×

Optimality conditions for vector optimization problems. (English) Zbl 1180.90289

The paper considers vector optimization problems (VOP) of the form \[ \min F(x) \text{ subject to } u_i(x) \leq 0, i\in\{1, \dots, m\}; v_j(x) =0, j\in\{1, \dots, n\}, \] where \(F:X \rightarrow \mathbb{R}^L\) and \(u_i, v_j: X\rightarrow \mathbb{R}\) and \(X\) is a Banach space.
Using nondifferentiable Abadie or generalized Zangwill constraint qualifications the authors derive necessary KKT conditions for weakly effficient solutions of (VOP) and sufficient conditions under Michel-Penot pseudoconvexity of \(F\) and Michel-Penot quasiconvexity of the constraints. Results for the special case of linear problems are also explicitely stated. Finally, a partial calmness condition is introduced and it is shown that under this condition weakly efficient solutions of the (VOP) and local minimia of a scalar penalty problem related to (VOP) are equivalent.

MSC:

90C29 Multi-objective and goal programming
90C46 Optimality conditions and duality in mathematical programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aghezzaf, B., Hachimi, M.: Generalized invexity and duality in multiobjective programming problems. J. Glob. Optim. 18, 91–101 (2000) · Zbl 0970.90087 · doi:10.1023/A:1008321026317
[2] Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-valued and Variational Analysis. Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005) · Zbl 1104.90044
[3] Chinchuluun, A., Pardalos, P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, 29–50 (2007) · Zbl 1146.90060 · doi:10.1007/s10479-007-0186-0
[4] Deng, S.: Characterizations of the nonemptiness and compactness of solution sets in convex vector optimization. J. Optim. Theory Appl. 96, 123–131 (1998) · Zbl 0897.90163 · doi:10.1023/A:1022615217553
[5] Flores-Bazãn, F.: Ideal, weakly efficient solutions for vector optimization problems. Math. Program. Ser. A 93, 453–475 (2002) · Zbl 1023.90056 · doi:10.1007/s10107-002-0311-4
[6] Liang, Z.A., Huang, H.X., Pardalos, P.M.: Efficiency conditions and duality for a class of multiobjective fractional programming problems. J. Glob. Optim. 27, 447–471 (2003) · Zbl 1106.90066 · doi:10.1023/A:1026041403408
[7] Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989) · Zbl 0688.90051
[8] Yuan, D.H., Chinchuluun, A., Liu, X.L., Pardalos, P.M.: Optimality conditions and duality for multiobjective programming involving (C;{\(\alpha\)};{\(\rho\)};d)-type I functions. In: Konnov, I.V., Luc, D.T., Rubinov, A.M. (eds.) Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, vol. 583, pp. 73–87. Springer, Berlin (2007) · Zbl 1132.49027
[9] Deng, S., Yang, X.Q.: Weak sharp minima in multicriteria linear programming. SIAM J. Optim. 15, 456–460 (2004) · Zbl 1114.90111 · doi:10.1137/S1052623403434401
[10] Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983) · Zbl 0582.49001
[11] Halkin, H.: Implicit functions and optimization problems without continuous differentiability of the data. SIAM J. Control 12, 229–236 (1974) · Zbl 0285.90070 · doi:10.1137/0312017
[12] Ioffe, A.D.: Necessary conditions in nonsmooth optimization. Math. Oper. Res. 9, 159–189 (1984) · Zbl 0548.90088 · doi:10.1287/moor.9.2.159
[13] Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969). Reprinted as Classics. Applied Mathematics, vol. 10. SIAM, Philadelphia (1994) · Zbl 0194.20201
[14] Mordukhovich, B.S.: On necessary conditions for an extremum in nonsmooth optimization. Sov. Math. Dokl. 283, 215–220 (1985) · Zbl 0586.49011
[15] Michel, P., Penot, J.-P.: Calcus sous-différentiel pour des fonctions Lipschitziennes et non Lipschitziennes. C. R. Acad. Sci. Paris Ser. I Math. 12, 269–272 (1984) · Zbl 0567.49008
[16] Michel, P., Penot, J.-P.: A generalized derivative for calm and stable functions. Diff. Integral Equ. 5, 433–454 (1992) · Zbl 0787.49007
[17] Treiman, J.S.: Lagrange multipliers for nonconvex generalized gradients with equality, inequality, and set constraints. SIAM J. Control Optim. 37, 1313–1329 (1999) · Zbl 0955.90126 · doi:10.1137/S0363012996306595
[18] 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
[19] 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
[20] Rockafellar, R.T.: Proximal subgradient, marginal functions, and augmented Lagrangians in nonsmooth optimization. Math. Oper. Res. 6, 427–437 (1981) · Zbl 0492.90073 · doi:10.1287/moor.6.3.424
[21] Mordukhovich, B.S.: Approximation Methods in Problems of Optimization and Control. Nauka, Moscow (1988). English translation, Wiley/Interscience · Zbl 0643.49001
[22] Birge, J.R., Qi, L.: Semiregularity and generalized subdifferentials with applications to optimization. Math. Oper. Res. 18, 982–1005 (1993) · Zbl 0806.49013 · doi:10.1287/moor.18.4.982
[23] Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) · Zbl 0193.18401
[24] Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995) · Zbl 0820.65032 · doi:10.1080/02331939508844060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.