Abstract
We consider a difficult class of optimization problems that we call a mathematical program with vanishing constraints. Problems of this kind arise in various applications including optimal topology design problems of mechanical structures. We show that some standard constraint qualifications like LICQ and MFCQ usually do not hold at a local minimum of our program, whereas the Abadie constraint qualification is sometimes satisfied. We also introduce a suitable modification of the standard Abadie constraint qualification as well as a corresponding optimality condition, and show that this modified constraint qualification holds under fairly mild assumptions. We also discuss the relation between our class of optimization problems with vanishing constraints and a mathematical program with equilibrium constraints.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
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
Bazaraa M.S. and Shetty C.M. (1976). Foundations of Optimization. Lecture Notes in Economics and Mathematical Systems 122. Springer, Berlin Heidelberg New York
Bendsøe M.P. (1989). Optimal shape design as a material distribution problem. Struct. Opt. 1: 193–202
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
Bendsøe M.P. and Sigmund O. (2003). Topology Optimization. Springer, Berlin Heidelberg New York
Chen Y. and Florian M. (1995). The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions. Optimization 32: 193–209
Dorn W., Gomory R. and Greenberg M. (1964). Automatic design of optimal structures. Journal de Mécanique 3: 25–52
Flegel M.L. and Kanzow Ch. (2005). On the Guignard constraint qualification for mathematical programs with equilibrium constraints. Optimization 54: 517–534
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)
Luo Z.-Q., Pang J.-S. and Ralph D. (1996). Mathematical Programs with Equilibrium Constraints. University Press, Cambridge
Nocedal J. and Wright S.J. (1999). Numerical Optimization. Springer, Berlin Heidelberg New York
Outrata J.V., Kočvara M. and Zowe J. (1998). Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht
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
Peterson D.W. (1973). A review of constraint qualifications in finite-dimensional spaces. SIAM Rev. 15: 639–654
Scholtes S. (2004). Nonconvex structures in nonlinear programming. Oper. Res. 52: 368–383
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
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Achtziger, W., Kanzow, C. Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications. Math. Program. 114, 69–99 (2008). https://doi.org/10.1007/s10107-006-0083-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-006-0083-3