Primal-dual extrapolation methods for monotone inclusions under local Lipschitz continuity
Zhaosong Lu, Sanyou Mei
TL;DR
The paper studies monotone inclusion problems of the form $0\in(F+B)(x)$ where $B$ is maximal monotone and $F$ is monotone and locally Lipschitz on ${\mathrm{cl}}(\mathrm{dom}\,B)$. It develops primal-dual extrapolation methods with backtracking line search to handle local Lipschitz continuity, providing a verifiable termination criterion and strong complexity guarantees: $\mathcal{O}(\log \varepsilon^{-1})$ iterations for the strongly monotone case ($\mu>0$) and $\mathcal{O}(\varepsilon^{-1}\log \varepsilon^{-1})$ for the non-strongly monotone case ($\mu=0$), substantially improving the previous $\mathcal{O}(\varepsilon^{-2})$ rate. The approach is extended to important problem classes, including convex conic optimization, conic constrained saddle point problems, and variational inequalities, yielding $\varepsilon$-KKT or $\varepsilon$-residual solutions under local Lipschitz continuity without requiring global Lipschitzness or bounded domains. Numerical results on a synthetic test problem demonstrate that primal-dual extrapolation accelerates convergence relative to existing first-order methods. These results broaden the applicability of fast first-order schemes to broad MI settings with only local regularity assumptions.
Abstract
In this paper we consider a class of monotone inclusion (MI) problems of finding a zero of the sum of two monotone operators, in which one operator is maximal monotone while the other is {\it locally Lipschitz} continuous. We propose primal-dual extrapolation methods to solve them using a point and operator extrapolation technique, whose parameters are chosen by a backtracking line search scheme. The proposed methods enjoy an operation complexity of ${\cal O}(\log ε^{-1})$ and ${\cal O}(ε^{-1}\log ε^{-1})$, measured by the number of fundamental operations consisting only of evaluations of one operator and resolvent of the other operator, for finding an $\varepsilon$-residual solution of strongly and non-strongly MI problems, respectively. The latter complexity significantly improves the previously best operation complexity ${\cal O}(\varepsilon^{-2})$. As a byproduct, complexity results of the primal-dual extrapolation methods are also obtained for finding an $\varepsilon$-KKT or $\varepsilon$-residual solution of convex conic optimization, conic constrained saddle point, and variational inequality problems under {\it local Lipschitz} continuity. We provide preliminary numerical results to demonstrate the performance of the proposed methods.