Lagrangian Penalization Scheme with Parallel Forward–Backward Splitting

Abstract

We propose a new iterative algorithm for the numerical approximation of the solutions to convex optimization problems and constrained variational inequalities, especially when the functions and operators involved have a separable structure on a product space, and exhibit some dissymmetry in terms of their component-wise regularity. Our method combines Lagrangian techniques and a penalization scheme with bounded parameters, with parallel forward–backward iterations. Conveniently combined, these techniques allow us to take advantage of the particular structure of the problem. We prove the weak convergence of the sequence generated by this scheme, along with worst-case convergence rates in the convex optimization setting, and for the strongly non-degenerate monotone operator case. Implementation issues related to the penalization of the constraint set are discussed, as well as applications in image recovery and non-Newtonian fluids modeling. A numerical illustration is also given, in order to prove the performance of the algorithm.