# Optimality Conditions in Differentiable Vector Optimization via Second-Order Tangent Sets

Optimality Conditions in Differentiable Vector Optimization via Second-Order Tangent Sets We provide second-order necessary and sufficient conditions for a point to be an efficient element of a set with respect to a cone in a normed space, so that there is only a small gap between necessary and sufficient conditions. To this aim, we use the common second-order tangent set and the asymptotic second-order cone utilized by Penot. As an application we establish second-order necessary conditions for a point to be a solution of a vector optimization problem with an arbitrary feasible set and a twice Fréchet differentiable objective function between two normed spaces. We also establish second-order sufficient conditions when the initial space is finite-dimensional so that there is no gap with necessary conditions. Lagrange multiplier rules are also given. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Optimality Conditions in Differentiable Vector Optimization via Second-Order Tangent Sets

, Volume 49 (2) – Mar 1, 2004
22 pages

/lp/springer_journal/optimality-conditions-in-differentiable-vector-optimization-via-second-jlDMTS0zLH
Publisher
Springer Journals
Subject
Philosophy
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-003-0782-6
Publisher site
See Article on Publisher Site

### Abstract

We provide second-order necessary and sufficient conditions for a point to be an efficient element of a set with respect to a cone in a normed space, so that there is only a small gap between necessary and sufficient conditions. To this aim, we use the common second-order tangent set and the asymptotic second-order cone utilized by Penot. As an application we establish second-order necessary conditions for a point to be a solution of a vector optimization problem with an arbitrary feasible set and a twice Fréchet differentiable objective function between two normed spaces. We also establish second-order sufficient conditions when the initial space is finite-dimensional so that there is no gap with necessary conditions. Lagrange multiplier rules are also given.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Mar 1, 2004

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