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Necessary Optimality Conditions for Semi-vectorial Bi-level optimization with convex Lower Level:Theoretical Results and Applications to the Quadratic Case

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Journal of the Operations Research Society of China 2021年第3期9卷 691-712页

作者： Julien Collonge University of New Caledonia New Caledonia 98800France

This paper explores related aspects to post-Pareto analysis arising from the multicriteria optimization *** consists of two main *** the first one,we give first-order necessary optimality conditions for a semi-vectorial bi-level optimization problem:the upper level is a scalar optimization problem to be solved by the leader,and the lower level is a multi-objective optimization problem to be solved by several followers acting in a cooperative way(greatest coalition multi-players game).For the lower level,we deal with weakly or properly Pareto(efficient)solutions and we consider the so-called optimistic problem,*** followers choose amongst Pareto solutions one which is the most favourable for the *** order to handle reallife applications,in the second part of the paper,we consider the case where each follower objective is expressed in a quadratic *** this setting,we give explicit first-order necessary optimality ***,some computational results are given to illustrate the paper.

关键词： Bi-level optimization multi-objective optimization Post-Pareto optimization multi-objective convex optimization Quadratic optimization

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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

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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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Stochastic optimization over a Pareto Set Associated with a Stochastic multi-objective optimization Problem

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JOURNAL OF optimization THEORY AND APPLICATIONS 2014年第2期162卷 405-427页

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

We deal with the problem of minimizing the expectation of a real valued random function over the weakly Pareto or Pareto set associated with a Stochastic multi-objective optimization Problem, whose objectives are expectations of random functions. Assuming that the closed form of these expectations is difficult to obtain, we apply the Sample Average Approximation method in order to approach this problem. We prove that the Hausdorff-Pompeiu distance between the weakly Pareto sets associated with the Sample Average Approximation problem and the true weakly Pareto set converges to zero almost surely as the sample size goes to infinity, assuming that our Stochastic multi-objective optimization Problem is strictly convex. Then we show that every cluster point of any sequence of optimal solutions of the Sample Average Approximation problems is almost surely a true optimal solution. To handle also the non-convex case, we assume that the real objective to be minimized over the Pareto set depends on the expectations of the objectives of the Stochastic optimization Problem, i.e. we optimize over the image space of the Stochastic optimization Problem. Then, without any convexity hypothesis, we obtain the same type of results for the Pareto sets in the image spaces. Thus we show that the sequence of optimal values of the Sample Average Approximation problems converges almost surely to the true optimal value as the sample size goes to infinity.

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

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