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The generalized Mangasarian-Fromowitz constraint qualification and optimality conditions for bilevel programs. (English) Zbl 1223.90061

The authors introduce a bilevel programming problem in terms of an upper level problem

minF(x,y)s.t.(x,y)X× m ,yΨ(x)( ULP )

where X is a closed subset of n and Ψ(x) is the solution set of the lower level problem defined as:

minf(x,y)s.t.yK(x)·

The functions F,f: n × m are continuous, X={xG(x)0,H(x)=0}, K(x)={yg(x,y)0,h(x,y)=0}, where the functions G: n k , H: n l , g: n × m p , h: n × m q are all continuous. Assuming feasibility of (ULP) the optimal value reformulation is

minF(x,y)s.t.xX,yK(x),f(x,y)φ(x)( OV )

where φ(x) is defined as: φ(x)=min{f(x,y)yK(x)}.

The authors contend that the problems (ULP) and (OV) are locally and globally equivalent. It is shown that the Mangasarian-Fromowitz constraint qualification in terms of basic generalized differentiation constructions of Mordukhovich fails and a weakened form of the same is used to derive Karush-Kuhn-Tucker optimality conditions for (ULP). They also suggest new sufficient conditions for the partial calmness based on a more general notion of the weak sharp minimum.

MSC:
90C30Nonlinear programming
90C46Optimality conditions, duality
References:
[1]Dempe, S., Zemkoho, A.B.: The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Submitted for publication
[2]Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic, Dordrecht (2002)
[3]Dempe, S., Dutta, J., Lohse, S.: Optimality conditions for bilevel programming problems. Optimization 55, 505–524 (2006) · Zbl 1156.90457 · doi:10.1080/02331930600816189
[4]Dempe, S., Kalashnikov, V.V., Kalashnykova, N.: Optimality conditions for bilevel programming problems. In: Dempe, S., Kalashnikov, V.V. (eds.) Optimization with Multivalued Mappings, vol. 2, pp. 3–28. Springer, New York (2006)
[5]Dutta, J., Dempe, S.: Bilevel programming with convex lower level problems. In: Dempe, S., Kalashnikov, V.V. (eds.) Optimization with Multivalued Mappings, vol. 2, pp. 51–71. Springer, New York (2006)
[6]Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995) · Zbl 0820.65032 · doi:10.1080/02331939508844060
[7]Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM Classics in Applied Mathematics, vol. 5. Wiley, New York (1983). Reprint, Philadelphia (1994)
[8]Ye, J.J., Zhu, D.L.: A note on optimality conditions for bilevel programming problems. Optimization 39, 361–366 (1997) · Zbl 0879.90173 · doi:10.1080/02331939708844290
[9]Ye, J.J.: Nondifferentiable multiplier rules for optimization and bilevel optimization problems. SIAM J. Optim. 15, 252–274 (2004) · Zbl 1077.90077 · doi:10.1137/S1052623403424193
[10]Ye, J.J.: Constraint qualifications and KKT conditions for bilevel programming problems. Math. Oper. Res. 31, 811–824 (2006) · Zbl 05279704 · doi:10.1287/moor.1060.0219
[11]Babahadda, H., Gadhi, N.: Necessary optimality conditions for bilevel optimization problems using convexificators. J. Glob. Optim. 34, 535–549 (2006) · Zbl 1090.49021 · doi:10.1007/s10898-005-1650-5
[12]Amahroq, T., Gadhi, N.: On the regularity condition for vector programming problems. J. Glob. Optim. 21, 433–441 (2001) · Zbl 1175.90409 · doi:10.1023/A:1012748412618
[13]Dempe, S., Dutta, J., Mordukhovich, B.S.: New necessary optimality conditions in optimistic bilevel programming. Optimization 56, 577–604 (2007) · Zbl 1172.90481 · doi:10.1080/02331930701617551
[14]Henrion, R., Outrata, J.: A subdifferential condition for calmness of multifunctions. J. Math. Anal. Appl. 258, 110–130 (2001) · Zbl 0983.49010 · doi:10.1006/jmaa.2000.7363
