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optimization over the Pareto outcome set associated with a convex bi-objective optimization problem: theoretical results, deterministic algorithm and application to the stochastic case

引用

JOURNAL OF GLOBAL optimization 2015年第3期62卷 481-505页

作者： Bonnel, Henri Collonge, Julien Univ New Caledonia ERIM Noumea 98851 New Caledonia

Our paper consists of two main parts. In the first one, we deal with the deterministic problem of minimizing a real valued function over the Pareto outcome set associated with a deterministic convex bi-objective optimization problem (BOP), in the particular case where depends on the objectives of (BOP), i.e. we optimize over the Pareto set in the outcome space. In general, the optimal value 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 . 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 of minimizing a real valued function 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 of problem . That is why we consider a sequence of Sample Average Approximation problems (SAA-, where is the sample size) whose optimal values converge almost surely to as the sample size goes to infinity. Assuming nondecreasing, we show that the convergence rate is exponential, and we propose a confidence interval for . Finally, some computational results are given to illustrate the paper.

关键词： optimization over the Pareto image set multi-objective deterministic optimization multi-objective stochastic optimization multi-objective convex optimization Sample average approximation method

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