2025/01/08 by Quoc Tran-Dinh, Tran-Dinh, Quoc, Nghia Nguyen-Trung +1 · 1 citation
Computer Science · Mathematics · #90-08 #90C06 #90C25 #Advanced Optimization Algorithms Research #FOS: Computer and information sciences #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Machine Learning (stat.ML) #Matrix Theory and Algorithms #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2501.04585
openalex publication_date 2025/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Following the first part of our project, this paper comprehensively studies two types of extragradient-based methods: anchored extragradient and Nesterov's accelerated extragradient for solving [non]linear inclusions (and, in particular, equations), primarily under the Lipschitz continuity and the co-hypomonotonicity assumptions. We unify and generalize a class of anchored extragradient methods for monotone inclusions to a wider range of schemes encompassing existing algorithms as special cases. We establish O(1/k) last-iterate convergence rates on the residual norm of the underlying mapping for this general framework and then specialize it to obtain convergence guarantees for specific instances, where k denotes the iteration counter. We extend our approach to a class of anchored Tseng's forward-backward-forward splitting methods to obtain a broader class of algorithms for solving co-hypomonotone inclusions. Again, we analyze O(1/k) last-iterate convergence rates for this general scheme and specialize it to obtain convergence results for existing and new variants. We generalize and unify Nesterov's accelerated extra-gradient method to a new class of algorithms that covers existing schemes as special instances while generating new variants. For these schemes, we can prove O(1/k) last-iterate convergence rates for the residual norm under co-hypomonotonicity, covering a class of nonmonotone problems. We propose another novel class of Nesterov's accelerated extragradient methods to solve inclusions. Interestingly, these algorithms achieve both O(1/k) and o(1/k) last-iterate convergence rates, and also the convergence of iterate sequences under co-hypomonotonicity and Lipschitz continuity. Finally, we provide a set of numerical experiments encompassing different scenarios to validate our algorithms and theoretical guarantees.