First and second order sufficient conditions for strict minimality in nonsmooth vector optimization. (English) Zbl 1033.90120

Summary: We present first and second order sufficient conditions for strict local minima of orders 1 and 2 to vector optimization problems with an arbitrary feasible set and a twice directionally differentiable objective function. With this aim, the notion of support function to a vector problem is introduced, in such a way that the scalar case and the multiobjective case, in particular, are contained. The obtained results extend the multiobjective ones to this case. Moreover, specializing to a feasible set defined by equality, inequality, and set constraints, first and second order sufficient conditions by means of Lagrange multiplier rules are established.


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


[1] Ben-Tal, A.; Zowe, J., A unified theory of first and second order conditions for extremum problems in topological vector spaces, Math. Programming Study, 19, 39-76 (1982) · Zbl 0494.49020
[2] Ben-Tal, A.; Zowe, J., Directional derivatives in nonsmooth optimization, J. Optim. Theory Appl., 47, 483-490 (1985) · Zbl 0556.90074
[3] Bigi, G.; Castellani, M., Second order optimality conditions for differentiable multiobjective problems, RAIRO Oper. Res., 34, 411-426 (2000) · Zbl 1039.90063
[4] Borwein, J. M., Stability and regular points of inequality systems, J. Optim. Theory Appl., 48, 9-52 (1986) · Zbl 0557.49020
[5] Cominetti, R., Metric regularity, tangent sets, and second-order optimality conditions, Appl. Math. Optim., 21, 265-287 (1990) · Zbl 0692.49018
[6] Corley, H. W., On optimality conditions for maximizations with respect to cones, J. Optim. Theory Appl., 46, 67-78 (1985) · Zbl 0542.90088
[7] Demyanov, V. F.; Rubinov, A. M., Constructive Nonsmooth Analysis (1995), Peter Lang: Peter Lang Frankfurt am Main · Zbl 0887.49014
[8] Flett, T. M., Differential Analysis (1980), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0442.34002
[9] Hestenes, M. R., Optimization Theory. The Finite Dimensional Case (1981), Krieger: Krieger New York
[10] Hiriart-Urruty, J. B.; Seeger, A., Calculus rules on a new set-valued second order derivative for convex functions, Nonlinear Anal., 13, 721-738 (1989) · Zbl 0705.26013
[11] Hoffmann, K. H.; Kornstaedt, H. J., Higher-order necessary conditions in abstract mathematical programming, J. Optim. Theory Appl., 26, 533-568 (1978) · Zbl 0373.90066
[12] Jiménez, B., Strict efficiency in vector optimization, J. Math. Anal. Appl., 265, 264-284 (2002) · Zbl 1010.90075
[13] Jiménez, B., Strict minimality conditions in nondifferentiable multiobjective programming, J. Optim. Theory Appl., 116, 99-116 (2003) · Zbl 1030.90116
[14] Jiménez, B.; Novo, V., A notion of local proper efficiency in Borwein sense in vector optimization, Aust. N. Z. Indust. Appl. Math. J., 45, 75-89 (2003) · Zbl 1161.90478
[15] Jiménez, B.; Novo, V., First and second order sufficient conditions for strict minimality in multiobjective programming, Numer. Funct. Anal. Optim., 23, 303-322 (2002) · Zbl 1025.90023
[16] Linnemann, A., Higher-order necessary conditions for infinite optimization, J. Optim. Theory Appl., 38, 483-512 (1982) · Zbl 0471.49019
[17] Maruyama, Y., Second-order necessary conditions for nonlinear optimization problems in Banach spaces by the use of Neustadt derivative, Math. Japon., 40, 509-522 (1994) · Zbl 0816.49017
[18] Studniarski, M., Necessary and sufficient conditions for isolated local minima of nonsmooth functions, SIAM J. Control Optim., 24, 1044-1049 (1986) · Zbl 0604.49017
[19] Tang, Y., Conditions for constrained efficient solutions of multiobjective problems in Banach spaces, J. Math. Anal. Appl., 96, 505-519 (1983) · Zbl 0527.49018
[20] Ward, D. E., Characterizations of strict local minima and necessary conditions for weak sharp minima, J. Optim. Theory Appl., 80, 551-571 (1994) · Zbl 0797.90101
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.