Epsilon efficiency
This paper considers the extension of \(\epsilon\)-optimality for scalar problems to vector maximization problems, or efficiency problems, which have m objective functions defined on a set \(X\subseteq {\mathbb{R}}^ n\). It is shown that the natural extension of the scalar \(\epsilon\)-optimality concepts (viz, given \(\epsilon >0\), given a solution set S, if \(x\in S\) there exists an efficient solution y with \(\| f(x)-f(y)\| \leq \epsilon\), and given an efficient solution y, there exists an \(x\in S\) with \(\| f(x)-f(y)\| \leq \epsilon)\) do not hold for some methods used. Six concepts of \(\epsilon\)-efficient sets are introduced and examined, to a very limited extent, in the context of five methods used for generating efficient points or near efficient points. In doing so, a distinction is drawn between methods in which the surrogate optimizations are carried out exactly, and those where terminal \(\epsilon\)-optimal solutions are obtained.
scientific article; zbMATH DE number 3916037