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Initially, second-order necessary optimality conditions and sufficient optimality conditions in terms of Hadamard type derivatives for the unconstrained scalar optimization problem $\phi \left(x\right)\to min$, $x\in {ℝ}^m$, are given. These conditions work with arbitrary functions $\phi {ℝ}^m\to \overline{ℝ}$, 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...

The present paper studies the following constrained vector optimization problem: ${min}_{C}f\left(x\right)$, $g\left(x\right)\in -K$, $h\left(x\right)=0$, where $f:{ℝ}^n\to {ℝ}^m$, $g:{ℝ}^n\to {ℝ}^p$ are locally Lipschitz functions, $h:{ℝ}^n\to {ℝ}^q$ is $C^1$ function, and $C\subset {ℝ}^m$ and $K\subset {ℝ}^p$ are closed convex cones. Two types of solutions are important for the consideration, namely $w$-minimizers (weakly efficient points) and $i$-minimizers (isolated minimizers of order 1). In terms of the Dini directional derivative first-order necessary conditions for a point $x^0$ to be a $w$-minimizer and first-order sufficient conditions for $x^0$...