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Locally Lipschitz vector optimization with inequality and equality constraints. (English) Zbl 1224.90154
Summary: The present paper studies the following constrained vector optimization problem: $$\min _Cf(x)$$, $$g(x)\in -K$$, $$h(x)=0$$, where $$f\:\mathbb R^n\to \mathbb R^m$$, $$g\:\mathbb R^n\to \mathbb R^p$$ are locally Lipschitz functions, $$h\:\mathbb R^n\to \mathbb R^q$$ is $$C^1$$ function, and $$C\subset \mathbb R^m$$ and $$K\subset \mathbb R^p$$ 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.

##### MSC:
 90C29 Multi-objective and goal programming 90C30 Nonlinear programming 90C46 Optimality conditions and duality in mathematical programming 49J52 Nonsmooth analysis
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##### References:
 [1] B. Aghezzaf, M. Hachimi: Second-order optimality conditions in multiobjective optimization problems. J. Optim. Theory Appl. 102 (1999), 37–50. · Zbl 1039.90062 [2] T. Amahroq, A. Taa: On Lagrange-Kuhn-Tucker multipliers for multiobjective optimization problems. Optimization 41 (1997), 159–172. · Zbl 0882.90114 [3] T. Antczak, K. Kisiel: Strict minimizers of order m in nonsmooth optimization problems. Commentat. Math. Univ. Carol. 47 (2006), 213–232. · Zbl 1150.90007 [4] A. Auslender: Stability in mathematical programming with nondifferentiable data. SIAM J. Control Optim. 22 (1984), 239–254. · Zbl 0538.49020 [5] D. Bednařík, K. Pastor: On second-order conditions in unconstrained optimization. Math. Program. 113 (2008), 283–298. · Zbl 1211.90276 [6] 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. · Zbl 0494.49020 [7] F.H. Clarke: Optimization and Nonsmooth Analysis. John Wiley & Sons, New York, 1983. · Zbl 0582.49001 [8] B.D. Craven: Nonsmooth multiobjective programming. Numer. Funct. Anal. Optim. 10 (1989), 49–64. · Zbl 0645.90076 [9] I. Ginchev, A. Guerraggio, M. Rocca: First-order conditions for C 0,1 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. · Zbl 1148.90011 [10] I. Ginchev, A. Guerraggio, M. Rocca: Second-order conditions in C 1,1 constrained vector optimization. Math. Program., Ser. B 104 (2005), 389–405. · Zbl 1102.90058 [11] I. Ginchev, A. Guerraggio, M. Rocca: From scalar to vector optimization. Appl. Math. 51 (2006), 5–36. · Zbl 1164.90399 [12] I. Ginchev, A. Guerraggio, M. Rocca: Second-order conditions in C 1,1 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. · Zbl 1101.90061 [13] Z. Li: The optimality conditions of differentiable vector optimization problems. J. Math. Anal. Appl. 201 (1996), 35–43. · Zbl 0862.35048 [14] L. Liu, P. Neittaanmäki, M. Křížek: Second-order optimality conditions for nondominated solutions of multiobjective programming with C 1,1 data. Appl. Math. 45 (2000), 381–397. · Zbl 0995.90085 [15] C. Malivert: First and second order optimality conditions in vector optimization. Ann. Sci. Math. Qué. 14 (1990), 65–79. · Zbl 0722.90065 [16] I. Maruşciac: On Fritz John type optimality criterion in multi-objective optimization. Anal. Numér. Théor. Approximation 11 (1982), 109–114.
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