Taiwanese Journal of Mathematics

SUBOPTIMALITY CONDITIONS FOR MATHEMATICAL PROGRAMS WITH EQUILIBRIUM CONSTRAINTS

Truong Q. Bao, Boris S. Mordukhovich, and Pankaj Gupta

Full-text: Open access

Abstract

In this paper we study mathematical programs with equilibrium constraints (MPECs) described by generalized equations in the extended form \begin{eqnarray*} 0\in G(x,y)+Q(x,y), \end{eqnarray*} where both mappings $G$ and $Q$ are set-valued. Such models arise, in particular, from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish new weak and strong suboptimality conditions for the general MPEC problems under consideration in finite-dimensional and infinite-dimensional spaces that do not assume the existence of optimal solutions. This issue is particularly important for infinite-dimensional optimization problems, where the existence of optimal solutions requires quite restrictive assumptions. Our techniques are mainly based on modern tools of variational analysis and generalized differentiation revolving around the fundamental extremal principle in variational analysis and its analytic counterpart known as the subdifferential variational principle.

Article information

Source
Taiwanese J. Math., Volume 12, Number 9 (2008), 2569-2592.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405196

Digital Object Identifier
doi:10.11650/twjm/1500405196

Mathematical Reviews number (MathSciNet)
MR2479072

Zentralblatt MATH identifier
1169.49024

Subjects
Primary: 49L52 49J5 90C29: Multi-objective and goal programming 90C48: Programming in abstract spaces

Keywords
mathematical programs with equilibrium constraints variational analysis nonsmooth optimization extremal principle subdifferential variational principle generalized differentiation coderivatives

Citation

Bao, Truong Q.; Mordukhovich, Boris S.; Gupta, Pankaj. SUBOPTIMALITY CONDITIONS FOR MATHEMATICAL PROGRAMS WITH EQUILIBRIUM CONSTRAINTS. Taiwanese J. Math. 12 (2008), no. 9, 2569--2592. doi:10.11650/twjm/1500405196. https://projecteuclid.org/euclid.twjm/1500405196


Export citation

References

  • T. Q. Bao, P. Gupta and B. S. Mordukhovich, Necessary conditions in multiobjective optimization with equilibrium constraints, J. Optim. Theory Appl., 135 (2007), 179-203.
  • T. Q. Bao and B. S. Mordukhovich, Existence of solutions and necessary conditions for set-valued optimization with equilibrium constraints, Appl. Math., 52 (2007), 453-472.
  • J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis, CMS Books in Mathematics 20, Springer, New York, 2005.
  • S. Dempe, J. Dutta and B. S. Mordukhovich, New necessary optimality conditions in optimistic bilevel programming, Optimization, 56 (2007), 577-604.
  • I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
  • M. Fabian, Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss, Acta Univ. Carolina, Ser. Math. Phys., 30 (1989), 51-56.
  • M. L. Flegel, C. Kanzow and J. V. Outrata, Optimality conditions for disjunctive programs with equilibrium constraints, Set-Valued Anal., 15 (2007), 139-162.
  • R. Gabasov, F. M. Kirillova and B. S. Mordukhovich, The $\ve$-maximum principle for suboptimal controls, Soviet Math. Dokl., 27 (1983), 95-99.
  • P. Gupta, S. Shiraishi and K. Yokoyama, Epsilon-optimality without constraint qualifications for multiobjective fractional programs, J. Nonlinear Convex Anal., 6 (2005), 347-357.
  • A. Hamel, An $\ve$-Lagrange multiplier rule for a mathematical programming problem on a Banach space, Optimization, 49 (2001), 137-150.
  • A. D. Ioffe and J.-P. Penot, Subdifferentials of performance functions and calculus of coderivatives of set-valued mappings, Serdica Math. J., 22 (1996), 359-384.
  • P. Q. Khanh and L. M. Luu, Some existence results for vector quasivariational inequality involving multifunctions and applications to traffic equilibrium problems, J. Global Optim., 32 (2005), 551-568.
  • A. B. Levy and B. S. Mordukhovich, Coderivatives in parametric optimization, Math. Prog., 99 (2004), 311-327.
  • P. Loridan and J. Morgan, New results on approximate solutions in two level optimization, Optimization, 20 (1989), 819-836.
  • Z.-Q. Luo, J.-S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, UK, 1996.
  • B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Grundlehren Series (Fundamental Principles of Mathematical Sciences) 330, Springer, Berlin, 2006.
  • B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, II: Applications, Grundlehren Series
  • (Fundamental Principles of Mathematical Sciences) Springer, Berlin, 2006.
  • B. S. Mordukhovich, N. M. Nam and N. D. Yen, Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming, Optimization, 55 (2006), 685-708.
  • B. S. Mordukhovich and J. V. Outrata, Coderivative analysis of quasivariational inequalities with applications to stability and optimization, SIAM J. Optim., 18 (2007), 389-412.
  • B. S. Mordukhovich and Y. Shao, Fuzzy calculus for coderivatives of multifunctions, Nonlinear Anal., 29 (1997), 605-626.
  • B. S. Mordukhovich and B. Wang, Necessary suboptimality and optimality conditions via variational principles, SIAM J. Control Optim., 41 (2002), 623-640.
  • J. V. Outrata, M. Kočhvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.
  • R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd edition, Springer, Berlin, 1993.
  • S. M. Robinson, Generalized equations and their solutions, I: Basic theory, Math. Progr. Study, 10 (1979), 128-141.
  • R. T. Rockafellar and R. J.-B.Wets, Variational Analysis, Grundlehren Series (Fundamental Principles of Mathematical Sciences) 317, Springer, Berlin, 1998.
  • A. Seeger, Approximate Euler-Lagrange inclusion, approximate transversality condition, and sensitivity analysis of convex parametric problems of calculus of variations, Set-Valued Anal., 2 (1994), 307-325.
  • M. I. Sumin, Suboptimal control of a semilinear elliptic equation with a phase constraint and a boundary control, Diff. Eq., 37 (2001), 281-300.
  • J.-C. Yao and O. Chadi, Pseudomonotone complementarity problems and variational inequalities, Handbook of Generalized Convexity and Generalized Monotonicity, pp. 501-558, Nonconvex Optimization, 76, Springer, New York, 2005.
  • J. J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, J. Math. Anal. Appl., 307 (2005), 350-369.
  • L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, Saunders, Philadelphia, Pennsylvania, 1969.