A smoothing moving balls approximation method for a class of conic-constrained difference-of-convex optimization problems
Jiefeng Xu, Ting Kei Pong, Nung-sing Sze
TL;DR
The paper addresses conic-constrained DC optimization by converting the conic constraint $G(x)\in\mathcal{K}$ into a single inequality using a base $\mathcal{B}$ of the polar cone, $g_{\mathcal{B}}(x)=\sigma_{\mathcal{B}}(G(x))\le 0$, and then smoothing this via a majorizing smoothing approximation $h_{\mu}$ to obtain $g_{\mu}(x)=h_{\mu}(G(x))$. It introduces the smoothing moving balls approximation (sMBA), which performs one MBA step per iteration on the smoothed problem, yielding subproblems with a single inequality that can be solved efficiently by 1D root finding when the proximal mapping of $P_1$ is tractable. The authors establish iteration complexity for obtaining an $\epsilon$-KKT point, provide convergence results in the convex case with local rates under Hölder growth, and demonstrate the practical impact through numerical studies on the choice of smoothing schedules $\{\mu_k\}$. The approach yields feasible iterates and scalable subproblems, offering a practical framework for a broad class of conic DC problems including nonlinear programs and semidefinite programs. Overall, the work advances first-order feasible methods for conic DC optimization by leveraging smoothing of the support function and the moving balls paradigm, with quantified complexity and convergence guarantees.
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 $ε$-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 Hölderian growth condition. Finally, we illustrate numerically the effects of different rules of evolving the smooth approximation functions on the rate of convergence.