18th French-German-Italian Conference on Optimization

The forthcoming 18th French-German-Italian Conference on Optimization will take place in Paderborn (Germany). This year it will host the minisymposia.

I look forward to attending the session Hierarchies of SDP Relaxations for Polynomial Systems. We will contribute the paper Semidefinite Characterization of Invariant Measures, joint work with V. Magron and D. Henrion.

This is the abstract of our contribution:

We consider the problem of characterizing measures which are invariant with respect to the dynamics of polynomial systems, under general semialgebraic set constraints. This characterization consists in numerical approximation of the moments and support of such invariant measures. First, we address the problem of approximating the density and hence the support of an invariant measure which is absolutely continuous with respect to the Lebesgue measure. Then, we focus on the characterization of the support of an invariant measure which is singular with respect to the Lebesgue measure. In both cases, our results apply for discrete-time and continuous-time polynomial systems. Each problem is handled through an adequate reformulation into a linear optimization problem over measures, solved in practice with two distinct hierarchies of finite-dimensional semidefinite moment-sum-of-square relaxations.

Under specific assumptions, the first moment hierarchy allows to extract a sequence of polynomials converging in L2-norm to the density of an absolutely continuous invariant measure. The second moment hierarchy allows to approximate as close as desired the support of a singular invariant measure with the level sets of the Christoffel functions associated to the moment matrices of this measure.

We also present some application examples together with numerical results for several dynamical systems admitting either absolutely continuous or singular invariant measures.

The conference will address all aspects of optimization and its applications, including:

• continuous optimization (smooth and nonsmooth)
• optimal control and calculus of variations
• optimization with PDE
• numerical methods for mathematical programming
• robust optimization
• mixed integer optimization
• differential inclusions and set-valued analysis
• stochastic optimization
• multicriteria optimization
• optimization techniques for industrial applications