### Remarks on second order generalized derivatives for differentiable functions with Lipschitzian Jacobian.

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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 $f:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$, $g:{\mathbb{R}}^{n}\to {\mathbb{R}}^{p}$ are locally Lipschitz functions, $h:{\mathbb{R}}^{n}\to {\mathbb{R}}^{q}$ is ${C}^{1}$ function, and $C\subset {\mathbb{R}}^{m}$ and $K\subset {\mathbb{R}}^{p}$ 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 ${x}^{0}$...

Initially, second-order necessary optimality conditions and sufficient optimality conditions in terms of Hadamard type derivatives for the unconstrained scalar optimization problem $\phi \left(x\right)\to min$, $x\in {\mathbb{R}}^{m}$, are given. These conditions work with arbitrary functions $\phi \phantom{\rule{0.222222em}{0ex}}{\mathbb{R}}^{m}\to \overline{\mathbb{R}}$, 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...

The existence of solutions to a scalar Minty variational inequality of differential type is usually related to monotonicity property of the primitive function. On the other hand, solutions of the variational inequality are global minimizers for the primitive function. The present paper generalizes these results to vector variational inequalities putting the Increasing Along Rays (IAR) property into the center of the discussion. To achieve that infinite elements in the image space $Y$ are introduced....

The existence of solutions to a scalar Minty variational inequality of differential type is usually related to monotonicity property of the primitive function. On the other hand, solutions of the variational inequality are global minimizers for the primitive function. The present paper generalizes these results to vector variational inequalities putting the Increasing Along Rays (IAR) property into the center of the discussion. To achieve that infinite elements in the image space are introduced. Under...

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