Multiobjective Optimization

Multiobjective optimization considers optimization problems involving more than one objective function to be optimized simultaneously. Multiobjective optimization problems arise in many fields, such as engineering, economics, and logistics, when optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. For example, developing a new component might involve minimizing weight while maximizing strength or choosing a portfolio might involve maximizing the expected return while minimizing the risk.

Typically, there does not exist a single solution that simultaneously optimizes each objective. Instead, there exists a (possibly infinite) set of Pareto optimal solutions. A solution is called nondominated or Pareto optimal if none of the objective functions can be improved in value without degrading one or more of the other objective values. Without additional subjective preference information, all Pareto optimal solutions are considered equally good.

In mathematical terms, a multiobjective optimization problem can be formulated as
\[
\begin{align}
\min &\left(f_1(x), f_2(x),\ldots, f_k(x) \right) \\
\text{s.t. } &x\in X,
\end{align}
\]
where the integer \(k\geq 2\) is the number of objectives and the set \(X\) is the feasible set of decision vectors. The feasible set is typically defined by some constraint functions. In addition, the vector-valued objective function is often defined as \(f:X\to\mathbb R^k, \ f(x)= (f_1(x),\ldots,f_k(x))^T\). An element \(x^*\in X\) is a feasible solution; a feasible solution \(x^1\in X\) is said to (Pareto) dominate another solution \(x^2\in X\), if

  • \(f_i(x^1)\leq f_i(x^2)\) for all indices \(i \in \left\{ {1,2,\dots,k } \right\}\) and
  • \(f_j(x^1) < f_j(x^2)\) for at least one index \(j \in \left\{ {1,2,\dots,k } \right\}\).

A solution \(x^1\in X\) is called Pareto optimal if there does not exist another solution that dominates it.

Online Resources

References