Journées de l'optimisation 2007 Optimizations days

It is well-known that a lot of problems in optimization and optimal control involve marginal functions and their subdifferentials since the sensitivity of these problems can be studied with the help of the behaviour of the subdifferentials of some associated marginal functions. Generally the infimum defining the marginal function is required to be attained near the point of interest. This paper is devoted to study, for a first important class of problems, how this condition can be removed. Here we will deal with locally Lipschitz value functions of the form M(x) := inf{g(y): y in G(x)} Where g is a real-valued function from a Banach space X into R and G is a multivalued mapping from X into a Banach space Y. The above infimum will not be required to be attained.

We present a new regularization scheme for MPCCs. We present comparisons with previously proposed schemes. Our discussion will refer to several stationarity conditions (S-, M-, W-stationarity, etc.). The existence of the Lagrange multipliers for the relaxed problem is proved, and we will compare the assumptions required to show the convergence of the regularization scheme to stationary points of the original MPEC; our new scheme requires weaker assumptions than previous schemes.

Minmax regret combinatorial optimization deals with problems where only parameters defining the objective function may be uncertain, but the set of feasible solutions is known precisely; it is required to find a feasible solution which is reasonably close to the optimal one (in terms of the objective function value) for all possible realizations of data. In this talk, we will discuss some recent results on the complexity of minmax regret combinatorial optimization problems with interval-data structure of uncertainty.

We consider global optimization of constrained non-convex problems in mixed variables, solved by a interval arithmetic approach. Tight bounds are obtained by a novel use of affine arithmetic combined with linear and quadratic reformulation. Early computational results are reported.