** 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

- International Society on Multiple Criteria Decision Making (MCDM)
- MCDM Newsletter
- Publications related to MCDM
- Software related to MCDM
- MCDMlib: Test Problems for Multiobjective Optimization

## References

- Marler, R. T. and Arora, J. S. 2004. Survey of multi-objective optimization methods for engineering.
*Structural and Multidisciplinary Optimization***26**, 369 - 395. - Optimization Online Other Topics area includes multi-criteria optimization
- Wikipedia entry on Multi-objective optimization
- Zitzler, E., Laumanns, M., and Bleuler, S. 2004. A Tutorial on Evolutionary Multiobjective Optimization in
*Metaheuristics for Multiobjective Optimization*, X. Gandibleux et al. eds., Springer-Verlag, Berlin, pp. 3 - 37.