Angelo Guerraggio

The Notion of Invexity in Vector Optimization: Smooth and Nonsmooth Case

After a survey of the main definitions of generalized convex and generalized invex vector functions, some other broad classes of generalized invex vector functions are introduced, both in the differentiable case and in the nonsmooth case. With reference to the said functions we extend some results of weak efficiency, efficiency and duality.

Various types of nonsmooth invex functions

Abstract The aim of this paper is to consider various proposals for the extension to a nonsmooth setting of the class of invex functions. In particular we consider functions endowed with generalized directional derivatives in the sense of Clarke, Demyanov and Rubinov, Pshenichnyi, Jeyakumar and Ye. From these extensions we derive some “parallel”, results with respect to the differentiable case and show the various relationships between the nonsmooth invex functions considered.

First-Order Conditions for C0,1 Constrained Vector Optimization

For a Fritz John type vector optimization problem with C0,1 data we define different type of solutions, give their scalar characterizations applying the so called oriented distance, and give necessary and sufficient first order optimality conditions in terms of the Dini derivative. While establishing the sufficiency, we introduce new type of efficient points referred to as isolated minimizers of first order, and show their relation to properly efficient points. More precisely, the obtained necessary conditions are necessary for weakly efficiency, and the sufficient conditions are both sufficient and necessary for a point to be an isolated minimizer of first order.

Second-order conditions in C 1,1 constrained vector optimization

We consider the constrained vector optimization problem minCf(x), g(x) ∈ −K, where f:ℝn→ℝm and g:ℝn→ℝp are C1,1 functions, and Cℝm and Kℝp are closed convex cones with nonempty interiors. Two type of solutions are important for our considerations, namely w-minimizers (weakly efficient points) and i-minimizers (isolated minimizers). We formulate and prove in terms of the Dini directional derivative second-order necessary conditions for a point x0 to be a w-minimizer and second-order sufficient conditions for x0 to be an i-minimizer of order two. We discuss the reversal of the sufficient conditions under suitable constraint qualifications of Kuhn-Tucker type. The obtained results improve the ones in Liu, Neittaanmäki, Křížek [21].

On the notion of tangent cone in mathematical programming

In the present paper, several different definitions of the contingent cone, of the attainable cone and of Clarkes tangent cone are examined in order to state some equivalence relations and the links among them

Minty Variational Inequality and Optimization: Scalar and Vector Case

Minty variational inequalities are considered as related to the scalar minimization problem in which the objective function is a primitive of the operator involved in the inequality itself. Well-posedness (in the sense of Tykhonov) of this primitive problem is proved as a consequence of the existence of a strict solution of a Minty variational inequality.

Second-Order Conditions in C1,1 Vector Optimization with Inequality and Equality Constraints

The present paper studies the following constrained vector optimization problem: minC f (x), g(x) ∈ −K, h(x) = 0, where f : ℝn → ℝm g : ℝn → ℝp are C 1,1 functions, h : ℝn → ℝq is C 2 function, and C ⊂ ℝm and K ⊂ ℝp are closed convex cones with nonempty interiors. Two type of solutions are important for the consideration, namely w-minimizers (weakly efficient points) and i-minimizers (isolated minimizers). In terms of the second-order Dini directional derivative second-order necessary conditions a point x 0 to be a w-minimizer and second-order sufficient conditions x 0 to be an i-minimizes of order two are formulated and proved. The effectiveness of the obtained conditions is shown on examples.

Second-order conditions for constrained vector optimization problems with ℓ-stable data

We consider the constrained vector optimization problem min C  f(x), g(x) ∈ −K, where C ⊂ ℝ m and K ⊂ ℝ p are pointed closed convex cones, and f : ℝ n  → ℝ m and g : ℝ n  → ℝ p are ℓ-stable at a point x 0 ∈ ℝ n . We give second-order sufficient and necessary conditions x 0 to be an i-minimizer (isolated minimizer) of order two, and second-order necessary conditions x 0 to be a w-minimizer (weakly efficient point). The obtained results improve the ones of Bednařík and Pastor [On second-order conditions in unconstrained optimization, Math. Program. Ser. A 113 (2008), pp. 283–298] (from unconstrained scalar problems to constrained vector problems) and Ginchev et al. [Second-order conditions in C1,1 constrained vector optimization, Math. Program. Ser. B 104 (2005), pp. 389–405], (from problems with C 1,1 data to problems with ℓ-stable data). In fact, the former paper introduces and studies the notion of a ℓ-stable at a point scalar function, and shows some possible applications. Here we generalize this notion to vector functions.

Stability of properly efficient points and isolated minimizers of constrained vector optimization problems

AbstractIn this paper the constrained vector optimization problem micCf(x), g(x) ∃ − K, is considered, where