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You might need to formulate problems with more than one objective, since a single objective with several constraints may not adequately represent the problem being faced. If so, there is a vector of objectives,

F(x) =
[F_{1}(x),
F_{2}(x),...,F_{m}(x)], | (1) |

Multiobjective optimization is concerned with the minimization of a vector
of objectives *F*(*x*) that can be the subject of
a number of constraints or bounds:

$$\begin{array}{l}\underset{x\in {\mathbb{R}}^{n}}{\mathrm{min}}F(x),\text{subjectto}\\ {G}_{i}(x)=0,\text{}i=1,\mathrm{...},{k}_{e};\text{}{G}_{i}(x)\le 0,\text{}i={k}_{e}+1,\mathrm{...},k;\text{}l\le x\le u.\end{array}$$

Note that because *F*(*x*) is a vector, if any
of the components of *F*(*x*) are competing, there
is no unique solution to this problem. Instead, the concept of noninferiority in Zadeh
[4] (also called Pareto optimality in Censor [1]
and Da Cunha and Polak [2]) must be used to characterize the objectives. A
noninferior solution is one in which an improvement in one objective requires a
degradation of another. To define this concept more precisely, consider a feasible
region, Ω, in the parameter space. *x* is an element of the
*n*-dimensional real numbers $$x\in {\mathbb{R}}^{n}$$ that satisfies all the constraints, that is,

$$\Omega =\left\{x\in {\mathbb{R}}^{n}\right\},$$

subject to

$$\begin{array}{l}{G}_{i}(x)=0,\text{}i=1,\mathrm{...},{k}_{e},\\ {G}_{i}(x)\le 0,\text{}i={k}_{e}+1,\mathrm{...},k,\\ l\le x\le u.\end{array}$$

This allows definition of the corresponding feasible region for the objective function space Λ:

$$\Lambda =\left\{y\in {\mathbb{R}}^{m}:y=F(x),x\in \Omega \right\}.$$

The performance vector *F*(*x*)
maps parameter space into objective function space, as represented in two dimensions in
the figure Figure 13-1, Mapping from Parameter Space into Objective Function Space.

**Figure 13-1, Mapping from Parameter Space into Objective Function
Space**

A noninferior solution point can now be defined.

**Definition:** Point $$x*\in \Omega $$ is a noninferior solution if for some neighborhood of *x** there
does not exist a Δ*x* such that $$\left(x*+\Delta x\right)\in \Omega $$ and

$$\begin{array}{l}{F}_{i}\left(x*+\Delta x\right)\le {F}_{i}(x*),\text{}i=1,\mathrm{...},m,\text{and}\\ {F}_{j}\left(x*+\Delta x\right){F}_{j}(x*)\text{foratleastone}j.\end{array}$$

In the two-dimensional representation of the figure Figure 13-2, Set of Noninferior Solutions, the set of noninferior solutions lies on the
curve between *C* and *D*. Points
*A* and *B* represent specific noninferior
points.

**Figure 13-2, Set of Noninferior Solutions**

*A* and *B* are clearly noninferior solution
points because an improvement in one objective,
*F*_{1}, requires a degradation in the other
objective, *F*_{2}, that is, *F*_{1B} < *F*_{1A},
*F*_{2B} > *F*_{2A}.

Since any point in Ω that is an inferior point represents a point in which improvement can be attained in all the objectives, it is clear that such a point is of no value. Multiobjective optimization is, therefore, concerned with the generation and selection of noninferior solution points.

Noninferior solutions are also called *Pareto optima*. A general goal in multiobjective
optimization is constructing the Pareto optima.