Bregman-like functions and proximal methods for variational problems with nonlinear constraints

Abstract

The use of Bregman functions, entropic ϕ-divergence or logarithmic-quadratic kernels, allows to construct a series of generalized proximal methods for the stable solution of convex optimization problems and variational inequalities with maximal monotone operators. The key advantage of these methods in comparison to the classical proximal regularization is that the auxiliary problems are structurally simpler than the original ones, in particular, they result in unconstrained inclusions. But, such methods with “interior point effect” were developed so far only for linearly constrained problems. In the present article, the use of Bregman functions with a modified “convergence sensing condition” enables us to construct an interior proximal method for solving variational inequalities (with multi-valued operators) on nonpolyhedral sets. The convergence results admit a successive approximation of the multi-valued operator (by means of the concept of ε-enlargements) and an inexact calculation of proximal iterates.

†Dedicated to the 65th birthday of Diethard Pallaschke.

Keywords:

• Bregman-like functions
• Variational problems
• Nonlinear constraints
• Proximal point methods

Keywords:

• AMS Subject Classifications: 90C25
• 90C51
• 49J40
• 65K05

Notes

†Dedicated to the 65th birthday of Diethard Pallaschke.

¹Some additional assumptions on are used, which are standard for Bregman-function-based methods (see for instance, 2,5 and assumptions A6, A7 below).

²For method (2), (3), in case K is bounded, this estimate has to be replaced by . No other essential modifications are required.

³ In the case A6 it is supposed that , where ∂_ϵ denotes the ϵ-subdifferential. Notice, that the inclusion is valid 2.