Multi-objective optimization: Difference between revisions

Revision as of 22:42, 8 April 2007

Revision as of 23:00, 8 April 2007

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== Introduction ==

In mathematical terms, the multiobjective problem statement can be written as:

<math>

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</math>

where <math> \mu_i </math> is the <math>i</math>-th objective function, <math>g</math> and <math>h</math> are the inequality and equality constraints, respectively, and <math>x</math> is the vector of optimization or decision variables. The solution to the above problem is a set of Pareto points. Pareto solutions are those for which improvement in one objective can only occur with the worsening of at least one other objective. Thus, instead of a unique solution to the problem (which is typically the case in traditional mathematical programming), the solution to thea abovemultiobjective problem is ana infinte(possibly infinite) set of Pareto points.

Pareto optimal solution: A design point in objective space <math>\mu^*</math> is termed [[pareto efficiency|Pareto optimal]] if there does not exist another feasible design objective vector <math>\mu</math> such that <math>\mu_i \leq \mu_i^*</math> for all <math>i \in \left\{ {1,2,...,n } \right\}</math>, and <math>\mu _j < \mu_j^*</math> for at least one index of <math>j</math>, <math>j \in \left\{ {1,2,...,n } \right\}</math>.

== Solution Methods ==