Existence and boundedness of the Kuhn-Tucker multipliers in nonsmooth multiobjective optimization. (English) Zbl 1201.90181
Given a nonsmooth multiobjective optimization problem with inequality constraints and an arbitrary set constraint, constraint qualifications are suggested that are both necessary and sufficient for the Kuhn-Tucker multiplier set to be nonempty and bounded at locally weak efficient solutions where the objective and constraint functions are locally Lipschitz. The constraint qualifications are based on upper convexificators satisfying some semiregularity conditions, generalizing the Clarke subdifferentials and the Michel-Penot subdifferentials of a locally Lipschitz function. For optimization problems without equality constraints, the constraint qualifications are of the Mangasarian-Fromovitz type.
The results are accompanied by several examples showing, among others, that, while the suggested constraint qualifications based on upper convexificators are satisfied at a locally weak efficient solution, they may fail to hold when the convexificators are replaced by the Clarke subdifferentials or the Michel-Penot subdifferentials.
The results are accompanied by several examples showing, among others, that, while the suggested constraint qualifications based on upper convexificators are satisfied at a locally weak efficient solution, they may fail to hold when the convexificators are replaced by the Clarke subdifferentials or the Michel-Penot subdifferentials.
Reviewer: Kathrin Klamroth (Wuppertal)
MSC:
| 90C29 | Multi-objective and goal programming |
| 90C46 | Optimality conditions and duality in mathematical programming |
Keywords:
upper convexificators; constraint qualifications; existence and boundedness of Kuhn-Tucker multipliers; nonsmooth multiobjective optimizationReferences:
| [1] | Gauvin, J., Tolle, J.W.: Differentiable stability in nonlinear programming. SIAM J. Control Optim. 15, 294–311 (1977) · Zbl 0374.90062 · doi:10.1137/0315020 |
| [2] | Gauvin, J.: A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming. Math. Program. 12, 136–138 (1977) · Zbl 0354.90075 · doi:10.1007/BF01593777 |
| [3] | Mangasarian, O.L., Fromovitz, S.: The Fritz-John necessary optimality condition in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1976) · Zbl 0149.16701 · doi:10.1016/0022-247X(67)90163-1 |
| [4] | Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) · Zbl 0582.49001 |
| [5] | Nguyen, V.H., Strodiot, J.-J., Mifflin, R.: On conditions to have bounded multipliers in locally Lipschitz programming. Math. Program. 18, 100–106 (1980) · Zbl 0437.90075 · doi:10.1007/BF01588302 |
| [6] | Pappalardo, M.: Error bounds for generalized Lagrange multipliers in locally Lipschitz programming. J. Optim. Theory Appl. 73, 205–210 (1992) · Zbl 0792.90059 · doi:10.1007/BF00940086 |
| [7] | Jourani, A.: Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems. J. Optim. Theory Appl. 81, 533–548 (1994) · Zbl 0810.90136 · doi:10.1007/BF02193099 |
| [8] | Demyanov, V.F.: Convexification and concavification of positively homogeneous functions by the same family of linear functions. Technical Report, Univ. of Pisa, No. 3.208.802, pp. 1–11 (1994) |
| [9] | Demyanov, V.F., Jeyakumar, V.: Hunting for smaller convex subdifferential. J. Glob. Optim. 10, 305–326 (1997) · Zbl 0872.90083 · doi:10.1023/A:1008246130864 |
| [10] | Jeyakumar, V., Luc, D.T.: Approximate Jacobian matrices for nonsmooth continuous maps and C 1-optimization. SIAM J. Control Optim. 36, 1815–1832 (1998) · Zbl 0923.49012 · doi:10.1137/S0363012996311745 |
| [11] | Jeyakumar, V., Luc, D.T.: Nonsmooth calculus, minimality, and monotonicity of convexificators. J. Optim. Theory Appl. 101, 599–621 (1999) · Zbl 0956.90033 · doi:10.1023/A:1021790120780 |
| [12] | Jeyakumar, V., Wang, W.: Approximate Hessian matrices and second-order optimality conditions for nonlinear programming problem with C 1-data. J. Aust. Math. Soc. B 40, 403–420 (1999) · Zbl 0954.90053 · doi:10.1017/S0334270000010985 |
| [13] | Jeyakumar, V., Yang, X.Q.: Approximate generalized Hessians and Taylor’s expansions for continuously Gateaux differentiable functions. Nonlinear Anal., Theory Methods Appl. 36, 353–368 (1999) · Zbl 0931.49011 · doi:10.1016/S0362-546X(98)00058-3 |
| [14] | Jeyakumar, V., Luc, D.T., Schaible, S.: Characterizations of generalized monotone nonsmooth continuous maps using approximate Jacobians. J. Convex Anal. 5, 119–132 (1998) · Zbl 0911.49014 |
| [15] | Dutta, J., Chandra, S.: Convexificators, generalized convexity, and optimality conditions. J. Optim. Theory Appl. 113, 41–64 (2002) · Zbl 1172.90500 · doi:10.1023/A:1014853129484 |
| [16] | Li, X.F., Zhang, J.Z.: Stronger Kuhn-Tucker type conditions in nonsmooth multiobjective optimization: locally Lipschitz case. J. Optim. Theory Appl. 127, 367–388 (2005) · Zbl 1116.90093 · doi:10.1007/s10957-005-6550-9 |
| [17] | Li, X.F., Zhang, J.Z.: Necessary optimality conditions in terms of convexificators in Lipschitz optimization. J. Optim. Theory Appl. 132 (2007) · Zbl 1143.90035 |
| [18] | Uderzo, A.: Convex approximators, convexificators, and exhausters: application to constrained extremum problem. In: Quasidifferentiability and Related Topics, Nonconvex Optimization and Application, vol. 43, pp. 297–327. Kluwer Academic, Dordrecht (2000) · Zbl 1016.90054 |
| [19] | Wang, X., Jeyakumar, V.: A sharp Lagrange multiplier rule for nonsmooth mathematical programming problems involving equality constraints. SIAM J. Optim. 10, 1136–1148 (2000) · Zbl 1047.90075 · doi:10.1137/S1052623499354540 |
| [20] | Luc, D.T.: A multiplier rule for multiobjective programming problems with continuous data. SIAM J. Optim. 13, 168–178 (2002) · Zbl 1055.90063 · doi:10.1137/S1052623400378286 |
| [21] | Bazaraa, M.S., Goode, J.J., Shetty, C.M.: Constraint qualifications revised. Manag. Sci. 18, 567–573 (1972) · Zbl 0257.90049 · doi:10.1287/mnsc.18.9.567 |
| [22] | Martin, D.H., Watkins, G.G.: Cores of tangent cones and Clarke’s tangent cone. Math. Oper. Res. 10, 565–575 (1985) · Zbl 0584.49008 · doi:10.1287/moor.10.4.565 |
| [23] | Ward, D.E.: Convex subcones of the contingent cone in nonsmooth calculus and optimization. Transl. Am. Math. Soc. 302, 661–682 (1987) · Zbl 0629.58007 · doi:10.1090/S0002-9947-1987-0891640-2 |
| [24] | Ward, D.E., Lee, G.M.: Generalized properly efficient solutions of vector optimization problems. Math. Methods Oper. Res. 53, 215–232 (2001) · Zbl 1031.90039 · doi:10.1007/s001860100112 |
| [25] | Michel, P., Penot, J.P.: A generalized derivative for calm and stable functions. Differ. Integral Equ. 5, 433–454 (1992) · Zbl 0787.49007 |
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