# Computing a Finite Size Representation of the Set of Approximate Solutions of an MOP - Computer Science > Numerical Analysis

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Abstract: Recently, a framework for the approximation of the entire set of$\epsilon$-efficient solutions denote by $E \epsilon$ of a multi-objectiveoptimization problem with stochastic search algorithms has been proposed. Itwas proven that such an algorithm produces - under mild assumptions on theprocess to generate new candidate solutions -a sequence of archives whichconverges to $E {\epsilon}$ in the limit and in the probabilistic sense. Theresult, though satisfactory for most discrete MOPs, is at least from thepractical viewpoint not sufficient for continuous models: in this case, the setof approximate solutions typically forms an $n$-dimensional object, where $n$denotes the dimension of the parameter space, and thus, it may come toperfomance problems since in practise one has to cope with a finite archive.Here we focus on obtaining finite and tight approximations of $E \epsilon$, thelatter measured by the Hausdorff distance. We propose and investigate a novelarchiving strategy theoretically and empirically. For this, we analyze theconvergence behavior of the algorithm, yielding bounds on the obtainedapproximation quality as well as on the cardinality of the resultingapproximation, and present some numerical results.

Author: ** Oliver Schuetze INRIA Futurs, Carlos A. Coello Coello INRIA Lille - Nord Europe, Emilia Tantar INRIA Lille - Nord Europe, El-Ghaz**

Source: https://arxiv.org/