Multi-objective Optimization - Introduction

Introduction

A multiobjective optimization problem is an optimization problem that involves multiple objective functions. In mathematical terms, a multiobjective optimization problem can be formulated as

\begin{align} \min &\left(f_1(x), f_2(x),\ldots, f_k(x) \right)^T \\ \text{s.t. } &x\in X, \end{align}

where the integer is the number of objectives and the set is the feasible set of decision vectors defined by constraint functions. In addition, the vector-valued objective function is often defined as

. If some objective function is to be maximized, it is equivalent to minimize its negative. The image of is denoted by

An element is called a feasible solution or a feasible decision. A vector for a feasible solution is called an objective vector or an outcome. In multiobjective optimization, there does not typically exist a feasible solution that minimizes all objective functions simultaneously. Therefore, attention is paid to Pareto optimal solutions, i.e., solutions that cannot be improved in any of the objectives without impairment in at least one of the other objectives. In mathematical terms, a feasible solution is said to (Pareto) dominate another solution, if

  1. for all indices and
  2. for at least one index .

A solution (and the corresponding outcome ) is called Pareto optimal, if there does not exist another solution that dominates it. The set of Pareto optimal outcomes is often called the Pareto front.

The Pareto front of a multiobjective optimization problem is bounded by a so-called nadir objective vector and an ideal objective vector, if these are finite. The nadir objective vector is defined as

and the ideal objective vector as

In other words, the components of a nadir and an ideal objective vector define upper and lower bounds for the objective function values of Pareto optimal solutions, respectively. In practice, the nadir objective vector can only be approximated as, typically, the whole Pareto optimal set is unknown.

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