Anchored Extra-Proximal Methods: Optimal Higher-Order Methods for Monotone Inclusion Problems

2026-09-24

AI Summary

This study addresses the challenge of optimizing higher-order deterministic oracle complexity for composite monotone inclusion problems by proposing an Anchored Extrapolated Proximal (AEP) algorithmic framework. The proposed method integrates Taylor approximation, bisection line search, and proximal updates with relative inexactness, achieving a higher-order extension through an anchored extrapolation mechanism. Theoretically, it is proven that the AEP framework attains an optimal oracle complexity of O(ε^{-2/(3p-1)}), which improves upon all previously known upper bounds and matches the worst-case theoretical lower bound. Consequently, this work provides the first optimal higher-order algorithm for solving this class of problems.

Abstract

We study the deterministic oracle complexity of finding approximate solutions to composite monotone inclusion problems, formed by the sum of a smooth single-valued monotone operator and a maximally monotone set-valued operator, under the tangent-residual criterion. We introduce the Anchored Extra-Proximal (AEP) framework, which combines an anchored extrapolation step with an inexact anchored proximal update satisfying a relative-error condition. The framework recovers the composite Fast Extragradient method in the first-order setting and yields natural second- and higher-order extensions by replacing the operator in the implicit update with its Taylor approximation at the extrapolated point. For every $p\geq 2$, assuming that the $(p-1)$th derivative of the single-valued operator is Lipschitz continuous, we combine this construction with a bisection line search to obtain a $p$th-order method that finds a point with tangent residual at most $\varepsilon$ in $\widetilde{O}(\varepsilon^{-2/(3p-1)})$ oracle calls. This improves all prior upper bounds for $p$th-order methods: in particular, it improves the previous best-known $\widetilde{O}(\varepsilon^{-1/p})$ tangent-residual complexity as well as the classical $O(\varepsilon^{-2/(p+1)})$ bound of higher-order hybrid proximal extragradient methods under the weaker duality-gap criterion. We complement this result with a worst-case lower bound of $Ω(\varepsilon^{-2/(3p-1)})$ for every deterministic algorithm in the $p$th-order oracle model, without restricting the algorithm to tensor steps or any other prescribed update structure. Thus, the proposed method attains the optimal dependence on $\varepsilon$, up to logarithmic factors, for all $p\geq2$.

Problem

• monotone inclusion problems
• oracle complexity
• higher-order methods
• tangent-residual criterion
• composite operators

Innovation

• Anchored Extra-Proximal Methods
• Monotone Inclusion Problems
• Higher-Order Methods
• Oracle Complexity
• Tangent Residual