An Algorithm Design Framework for Linearly Constrained Convex Optimization: Proximal and Contractive Perspectives

This article presents an algorithm design framework for general linearly constrained convex optimization problems. Our approach offers a flexible framework to analyze and design various methods, drawing from proximal and contractive viewpoints. This framework utilizes a predictor–corrector mechanism, in which the predictor's role is to generate a predicted solution through proximally contractive mapping, and the corrector's objective is to refine this predicted solution via an inertial update rule. We proceed to provide a convergence analysis for our algorithm design framework by using proximal-point and contraction inequality techniques. Our analysis demonstrates that methods developed within this framework are guaranteed to converge and achieve an ergodic sublinear convergence rate of $\mathit{O}(1/{\mathit{k}})$ in the worst-case scenario. Moreover, our approach is capable of ensuring convergence with a nonergodic rate for the linearly equality-constrained convex optimization problems. Finally, we have verified the effectiveness of our framework and design guidelines via some Lagrangian-based methods for one-block, two-block, and multiblock problems.