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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: ℝ n → ℝ m, g: ℝ n → ℝ p are locally Lipschitz functions, h: ℝ n → ℝ q is C1 function, and C ⊂ ℝ m and K ⊂ 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 x0 to be a w-minimizer and first-order sufficient conditions for x0 to be an i-minimizer are obtained. Their effectiveness is illustrated on an example. A comparison with some known results is done. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a class of nondifferentiable multiobjective fractional programming problems in which every component of objective functions contains a term involving the support function of a compact convex set. For a differentiable function, we introduce the definition of higher-order $(C,\alpha,\gamma,\rho,d)$-convex function. A nontrivial example is also constructed which is in this class but not $(F,\alpha,\gamma,\rho,d)$-convex. Based on the $(C,\alpha,\gamma,\rho,d)$-convexity, sufficient optimality conditions for an efficient solution of the nondifferentiable multiobjective fractional programming problem are established. Further, a higher-order Mond-Weir type dual is formulated for this problem and appropriate duality results are proved under higher-order $(C,\alpha,\gamma,\rho,d)$-assumptions. [ABSTRACT FROM AUTHOR]

In this paper, a class of functions called B-arcwise connected (BCN) and strictly B-arcwise connected (STBCN) functions are introduced by relaxing definitions of arcwise connected function (CN) and B-vex function. The differential properties of B-arcwise connected function (BCN) are studied. Their two extreme properties are proved. The necessary and sufficient optimality conditions are obtained for the nondifferentiable nonlinear semi-infinite programming involving B-arcwise connected (BCN) and strictly B-arcwise connected (STBCN) functions. Mond-Weir type duality results have also been established. [ABSTRACT FROM AUTHOR]

A new approach for obtaining the second order sufficient conditions for non-linear mathematical programming problems which makes use of second order derivative is presented. In the so-called second order η-approximation method, an optimization problem associated with the original nonlinear programming problem is constructed that involves a second order η-approximation of both the objective function and the constraint function constituting the original problem. The equivalence between the nonlinear original mathematical programming problem and its associated second order η-approximated optimization problem is established under second order invexity assumption imposed on the functions constituting the original optimization problem. [ABSTRACT FROM AUTHOR]

In this work we introduce two approximate duality approaches for vector optimization problems. The first one by means of approximate solutions of a scalar Lagrangian, and the second one by considering $$(C,\varepsilon )$$ -proper efficient solutions of a recently introduced set-valued vector Lagrangian. In both approaches we obtain weak and strong duality results for $$(C,\varepsilon )$$ -proper efficient solutions of the primal problem, under generalized convexity assumptions. Due to the suitable limit behaviour of the $$(C,\varepsilon )$$ -proper efficient solutions when the error $$\varepsilon $$ tends to zero, the obtained duality results extend and improve several others in the literature. [ABSTRACT FROM AUTHOR]

We propose higher-order radial sets and corresponding derivatives of a set-valued map and prove calculus rules for sums and compositions, which are followed by direct applications in discussing optimality conditions for several particular optimization problems. Our main results are both necessary and sufficient higher-order conditions for weak efficiency in a general set-valued vector optimization problem without any convexity assumptions. Many examples are provided to explain advantages of our results over a number of existing ones in the literature. [ABSTRACT FROM AUTHOR]