We consider the Frank-Wolfe algorithm for solving variational inequalities over compact, convex sets under a monotone \(C^1\) operator and vanishing, nonsummable step sizes. We introduce a continuous-time interpolation of the discrete iteration and use tools from dynamical systems theory to analyze its asymptotic behavior. This allows us to derive convergence results for the original discrete … Read more
Many works in convex optimization provide rates for achieving a small primal gap. However, this quantity is typically unavailable in practice. In this work, we show that solving a regularized surrogate with algorithms based on simple primal-dual averaging provides non-asymptotic convergence guarantees for a computable optimality certificate. We first analyze primal and dual methods based … Read more
The Conditional Gradient method offers a computationally efficient, projection-free framework for constrained problems; however, in nonconvex settings it may converge to stationary points of low quality. We propose the Voronoi Conditional Gradient (VCG) method, a geometric heuristic that systematically explores the feasible region by constructing adaptive Voronoi partitions from previously discovered stationary points. VCG incrementally … Read more
This paper studies the primal-dual convergence and iteration-complexity of proximal bundle methods for solving nonsmooth problems with convex structures. More specifically, we develop a family of primal-dual proximal bundle methods for solving convex nonsmooth composite optimization problems and establish the iteration-complexity in terms of a primal-dual gap. We also propose a class of proximal bundle … Read more