Interpolation Conditions for Linear Operators and Applications to Performance Estimation Problems

SIAM Journal on Optimization, Volume 34, Issue 3, Page 3033-3063, September 2024.
Abstract. The performance estimation problem methodology makes it possible to determine the exact worst-case performance of an optimization method. In this work, we generalize this framework to first-order methods involving linear operators. This extension requires an explicit formulation of interpolation conditions for those linear operators. We consider the class of linear operators [math], where matrix [math] has bounded singular values, and the class of linear operators, where [math] is symmetric and has bounded eigenvalues. We describe interpolation conditions for these classes, i.e., necessary and sufficient conditions that, given a list of pairs [math], characterize the existence of a linear operator mapping [math] to [math] for all [math]. Using these conditions, we first identify the exact worst-case behavior of the gradient method applied to the composed objective [math], and observe that it always corresponds to [math] being a scaling operator. We then investigate the Chambolle–Pock method applied to [math], and improve the existing analysis to obtain a proof of the exact convergence rate of the primal-dual gap. In addition, we study how this method behaves on Lipschitz convex functions, and obtain a numerical convergence rate for the primal accuracy of the last iterate. We also show numerically that averaging iterates is beneficial in this setting.