## Invited Session Wed.1.H 1029

#### Wednesday, 10:30 - 12:00 h, Room: H 1029

**Cluster 15: Multi-objective optimization** [...]

### Vector optimization: Post pareto analysis

**Chair: Henri Bonnel**

**Wednesday, 10:30 - 10:55 h, Room: H 1029, Talk 1**

**Jacqueline Morgan**

Semivectorial bilevel convex optimal control problems: Existence results

**Coauthor: Henri Bonnel**

**Abstract:**

We consider a bilevel optimal control problem where the upper level, to be solved by a leader, is a scalar optimal control problem and the lower level, to be solved by several followers, is a multiobjective convex optimal control problem. We deal with the so-called optimistic case, when the followers are assumed to choose a best choice for the leader among their best responses, as well with the so-called pessimistic case, when the best response chosen by the followers can be the worst choice for the leader. We present sufficient conditions on the data for existence of solutions to both the optimistic and pessimistic optimal control problems, with particular attention to the linear-quadratic case.

**Wednesday, 11:00 - 11:25 h, Room: H 1029, Talk 2**

**Henri Bonnel**

Semivectorial bilevel optimal control problems: Optimality conditions

**Coauthor: Jacqueline Morgan**

**Abstract:**

We deal with a bilevel optimal control problem where the upper level is a scalar optimal control problem to be solved by the leader, and the lower level is a multi-objective convex optimal control problem to be solved by several followers acting in a cooperative way inside the greatest coalition and choosing amongst the Pareto optimal controls. This problem belongs to post-Pareto analysis area because generalizes the problem of optimizing a scalar function over a Pareto set. We obtain optimality conditions for the so-called optimistic case when the followers choose among their best responses one which is a best choice for the follower, as well as for the so-called pessimistic case, when the best response chosen by the followers can be the worst case for the leader.

**Wednesday, 11:30 - 11:55 h, Room: H 1029, Talk 3**

**Julien Collonge**

Optimization over the Pareto set associated with a multi-objective stochastic convex optimization problem

**Coauthor: Henri Bonnel**

**Abstract:**

We deal with the problem of minimizing the expectation of a scalar valued function over the Pareto set associated with a multi-objective stochastic convex optimization problem. Every objective is an expectation and will be approached by a sample average approximation function (SAA-N), where *N* is the sample size.

In order to show that the Hausdorff distance between the SAA-N weakly Pareto set and the true weakly Pareto set converges to zero almost surely as N goes to infinity, we need to assume that all the objectives are strictly convex. Then we show that every cluster point of any sequence of SAA-N optimal solutions (*N=1,2, … *) is a true optimal solution.

To weaken the strict convexity hypothesis to convexity, we need to work in the outcome space. Then, under some reasonnable and suitable assumptions, we obtain the same type of results for the image of the Pareto sets. Thus, assuming that the function to minimize over the true Pareto set is expressed as a function of other objectives, we show that the sequence of SAA-N optimal values (*N=1,2, … *) converges almost surely to the true optimal value.

A numerical example is presented.