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CORMSIS Seminar by Henri Bonnel
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Event
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- Time:
- 11:00 - 12:00
- Date:
- 30 September 2013
- Venue:
- Room 8031 (8c), Building 54

For more information regarding this event, please email Prof. Joerg Fliege at J.Fliege@soton.ac.uk .

## Event details

You are invited to a CORMSIS seminar on Monday 30th Sep at 11:00, which will be presented by Henri Bonnel, a guest of Joerg Fliege, from University of New Caledonia. Tea/coffee will be available before the seminar.

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.

## Speaker information

Henri Bonnel,University of New Caledonia