Multi Objective Optimization

For mathematical optimization problems where more than one objective function needs to be optimized simultaneously.

• applied when optimal decisions need to be taken in presence of trade offs between conflicting objectives
• not guaranteed that single solution simultaneously optimizes each objetive.
  • objective functions are said to be conflicting, solution called nondominated
    • if this is the case, with no subjective presence/information, may exist an infinite number of pareto optimal solutions

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Formulation

$$\underset{x\in X}{\min }(f_1(x),f_2(x),\dots ,f_k(x))$$

where $k\in \mathbb{Z}$ is $\ge 2$ is num of objectives, set $X$ is is feasible set of decision vectors $X\subseteq {\mathbb{R}}^n$, however depends on n dim application domain.

• objective function usually defined as

$$\begin{array}{rl}f:X& \to {\mathbb{R}}^k\\ x& \mapsto \left(\begin{array}{c}{f}_1(x)\\ ⋮\\ f_k(x)\end{array}\right)\end{array}$$