Nonlinear perspective functions.

par Francisco Silva

vendredi 24 nov. 2023, 11:00 → 12:00 Europe/Paris

XLIM1 (XLIM)

The notion of perspective function, defined in terms of a convex and lower semicontinuous base function and a linear scaling, has been introduced by R.T. Rockafellar in 1966. Under some constant sign conditions over the conjugate of the base function, an extension of perspective functions, with nonlinear scaling, has been introduced by P, Maréchal in 2006. In this talk, we propose a constructive approach for perspective functions with nonlinear scaling, without any sign constraint on the conjugate of the base function, by considering the convex envelope of preperspective functions. If time allows, we will also discuss the computation of proximity operators of a class of perspective functions with nonlinear scaling. The latter play an important role in the implementation of first order methods to approximate solutions to convex optimization problems involving nonlinear perspective functions, and finds applications in optimal transport, gradient flows in probability spaces, and optimal control problems of McKean-Vlasov systems.
 

Based on joint works with L. Briceño-Arias and P.-L. Combettes.