×

Characterizations of linear suboptimality for mathematical programs with equilibrium constraints. (English) Zbl 1169.90021

Summary: The paper is devoted to the study of a new notion of linear suboptimality in constrained mathematical programming. This concept is different from conventional notions of solutions to optimization-related problems, while seems to be natural and significant from the viewpoint of modern variational analysis and applications. In contrast to standard notions, it admits complete characterizations via appropriate constructions of generalized differentiation in nonconvex settings. In this paper we mainly focus on various classes of mathematical programs with equilibrium constraints (MPECs), whose principal role has been well recognized in optimization theory and its applications. Based on robust generalized differential calculus, we derive new results giving pointwise necessary and sufficient conditions for linear suboptimality in general MPECs and its important specifications involving variational and quasivariational inequalities, implicit complementarity problems, etc.

MSC:

90C30 Nonlinear programming
49J52 Nonsmooth analysis
49J53 Set-valued and variational analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Borwein J.M. and Strójwas H.M. (1985). Tangential approximations. Nonlinear Anal. 9: 1347–1366 · Zbl 0613.49016 · doi:10.1016/0362-546X(85)90095-1
[2] Borwein J.M. and Zhu Q.J. (2005). Techniques of Variational Analysis. Canadian Mathematical Society Series, Springer, New York · Zbl 1076.49001
[3] Bounkhel M. and Thibault L. (2002). On various notions of regularity of sets in nonsmooth analysis. Nonlinear Anal. 48: 223–246 · Zbl 1012.49013 · doi:10.1016/S0362-546X(00)00183-8
[4] Burke J.V. and Deng S. (2005). Weak sharp minima revisted, II: application to linear regularity and error bounds. Math. Prog. 104: 235–261 · Zbl 1124.90349 · doi:10.1007/s10107-005-0615-2
[5] Deville R., Godefroy G. and Zizler V. (1997). Smoothness and Renorming in Banach spaces. Wiley, New York · Zbl 0782.46019
[6] Dontchev A.L. and Rockafellar R.T. (1996). Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 7: 1087–1105 · Zbl 0899.49004 · doi:10.1137/S1052623495284029
[7] Fabian M. and Mordukhovich B.S. (2003). Sequential normal compactness versus topological normal compactness in variational analysis. Nonlinear Anal. 54: 1057–1067 · Zbl 1038.46016 · doi:10.1016/S0362-546X(03)00126-3
[8] Kruger A.Y. (2002). Strict (,{\(\delta\)})-semidifferentials and extremality conditions. Optimization 51: 539–554 · Zbl 1053.49014 · doi:10.1080/0233193021000004967
[9] Kruger A.Y. (2004). Weak stationarity: eliminating the gap between necessary and sufficient conditions. Optimization 53: 147–164 · Zbl 1144.90507 · doi:10.1080/02331930410001695292
[10] Kruger A.Y. and Mordukhovich B.S. (1980). Extremal points and the Euler equation in nonsmooth optimization. Dokl. Akad. Nauk BSSR 24: 684–687 · Zbl 0449.49015
[11] Levy A.B., Poliquin R.A. and Rockafellar R.T. (2000). Stability of locally optimal solutions. SIAM J. Optim. 10: 580–604 · Zbl 0965.49018 · doi:10.1137/S1052623498348274
[12] Luo Z.Q., Pang J.-S. and Ralph D. (1996). Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge · Zbl 0870.90092
[13] Mordukhovich, B.S.: Sensitivity analysis in nonsmooth optimization. In: Field, D.A., Komkov, V. (eds.) Theoretical Aspects of Industrial Design, SIAM Proc. Appl. Math. vol. 58, 32–46 (1992) · Zbl 0769.90075
[14] Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Grundlehren Series (Fundamental Principles of Mathematical Sciences) vol. 330, Springer, Berlin (2006)
[15] Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences) vol. 331, Springer, Berlin (2006)
[16] Mordukhovich B.S. and Outrata J.V. (2001). Second-order subdifferentials and their applications. SIAM J. Optim. 12: 139–169 · Zbl 1011.49016 · doi:10.1137/S1052623400377153
[17] Mordukhovich, B.S., Outrata, J.V., Červinka, M.: Equilibrium problems with complementarity constraints: case study with applications to oligopolistic markets. Optimization (to appear) · Zbl 1132.91342
[18] Outrata J.V. (1999). Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res. 24: 627–644 · Zbl 1039.90088 · doi:10.1287/moor.24.3.627
[19] Outrata J.V., Kočvara M. and Zowe J. (1998). Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht · Zbl 0947.90093
[20] Poliquin R.A. and Rockafellar R.T. (1998). Tilt stability of a local minimum. SIAM J. Optim. 8: 287–299 · Zbl 0918.49016 · doi:10.1137/S1052623496309296
[21] Polyak B.T. (1987). Introduction to Optimization. Optimization Software, New York · Zbl 0708.90083
[22] Robinson S.M. (1979). Generalized equations and their solutions, I: Basic theory. Math. Progr. Study 10: 128–141 · Zbl 0404.90093
[23] Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren Series (Fundamental Principles of Mathematical Sciences) vol. 317, Springer, Berlin (1998)
[24] Ye J.J. (1999). Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J. Optim. 10: 943–962 · Zbl 1005.49019 · doi:10.1137/S105262349834847X
[25] Ye J.J. and Ye X.Y. (1997). Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22: 977–997 · Zbl 1088.90042 · doi:10.1287/moor.22.4.977
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.