Abstract
The present paper studies the following constrained vector optimization problem: \(\mathop {\min }\limits_C f(x),g(x) \in - K,h(x) = 0\), where f: ℝn → ℝm, g: ℝn → ℝp are locally Lipschitz functions, h: ℝn → ℝq is C 1 function, and C ⊂ ℝm and K ⊂ ℝp are closed convex cones. Two types of solutions are important for the consideration, namely w-minimizers (weakly efficient points) and i-minimizers (isolated minimizers of order 1). In terms of the Dini directional derivative first-order necessary conditions for a point x 0 to be a w-minimizer and first-order sufficient conditions for x 0 to be an i-minimizer are obtained. Their effectiveness is illustrated on an example. A comparison with some known results is done.
Similar content being viewed by others
References
B. Aghezzaf, M. Hachimi: Second-order optimality conditions in multiobjective optimization problems. J. Optim. Theory Appl. 102 (1999), 37–50.
T. Amahroq, A. Taa: On Lagrange-Kuhn-Tucker multipliers for multiobjective optimization problems. Optimization 41 (1997), 159–172.
T. Antczak, K. Kisiel: Strict minimizers of order m in nonsmooth optimization problems. Commentat. Math. Univ. Carol. 47 (2006), 213–232.
A. Auslender: Stability in mathematical programming with nondifferentiable data. SIAM J. Control Optim. 22 (1984), 239–254.
D. Bednařík, K. Pastor: On second-order conditions in unconstrained optimization. Math. Program. 113 (2008), 283–298.
A. Ben-Tal, J. Zowe: A unified theory of first and second order conditions for extremum problems in topological vector spaces. Math. Program. Study 18 (1982), 39–76.
F.H. Clarke: Optimization and Nonsmooth Analysis. John Wiley & Sons, New York, 1983.
B.D. Craven: Nonsmooth multiobjective programming. Numer. Funct. Anal. Optim. 10 (1989), 49–64.
I. Ginchev, A. Guerraggio, M. Rocca: First-order conditions for C0,1 constrained vector optimization. In: Variational Analysis and Applications. Proc. 38th Conference of the School of Mathematics “G. Stampacchia” in Memory of G. Stampacchia and J.-L. Lions, Erice, Italy, June 20–July 1, 2003 (F. Giannessi, A. Maugeri, eds.). Springer, New York, 2005, pp. 427–450.
I. Ginchev, A. Guerraggio, M. Rocca: Second-order conditions in C1,1 constrained vector optimization. Math. Program., Ser. B 104 (2005), 389–405.
I. Ginchev, A. Guerraggio, M. Rocca: From scalar to vector optimization. Appl. Math. 51 (2006), 5–36.
I. Ginchev, A. Guerraggio, M. Rocca: Second-order conditions in C1,1 vector optimization with inequality and equality constraints. In: Recent Advances in Optimization. Proc. 12th French-German-Spanish Conference on Optimization, Avignon, France, September 20–24, 2004. Lecture Notes in Econom. and Math. Systems, Vol. 563 (A. Seeger, ed.). Springer, Berlin, 2006, pp. 29–44.
Z. Li: The optimality conditions of differentiable vector optimization problems. J. Math. Anal. Appl. 201 (1996), 35–43.
L. Liu, P. Neittaanmäki, M. Křížek: Second-order optimality conditions for nondominated solutions of multiobjective programming with C1,1 data. Appl. Math. 45 (2000), 381–397.
C. Malivert: First and second order optimality conditions in vector optimization. Ann. Sci. Math. Qué. 14 (1990), 65–79.
I. Maruşciac: On Fritz John type optimality criterion in multi-objective optimization. Anal. Numér. Théor. Approximation 11 (1982), 109–114.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ginchev, I., Guerraggio, A. & Rocca, M. Locally Lipschitz vector optimization with inequality and equality constraints. Appl Math 55, 77–88 (2010). https://doi.org/10.1007/s10492-010-0003-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10492-010-0003-y