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Is bilevel programming a special case of a mathematical program with complementarity constraints? (English) Zbl 1235.90145
Summary: Bilevel programming problems are often reformulated using the Karush-Kuhn-Tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints (MPCC). Clearly, both problems are closely related. But the answer to the question posed is “no” even in the case when the lower level programming problem is a parametric convex optimization problem. This is not obvious and concerns local optimal solutions. We show that global optimal solutions of the MPCC correspond to global optimal solutions of the bilevel problem provided the lower-level problem satisfies the Slater’s constraint qualification. We also show by examples that this correspondence can fail if the Slater’s constraint qualification fails to hold at lower-level. When we consider the local solutions, the relationship between the bilevel problem and its corresponding MPCC is more complicated. We also demonstrate the issues relating to a local minimum through examples.

MSC:
90C30 Nonlinear programming
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[1] Anitescu M.: On using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints. SIAM J. Optim. 15, 1203–1236 (2006) · Zbl 1097.90050 · doi:10.1137/S1052623402401221
[2] Bard J.F.: Practical Bilevel Optimization: Algorithms and Applications. Kluwer, Dordrecht (1998) · Zbl 0943.90078
[3] DeMiguel, A.-V., Friedlander, M.P., Nogales, F.J., Scholtes, S.: An interior-point method for MPECS based on strictly feasible relaxations. Technical report, Department of Decision Sciences, London Business School (2004)
[4] Dempe S.: Foundations of Bilevel Programming. Kluwer, Dordrecht (2002) · Zbl 1038.90097
[5] Dempe S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52, 333–359 (2003) · Zbl 1140.90493 · doi:10.1080/0233193031000149894
[6] Dempe, S., Kalashnikov, V. (eds): Optimization with Multivalued Mappings: Theory, Applications and Algorithms. Springer, Berlin (2006) · Zbl 1097.90001
[7] Fukushima, M., Pang, J.-S.: Convergence of a smoothing continuation method for mathematical programs with complementarity constraints Ill-posed Variational Problems and Regularization Techniques, no. 477, Springer, Berlin (1999) · Zbl 0944.65070
[8] Gauvin J.: A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming. Math. Program. 12, 136–139 (1977) · Zbl 0354.90075 · doi:10.1007/BF01593777
[9] Jongen H.Th., Weber G.-W.: Nonlinear optimization: characterization of structural optimization. J. Glob. Optim. 1, 47–64 (1991) · Zbl 0745.90067 · doi:10.1007/BF00120665
[10] Kojima M.: Strongly stable stationary solutions in nonlinear programs. In: Robinson, S.M. (eds) Analysis and Computation of Fixed Points, pp. 93–138. Academic Press, New York (1980)
[11] Luo Z.-Q., Pang J.-S., Ralph D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)
[12] Mersha, A.G.: Solution methods for bilevel programming problems. Ph.D. thesis, TU Bergakademie Freiberg (2008)
[13] Migdalas, A., Pardalos, P.M., Värbrand, P. (eds): Multilevel Optimization: Algorithms and Applications. Kluwer, Dordrecht (1998)
[14] Mirrlees J.A.: The theory of moral hazard and unobservable bevaviour: part I. Rev. Econ. Stud. 66, 3–21 (1999) · Zbl 0958.91027 · doi:10.1111/1467-937X.00075
[15] Outrata J.: On the numerical solution of a class of Stackelberg problems. ZOR Math. Methods Oper. Res. 34, 255–277 (1990) · Zbl 0714.90077 · doi:10.1007/BF01416737
[16] Outrata J., Kočvara M., Zowe J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht (1998) · Zbl 0947.90093
[17] Robinson S.M.: Generalized equations and their solutions, part II: applications to nonlinear programming. Math. Program. Stud. 19, 200–221 (1982) · Zbl 0495.90077 · doi:10.1007/BFb0120989
[18] Scholtes S., Stöhr M.: How stringent is the linear independence assumption for mathematical programs with stationarity constraints?. Math. Oper. Res. 26, 851–863 (2001) · Zbl 1082.90580 · doi:10.1287/moor.26.4.851.10007
[19] Ye J.J., Zhu D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995) · Zbl 0820.65032 · doi:10.1080/02331939508844060
[20] Ye J.J., Zhu D.L., Zhu Q.J.: Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. 7, 481–507 (1997) · Zbl 0873.49018 · doi:10.1137/S1052623493257344
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