Non-smooth DC-constrained optimization: constraint qualification and minimizing methodologies

This work concerns the study of a constraint qualification for non-smooth DC-constrained optimization problems, as well as the design and convergence analysis of minimizing algorithms to address the task of computing a stationary/critical point for problems of this class. Specialized algorithms for DC programming approximate the non-convex optimization problem by a sequence of convex subproblems, obtained by linearizing the second components of the involved DC (difference of convex) functions. We propose new approaches that define trial points as inexact solutions of such convex subproblems. This is a property of practical interest that substantially reduces the computational burden to compute a stationary/critical point of non-smooth DC-constrained optimization problems. One variant of the proposed algorithmic patterns is numerically assessed on a DC reformulation of an energy management problem considering a smart-grid controlled by a local actor (follower) and its interaction with a global actor (leader) in the power system.