On the implicit programming approach in a class of mathematical programs with equilibrium constraints

• Autorzy: Outrata J. V. , Cervinka M.
• Języki publikacji: EN

Abstrakty

EN

In the paper we analyze the influence of implicit programming hypothesis and presence of state constraints on first order optimality conditions to mathematical programs with equilibrium constraints. In the absence of state constraints, we derive sharp stationarity conditions, provided the strong regularity condition holds. In the second part of the paper we suggest an exact penalization of state constraints and test the behavior of standard bundle trust region algorithm on academic examples.

Słowa kluczowe

EN

mathematical problem with equilibrium constraint; state constraints; implicit programming; calmness; exact penalization

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2009
• Tom: Vol. 38, no 4B
• Strony: 1557--1574
• Opis fizyczny: Bibliogr. 24 poz.

Twórcy

autor

Outrata J. V.

autor

Cervinka M.

• Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod Vodarenskou vezi 4, Prague 182 08, Czech Republic,

Bibliografia

• BURKE, J.V. (1991) Calmness and exact penalization. SIAM J. Control Op-tim. 29, 493-497.
• CLARKE, F.H. (1983) Optimization and Nonsmooth Analysis. Wiley, New York.
• DEMPE, S. (2002) Foundations of Bi-level Programming. Kluwer Academic Publishers, Dordrecht, The Netherlands.
• DONTCHEV, A.D. and ROCKAFELLAR, R.T. (1996) Characterization of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 7, 1087-1105.
• HENRION, R. and OUTRATA, J.V. (2001) A subdifferential criterion for calmness of multifunctions. J. Math. Anal. Appl. 258, 110-130.
• HENRION, R. and ROEMISCH, W. (2007) On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling. Applications of Mathematics 52, 6, 473-494.
• HENRION, R., JOURANI, A. and OUTRATA, J.V. (2002) On the calmness of a class of multifunctions. SIAM J. Optim. 13, 603-618.
• HENRION, R., OUTRATA, J.V. and SUROWIEC, T. (2009) On the co-derivative of normal cone mappings to inequality systems. Nonlinear Analysis: Theory, Methods and Applications 71, 1213-1226.
• IOFFE, A.D. and OUTRATA, J.V. (2008) On metric and calmness qualification conditions in subdifferential calculus. Set- Valued Analysis 16, 199-227.
• KOCVARA, M. and OUTRATA, J.V. (2004) Optimization problems with equilibrium constraints and their numerical solution. Math. Prog. B 101, 119-150.
• LUO, Z.-Q., PANG, J.-S. and RALPH, D. (1996) Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge, UK.
• MORDUKHOVICH, B.S. (1988) Approximation Methods in Problems of Optimization and Control. Nauka, Moscow (in Russian).
• MORDUKHOVICH, B.S. (1994) Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183, 250-288.
• MORDUKHOVICH, B.S. (2006) Variational Analysis and Generalized Differentiation: I (Basic Theory) and II (Applications). Springer-Verlag, Berlin, Heidelberg.
• MORDUKHOVICH, B.S. and OUTRATA, J.V. (2001) Second order subdifferentials and their applications. SIAM J. Optim. 12, 139-169.
• MORDUKHOVICH, B.S. and OUTRATA, J.V. (2007) Coderivative analysis of quasivariational inequalities with applications to stability and optimization. SIAM J. Optim. 18, 389-412.
• MURPHY, F.H., SHERALI, H.D. and SOYSTER, A.L. (1982) A mathematical programming approach for determining oligopolistic market equilibrium. Math. Prog. 24, 92-106.
• OUTRATA, J.V. (1990) On the numerical solution of a class of Stackelberg problems. Zeit. f. Operationsresearch 34, 255-277.
• OUTRATA, J.V. (1999) Optimality conditions for a class of mathematical programs with equilibrium constraints. Mathematics of Operations Research 24, 627-644.
• OUTRATA, J.V. (2000) Generalized mathematical program with equilibrium constraints. SIAM J. Control Optim. 38, 1623-1638.
• OUTRATA, J.V. and SUN, D. (2008) On the coderivative of the projection operator onto the second-order cone. Set-Valued Analysis 16, 999-1014.
• OUTRATA, J.V., KOCVARA, M. and ZOWE, J. (1998) Nonsmooth Approach to Optimization Problems with Equilibrium Contraints. Kluwer, Dordrecht.
• RALPH, D. and DEMPE, S. (1995) Directional derivatives of the solution of a parametric nonlinear program. Math. Prog. 70, 159-172.
• ROBINSON, S.M. (1980) Strongly regular generalized equations. Mathematics of Operations Research 5, 43-62.
• ROBINSON, S.M. (1987) Local structure of feasible sets in nonlinear programming, Part III: Stability and Sensitivity. Mathematical Programming Study 30, 45-66.
• ROBINSON, S.M. (1991) An implicit-function theorem for a class of nonsmooth functions. Mathematics of Operations Research 16, 292-309.
• ROCKAFELLAR, R.T. and WETS, R. (1998) Variational Analysis. Springer Verlag, Berlin.
• SCHRAMM, H. and ZOWE, J. (1992) A version of bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2, 121-152.
• YE, J.J. (2000) Constraint qualifications and necesary optimality conditions for optimization problems with variational inequality constraints. SIAM J. Optim. 10, 943-962.
• YE, J.J. and YE, X.Y. (1997) Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977-997.