Sensitivity analysis in parametric vector optimization in Banach spaces viaτw-contingent derivatives

Abstract

This paper is concerned with sensitivity analysis in parametric vector optimization problems via τ wcontingent derivatives. Firstly, relationships between the τ w-contingent derivative of the Borwein proper perturbation map and the τ w-contingent derivative of feasible map in objective space are considered. Then, the formulas for estimating the τ w-contingent derivative of the Borwein proper perturbation map via the τ w-contingent of the constraint map and the Hadamard derivative of the objective map are obtained.

On generalized $\tau^w$-contingent epiderivatives in parametric vector optimization problems

“…Definition 2.6. [33] G is called weak directional compact at (p,ȳ) ∈ grG in the direction p ∈ P if for every sequence t n ↓ 0 and for any sequence p n → p ∈ P, any sequence y n in Y withȳ +t n y n ∈ G(p + t n p n ) for each n includes a weak convergent subsequence. If G is weak directional compact at (p,ȳ) for every p ∈ P, then G is said to be weak directional compact at (p,ȳ).…”

“…In the dual space approach, we refer to the books [16,17] and the recent papers [6,8,18]. In primal space approach, the first-order derivatives of perturbation maps were studied in [3,5,7,13,19,26,33]. To get more information, the higher-order derivatives of perturbation maps have been investigated in [1,9,27,29].…”

“…In [22], τ w -contingent epiderivatives, based on the consideration of the weak topology in the image space, were introduced, and applied to the optimality conditions for a set-valued optimization problem. The formulas for estimating the τ w -contingent derivative of the Borwein proper perturbation map via the τ w -contingent of the constraint map and the Hadamard derivative of the objective map were considered in [33]. It should be noted that contingent epiderivatives [12] as well as τ w -contingent epiderivatives may not exist in some cases.…”