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

Abstract

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 C^1,1 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.

Keywords:

• vector optimization
• ℓ-stable functions
• second-order conditions

AMS Subject Classifications::

• 90C29
• 90C30
• 90C46
• 49J52