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.