[15]Henrion, R., Jourani, A., Outrata, J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13, 603–618 (2002) · Zbl 1028.49018 · doi:10.1137/S1052623401395553
[16]Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory, II: Applications. Springer, Berlin (2006)
[17]Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
[18]Zheng, X.Y., Ng, K.F.: Calmness for L-subsmooth multifunctions in Banach spaces. SIAM J. Optim. 19, 1648–1673 (2009) · Zbl 1188.49019 · doi:10.1137/080714129
[19]Mangasarian, O.L., Fromowitz, S.: The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967) · Zbl 0149.16701 · doi:10.1016/0022-247X(67)90163-1
[20]Gauvin, J., Dubeau, F.: Differential properties of the marginal function in mathematical programming. Math. Program. Stud. 18, 101–119 (1982)
[21]Mordukhovich, B.S., Nam, N.M.: Variational stability and marginal functions via generalized differentiation. Math. Oper. Res. 30, 800–816 (2005) · Zbl 05279643 · doi:10.1287/moor.1050.0147
[22]Klatte, D., Kummer, B.: Stability properties of infima and optimal solutions of parametric optimization problems. Nondifferentiable optimization: motivations and applications. In: Proc. IIASA Workshop, Sopron/Hung, 1984. Lect. Notes Econ. Math. Syst, vol. 255, pp. 215–229. Springer, Berlin (1985)
[23]Rockafellar, R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183–238 (1993) · Zbl 0779.49024 · doi:10.1137/1035044
[24]Mangasarian, O.L.: Nonlinear Programming. SIAM Classics in Applied Mathematics, vol. 10. McGraw-Hill, New York (1969). Reprint, Philadelphia (1994)
[25]Dempe, S., Dinh, N., Dutta, J.: Optimality conditions for a simple convex bilevel programming problem. In: Burachik, R.S., Yao, J.-C. (eds.) Variational Analysis and Generalized Differentiation in Optimization and Control. Springer, Berlin (2010)
[26]Dempe, S., Zemkoho, A.B.: On the Karush-Kuhn-Tucker reformulation of the bilevel optimization problem. Submitted for publication
[27]Henrion, R., Outrata, J., Surowiec, T.: On the co-derivative of normal cone mappings to inequality systems. Nonlinear Anal. 71, 1213–1226 (2009) · Zbl 1176.90568 · doi:10.1016/j.na.2008.11.089
[28]Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. I: Sufficient optimality conditions. J. Optim. Theory Appl. 142, 147–163 (2009) · Zbl 1198.90391 · doi:10.1007/s10957-009-9518-3
[29]Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. II: Necessary optimality conditions. J. Optim. Theory Appl. 142, 165–183 (2009) · Zbl 1205.90294 · doi:10.1007/s10957-009-9521-8
[30]Dempe, S., Zemkoho, A.B.: Bilevel road pricing: Theoretical analysis and optimality conditions. Submitted for publication
[31]Ye, J.J.: New uniform parametric error bounds. J. Optim. Theory Appl. 98, 197–219 (1998) · Zbl 0908.90248 · doi:10.1023/A:1022649217032
[32]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
[33]Heerda, J., Kummer, B.: Characterization of calmness for Banach space mappings. Preprint, Humboldt-Universität zu Berlin (2006)
[34]Kojima, M.: Strongly stable stationary solutions in nonlinear programs. Analysis and computation of fixed points. Proc. Symp., Univ. Wis. 1979, 93–138 (1980)
[35]Ralph, D., Dempe, S.: Directional derivatives of the solution of a parametric nonlinear program. Math. Program. 70, 159–172 (1995)
[36]Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183, 250–288 (1994) · Zbl 0807.49016 · doi:10.1006/jmaa.1994.1144
[37]Dempe, S., Vogel, S.: The generalized Jacobian of the optimal solution in parametric optimization. Optimization 50, 387–405 (2001) · Zbl 1054.90074 · doi:10.1080/02331930108844570