An Inexact Deflected Subgradient Algorithm in Infinite Dimensional spaces

TL;DR

This paper introduces a duality framework and a deflected subgradient algorithm for solving complex nonsmooth, nonconvex optimization problems in infinite-dimensional spaces, ensuring convergence to solutions.

Contribution

It develops a duality scheme with a less restrictive coercivity condition and proposes a deflected subgradient method with proven convergence in infinite-dimensional spaces.

Findings

Strong duality established for a broad class of problems.

Convergence of primal and dual sequences proven.

Method applicable to nonsmooth, nonconvex optimization in Banach spaces.

Abstract

We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems in a reflexive Banach space. We establish strong duality for a very general type of augmented Lagrangian, in which we assume a less restrictive type of coercivity on the augmenting function. We solve the dual problem (in a Hilbert space) using a deflected subgradient method via this general augmented Lagrangian. We provide two choices of step-size for the method. For both choices, we prove that every weak accumulation point of the primal sequence is a primal solution. We also prove strong convergence of the dual sequence.