Linear Fractional and Convex Quadratic Vector Optimization Problems

Abstract

Let $K \subseteq {\mathbb{R}}^{n}$ be a nonempty closed convex set, $\varphi = ({\varphi }_{1},\ldots,{\varphi }_{m}) : \Omega \rightarrow {\mathbb{R}}^{m}$ a continuously differentiable function defined on an open set $\Omega \subseteq {\mathbb{R}}^{n}$ which contains K as a subset. The standard vector optimization problem given by the constraint setK and the vector objective function φ is written formally as follows:

$$\mathrm{(VP)}\qquad \qquad \qquad \qquad \mathrm{Minimize}\ \;\varphi (x)\quad \mathrm{subject\ to}\quad x \in K.\qquad \qquad \qquad \qquad$$

As usual, we denote by ${\mathbb{R}}_{+}^{m}$ the nonnegative orthant in ${\mathbb{R}}^{m}$ and by $\mathrm{int}\,{\mathbb{R}}_{+}^{m}$ the interior of that orthant.