Interpolation Conditions for Linear Operators and Applications to Performance Estimation Problems

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 $\mathcal{M}:x \mapsto Mx$ where matrix $M$ has bounded singular values, and the class of linear operators where $M$ 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 $\{(x_i,y_i)\}$, characterize the existence of a linear operator mapping $x_i$ to $y_i$ for all $i$. Using these conditions, we first identify the exact worst-case behavior of the gradient method applied to the composed objective $h\circ \mathcal{M}$, and observe that it always corresponds to $\mathcal{M}$ being a scaling operator. We then investigate the Chambolle-Pock method applied to $f+g\circ \mathcal{M}$, 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.