A smoothing moving balls approximation method for a class of conic-constrained difference-of-convex optimization problems

Abstract

In this paper, we consider the problem of minimizing a difference-of-convex objective over a nonlinear conic constraint, where the cone is closed, convex, pointed and has a nonempty interior. We assume that the support function of a compact base of the polar cone exhibits a majorizing smoothing approximation, a condition that is satisfied by widely studied cones such as ${\mathbb R}^m_-$ and ${\cal S}^m_-$. Leveraging this condition, we reformulate the conic constraint equivalently as a single constraint involving the aforementioned support function, and adapt the moving balls approximation (MBA) method for its solution. In essence, in each iteration of our algorithm, we approximate the support function by a smooth approximation function and apply one MBA step. The subproblems that arise in our algorithm always involve only one single inequality constraint, and can thus be solved efficiently via one-dimensional root-finding procedures. We design explicit rules to evolve the smooth approximation functions from iteration to iteration and establish the corresponding iteration complexity for obtaining an $(\epsilon_1, \epsilon_2, \epsilon_1 \sqrt{\epsilon_2})$-Karush-Kuhn-Tucker point. In addition, in the convex setting, we establish convergence of the sequence generated, and study its local convergence rate under a standard H\"olderian growth condition. Finally, we perform numerical experiments to illustrate the performance of our algorithm.