Definition: We say that a vector of decision variables $ \hat x^{*} \in F$ is Pareto optimal if there does not exist another $ \hat x \in F$ such that $ f(\hat x) \le f(\hat x^{*})$ for all $ i = 1, \hdots k $ and $ f_{j}(\hat x)< f_{j}(\hat x)$ for at least one j. In words, this deļ¬nition says that $ \hat x^{*}$ is Pareto optimal if there exists no feasible vector of decision variables $ \hat x^{*} \in F$ which would decrease some criterion without causing a simultaneous increase in at least one other criterion. Unfortunately, this concept almost always gives not a single solution, but rather a set of solutions called the Pareto optimal set. The vectors $ \hat x^{*}$ corresponding to the solutions included in the Pareto optimal set are called nondominated. The plot of the objective functions whose nondominated vectors are in the Pareto optimal set is called the Pareto front.


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