dc.contributor.author Bonnel, Henri dc.contributor.author Collonge, J. dc.date.accessioned 2017-01-30T14:55:37Z dc.date.available 2017-01-30T14:55:37Z dc.date.created 2015-12-10T04:26:05Z dc.date.issued 2014 dc.identifier.citation Bonnel, H. and Collonge, J. 2014. Optimization over the Pareto outcome set associated with a convex bi-objective optimization problem: theoretical results, deterministic algorithm and application to the stochastic case. Journal of Global Optimization. 62 (3): pp. 481-505. dc.identifier.uri http://hdl.handle.net/20.500.11937/41837 dc.identifier.doi 10.1007/s10898-014-0257-0 dc.description.abstract Our paper consists of two main parts. In the first one, we deal with the deterministic problem of minimizing a real valued function (Formula presented.) over the Pareto outcome set associated with a deterministic convex bi-objective optimization problem (BOP), in the particular case where (Formula presented.) depends on the objectives of (BOP), i.e. we optimize over the Pareto set in the outcome space. In general, the optimal value (Formula presented.) of such a kind of problem cannot be computed directly, so we propose a deterministic outcome space algorithm whose principle is to give at every step a range (lower bound, upper bound) that contains (Formula presented.). Then we show that for any given error bound, the algorithm terminates in a finite number of steps. In the second part of our paper, in order to handle also the stochastic case, we consider the situation where the two objectives of (BOP) are given by expectations of random functions, and we deal with the stochastic problem (Formula presented.) of minimizing a real valued function (Formula presented.) over the Pareto outcome set associated with this Stochastic bi-objective Optimization Problem (SBOP). Because of the presence of random functions, the Pareto set associated with this type of problem cannot be explicitly given, and thus it is not possible to compute the optimal value (Formula presented.) of problem (Formula presented.). That is why we consider a sequence of Sample Average Approximation problems (SAA-(Formula presented.), where (Formula presented.) is the sample size) whose optimal values converge almost surely to (Formula presented.) as the sample size (Formula presented.) goes to infinity. Assuming (Formula presented.) nondecreasing, we show that the convergence rate is exponential, and we propose a confidence interval for (Formula presented.). Finally, some computational results are given to illustrate the paper. dc.publisher Kluwer Academic Publishers dc.title Optimization over the Pareto outcome set associated with a convex bi-objective optimization problem: theoretical results, deterministic algorithm and application to the stochastic case dc.type Journal Article dcterms.source.issn 0925-5001 dcterms.source.title Journal of Global Optimization curtin.department Department of Mathematics and Statistics curtin.accessStatus Fulltext not available
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