Moment and SDP relaxation techniques for smooth approximations of problems involving nonlinear differential equations

Martin Mevissen (1) , Jean-Bernard Lasserre (2) , Didier Henrion (2)

1 Department of Mathematical and Computing Sciences
2 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande

Combining recent moment and sparse semidefinite programming (SDP) relaxation techniques, we propose an approach to find smooth approximations for solutions of problems involving nonlinear differential equations. Given a system of nonlinear differential equations, we apply a technique based on finite differences and sparse SDP relaxations for polynomial optimization problems (POP) to obtain a discrete approximation of its solution. In a second step we apply maximum entropy estimation (using moments of a Borel measure associated with the discrete solution) to obtain a smooth closed-form approximation. The approach is illustrated on a variety of linear and nonlinear ordinary differential equations (ODE), partial differential equations (PDE) and optimal control problems (OCP), and preliminary numerical results are reported.

Mots clés: semidefinite programming; nonlinear optimal control; maximum entropy estimation; approximation methods; nonlinear optimization; moment methods

Domaines: Optimisation et contrôle [math.OC]

Martin Mevissen, Jean-Bernard Lasserre, Didier Henrion. Moment and SDP relaxation techniques for smooth approximations of problems involving nonlinear differential equations. World IFAC Congress (IFAC 2011), Aug 2011, Milan, Italy. p. 10887-10892. ⟨hal-00462301v2⟩