What Is Multiobjective Optimization?
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) = [F1(x),F2(x),...,Fm(x)], | (1) |
Multiobjective optimization is concerned with the minimization of a vector of objectivesF(x) that can be the subject of a number of constraints or bounds:
Note that becauseF(x) is a vector, if any of the components ofF(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.xis an element of then-dimensional real numbers that satisfies all the constraints, that is,
subject to
This allows definition of the corresponding feasible region for the objective function space Λ:
The performance vectorF(x) maps parameter space into objective function space, as represented in two dimensions in the figureFigure 14-1, Mapping from Parameter Space into Objective Function Space.
Figure 14-1, Mapping from Parameter Space into Objective Function Space
A noninferior solution point can now be defined.
Definition:Point is anoninferior solution if for some neighborhood ofx* there does not exist a Δxsuch that and
In the two-dimensional representation of the figureFigure 14-2, Set of Noninferior Solutions, the set of noninferior solutions lies on the curve betweenCandD. PointsAandBrepresent specific noninferior points.
Figure 14-2, Set of Noninferior Solutions
AandBare clearly noninferior solution points because an improvement in one objective,F1, requires a degradation in the other objective,F2, that is,F1B<F1A,F2B>F2A.
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 calledPareto optima. A general goal in multiobjective optimization is constructing the Pareto optima.