Convex programming under a linear optimization oracle The Complexity of Large-scale Convex Programming under a Linear Optimization Oracle

This paper considers a general class of iterative optimization algorithms, referred to as linearoptimization-based convex programming (LCP) methods, for solving large-scale convex programming (CP) problems. The LCP methods, covering the classic conditional gradient (CndG) method (a.k.a., Frank-Wolfe method) as a special case, can only solve a linear optimization subproblem at each iteration. In this paper, we first establish a series of lower complexity bounds for the LCP methods to solve different classes of CP problems, including smooth, nonsmooth and certain saddle-point problems. We then formally establish the theoretical optimality or nearly optimality, in the largescale case, for the CndG method and its variants to solve different classes of CP problems. We also introduce several new optimal LCP methods, obtained by properly modifying Nesterov’s accelerated gradient method, and demonstrate their possible advantages over the classic CndG for solving certain classes of large-scale CP problems.