CORMSIS Seminar by Henri Bonnel Event

Post-Pareto Analysis for Multiobjective Stochastic Problems

The solution set (Pareto or efficient set) of a multiobjective  optimization problem is often very large (infinite and even  unbounded). The grand coalition of a cooperative game can be written  as a multiobjective optimal control problem. Assuming that this game  is supervised by a decision maker (DM), the DM can use his own (scalar) objective for choosing a solution. Of course this solution  must satisfy all the players of the grand coalition, hence must be a  Pareto solution. Another interest for the study of the  problem of  optimizing a scalar function over a Pareto set  is that it may be  possible to avoid the generation of all the Pareto set.

For multiobjective mathematical programming problems (finite  dimensional optimization) there are many contributions in this field  (see e.g. [5] for an extensive bibliography). Some recent results for  the case of the multiobjective control problems can be found in [1,4]. Generalization of this problem for the semivectorial bilevel problems  has been studied in [3].

My talk deals with a different setting: mulitobjective stochastic  optimization, and it is based on the paper [1]. Thus I will consider  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, the  Sample Average Approximation method is applied in order to approach  this problem.

I will show that the Hausdorff-Pompeiu distance between the Sample  Average Approximation of size N weakly Pareto sets and the true weakly  Pareto set converges to zero almost surely as the sample size N goes  to infinity, assuming that our Stochastic Multi-Objective Optimization  Problem is strictly convex. Also every cluster point of any sequence  of Sample Average Approximation  optimal solutions is almost surely a  true optimal solution.

To handle also the non-convex case, it is assumed 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. it is considered the problem of optimizing a scalar  function over the Pareto outcome space of the Stochastic Optimization  Problem. Then, without any convexity hypothesis, some similar results  hold for the Pareto sets in the outcome spaces. Finally I will  show  that the sequence of Sample Average Approximation optimal values  converges almost surely to the true optimal value as the sample size  goes to infinity.

REFERENCES

[1]  Henri Bonnel. Post-Pareto Analysis for Multiobjective  Parabolic Control Systems. Ann. Acad. Rom. Sci. Ser. Math. Appl., 5: 13-34, 2013.

[2]  H. Bonnel and J. Collonge. Stochastic Optimization over a  Pareto Set Associated with a Stochastic Multi-objective Optimization  Problem. Journal of Optimization Theory and Applications, (online  first, DOI 10.1007/s10957-013-0376-8) 2013.

[3]  H. Bonnel and J. Morgan. Semivectorial Bilevel Convex Optimal  Control Problems: An Existence Result. SIAM Journal on Control and  Optimization, 50, (6): 3224-3241, 2012.

[4]  H. Bonnel and Y. Kaya. Optimization Over the Efficient Set  of Multi-objective Convex Optimal Control Problems. Journal of  Optimization Theory and Applications, 147, (1): 93-112, 2010.

[5]  Y. Yamamoto. Optimization over the efficient set : an  overview. J. Global Optim., 22: 285-317, 2002.