Multiobjective Optimization Problems

Why

Often we care about multiple criteria at once.¹

Definition

A multiobjective optimization problem is a pair $(X, f: X \to \R ^d)$. As before, $X$ is the constraint set and $f$ is called the objective function. Since $f$ is vector valued, and there is no natural order on $\R ^d$, there may exist $x \in X$ with non-comparable images under $f$.

Scalarization

The $\alpha \in \R ^d$ scalarization of a multiobjective optimization problem $(X, f)$ is the optimization problem $(X, g)$ where $g: X \to \R $ is defined by $g(x) = \alpha ^\tp f(x)$. We call $g$ the scalarized objective.

1. Future editions will modify. ↩︎