Variational Analysis and Set Optimization

Chapter 9

Vectorial Penalization in Multi-objective Optimization

Christian Günther

Martin Luther University Halle-Wittenberg, Faculty of Natural Sciences II, Institute for Mathematics, 06099 Halle (Saale), Germany.

9.1    Introduction

In multi-objective optimization, one often tries to minimize a vector-valued objective function, say

$f=\left(f_1,\mathrm{\dots },f_m\right):E\to {\text{ℝ}}^m\left(m\in \text{ℕ}\right),$

over a certain feasible set Ω in a normed space E, i.e., the aim is to study the optimization problem

$\{\begin{array}{l}f(x)=(f_1(x),\mathrm{\dots },f_m(x))\to \min \text{w}\text{.r}\text{.t}.{\text{ℝ}}_{+}^m\\ x\in \text{Ω},\end{array}$

(𝒫Ω

where, in this case, the finite-dimensional Euclidean space ℝ^m is partially ordered by the natural ordering cone

${ℝ}_{+}^m≔\{y=(y_1\mathrm{\dots },y_m)\in {ℝ}^m|\forall i\in \{1,\mathrm{\dots },m\}:yi\ge 0\}.$

The componentwise approach is known to be appropriate for studying vector-valued ...