. In the last part of the paper a modified Abadie constraint qualification tailored to the special structure of the problem is introduced and its usefulness for deriving optimality conditions is shown.

90C30 Nonlinear programming
90C46 Optimality conditions and duality in mathematical programming
Full Text: DOI
[1] Achtziger, W.: On optimality conditions and primal-dual methods for the detection of singular optima. In: Cinquini, C., Rovati, M., Venini, P., Nascimbene, R. (eds.) Proceedings of the Fifth World Congress of Structural and Multidisciplinary Optimization, Lido di Jesolo, Italy, Paper 073, 1–6. Italian Polytechnic Press, Milano, Italy (2004)
[2] Achtziger, W., Kanzow, C.: Mathematical programs with vanishing constraints: Optimality conditions and constraint qualifications. Preprint 263, Institute of Applied Mathematics and Statistics, University of Würzburg, Würzburg, Germany, November 2005 · Zbl 1151.90046
[3] Bazaraa M.S. and Shetty C.M. (1976). Foundations of Optimization. Lecture Notes in Economics and Mathematical Systems 122. Springer, Berlin Heidelberg New York · Zbl 0334.90049
[4] Bendsøe M.P. (1989). Optimal shape design as a material distribution problem. Struct. Opt. 1: 193–202 · doi:10.1007/BF01650949
[5] Bendsøe M.P. and Kikuchi N. (1988). Generating optimal topologies in optimal design using a homogenization method. Comput. Meth. Appl. Mech. Eng. 71: 197–224 · Zbl 0671.73065 · doi:10.1016/0045-7825(88)90086-2
[6] Bendsøe M.P. and Sigmund O. (2003). Topology Optimization. Springer, Berlin Heidelberg New York · Zbl 1059.74001
[7] Chen Y. and Florian M. (1995). The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions. Optimization 32: 193–209 · Zbl 0812.00045 · doi:10.1080/02331939508844048
[8] Dorn W., Gomory R. and Greenberg M. (1964). Automatic design of optimal structures. Journal de Mécanique 3: 25–52
[9] Flegel M.L. and Kanzow Ch. (2005). On the Guignard constraint qualification for mathematical programs with equilibrium constraints. Optimization 54: 517–534 · Zbl 1147.90397 · doi:10.1080/02331930500342591
[10] Kohn, R.V., Strang, G.: Optimal design and relaxation of variational problems. Commun. Pure Appl. Math. (New York) 39, 1–25 (part I), 139–182 (part II), 353–357 (part III) (1986) · Zbl 0609.49008
[11] Luo Z.-Q., Pang J.-S. and Ralph D. (1996). Mathematical Programs with Equilibrium Constraints. University Press, Cambridge
[12] Nocedal J. and Wright S.J. (1999). Numerical Optimization. Springer, Berlin Heidelberg New York · Zbl 0930.65067
[13] Outrata J.V., Kočvara M. and Zowe J. (1998). Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht · Zbl 0947.90093
[14] Pang J.-S. and Fukushima M. (1999). Complementarity constraint qualifications and simplified conditions for mathematical programs with equilibrium constraints. Comput. Opt. Appl. 13: 111–136 · Zbl 1040.90560 · doi:10.1023/A:1008656806889
[15] Peterson D.W. (1973). A review of constraint qualifications in finite-dimensional spaces. SIAM Rev. 15: 639–654 · Zbl 0255.90049 · doi:10.1137/1015075
[16] Scholtes S. (2004). Nonconvex structures in nonlinear programming. Oper. Res. 52: 368–383 · Zbl 1165.90597 · doi:10.1287/opre.1030.0102
[17] Zhou M. and Rozvany G.I.N. (1991). The COC algorithm, part II: Topological, geometry and generalized shape optimization. Comput. Meth. Appl. Mech. Eng. 89: 197–224 · doi:10.1016/0045-7825(91)90046-9
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.