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Keywords:
scalar and vector optimization; ${\mathcal C}^{1,1}$ functions; Hadamard and Dini derivatives; second-order optimality conditions; Lagrange multipliers.
scalar and vector optimization; ${\mathcal C}^{1,1}$ functions; Hadamard and Dini derivatives; second-order optimality conditions; Lagrange multipliers.
Summary:
Initially, second-order necessary optimality conditions and sufficient optimality conditions in terms of Hadamard type derivatives for the unconstrained scalar optimization problem $\phi (x)\rightarrow \min $, $x\in \mathbb{R}^m$, are given. These conditions work with arbitrary functions $\phi \:\mathbb{R}^m \rightarrow \overline{\mathbb{R}}$, but they show inconsistency with the classical derivatives. This is a base to pose the question whether the formulated optimality conditions remain true when the “inconsistent” Hadamard derivatives are replaced with the “consistent” Dini derivatives. It is shown that the answer is affirmative if $\phi $ is of class ${\mathcal C}^{1,1}$ (i.e., differentiable with locally Lipschitz derivative). Further, considering ${\mathcal C}^{1,1}$ functions, the discussion is raised to unconstrained vector optimization problems. Using the so called “oriented distance” from a point to a set, we generalize to an arbitrary ordering cone some second-order necessary conditions and sufficient conditions given by Liu, Neittaanmäki, Křížek for a polyhedral cone. Furthermore, we show that the conditions obtained are sufficient not only for efficiency but also for strict efficiency.
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