From Scalar to Vector Optimization

Initially, second-order necessary and sufficient optimality conditions in terms of Hadamard type derivatives for the unconstrained scalar optimization problem φ(x)→ min, x ∈ R, are given. These conditions work with arbitrary functions φ, 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 φ is of class C (i.e. differentiable with locally Lipschitz derivative). Further, considering C 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 necessary and sufficient second-order optimality conditions given by Liu, Neittaanmäki, Kř́ı̌rek for a polyhedral cone. Furthermore, we show that the obtained conditions are sufficient not only for efficiency, but also for strict efficiency.