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.
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Jiménez, B., Novo, V. Optimality Conditions in Differentiable Vector Optimization via Second-Order Tangent Sets. Appl Math Optim 49, 123–144 (2004). https://doi.org/10.1007/s00245-003-0782-6
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DOI: https://doi.org/10.1007/s00245-003-0782-6