Accelerated Schemes For A Class of Variational Inequalities

Yunmei Chen; Guanghui Lan; Yuyuan Ouyang

Home/Archive/arXiv

arXiv · 1403.4164

Accelerated Schemes For A Class of Variational Inequalities

Yunmei Chen·Guanghui Lan·Yuyuan Ouyang

View original source ↗Read source document ↗Export metadata ↓

Abstract

We propose a novel method, namely the accelerated mirror-prox (AMP) method, for computing the weak solutions of a class of deterministic and stochastic monotone variational inequalities (VI). The main idea of this algorithm is to incorporate a multi-step acceleration scheme into the mirror-prox method. For both deterministic and stochastic VIs, the developed AMP method computes the weak solutions with optimal rate of convergence. In particular, if the monotone operator in VI consists of the gradient of a smooth function, the rate of convergence of the AMP method can be accelerated in terms of its dependence on the Lipschitz constant of the smooth function. For VIs with bounded feasible sets, the estimate of the rate of convergence of the AMP method depends on the diameter of the feasible set. For unbounded VIs, we adopt the modified gap function introduced by Monteiro and Svaiter for solving monotone inclusion, and demonstrate that the rate of convergence of the AMP method depends on the distance from the initial point to the set of strong solutions.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Yunmei Chen, Guanghui Lan, Yuyuan Ouyang. 2014-03-17. Accelerated Schemes For A Class of Variational Inequalities. https://arxiv.org/abs/1403.4164

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Strategic Inference in Stackelberg Games: Optimal Control for Revealing Adversary Intent

We study a continuous-time stochastic Stackelberg game in which a leader seeks to accomplish a primary objective while inferring a hidden parameter of a rational follower. The follower solves an entropy-regularized linear-quadratic tracking problem and responds to the leader's trajectory with a randomized policy. Anticipating this response, the leader designs informative controls to maximize the estimation efficiency for the follower's latent intent, through maximum likelihood estimation. Unlike prior work on discrete-time or finite-candidate inverse learning, our framework enables continuous parameter inference without prior assumptions and endogenizes the information source through the follower's strategic feedback. We derive semi-explicit solutions, prove well-posedness, and develop recurrent neural network algorithms to approximate the leader's path-dependent control. Numerical experiments demonstrate how the leader balances task performance and information gain, highlighting the practical value of our approach for adversarial strategic inference.

math.OC↗

arXiv2026

Stratification for Nonlinear Semidefinite Programming

This paper introduces a stratification framework for nonlinear semidefinite programming (NLSDP) that reveals and utilizes the geometry behind the nonsmooth KKT system. Based on the index stratification of $\mathbb{S}^n$ and its lift to the primal-dual space, a stratified variational analysis is developed. Specifically, we define the stratum-restricted regularity property, characterize it by the verifiable weak second order condition (W-SOC) and weak strict Robinson constraint qualification (W-SRCQ), and interpret the W-SRCQ geometrically via transversality, with stability along strata. The interactions of these properties across neighboring strata are further examined, leading to the conclusion that classical strong-form regularity conditions correspond to the local uniform validity of stratum-restricted counterparts. On the algorithmic side, a stratified Gauss--Newton method with normal steps and a correction mechanism is proposed for globally solving the KKT equation through a least-squares merit function. We demonstrate that the algorithm converges globally to directional stationary points. Moreover, under the second order sufficient condition (SOSC) and the strict Robinson constraint qualification (SRCQ) at an accumulation point, with a suitable correction threshold, the whole sequence converges superlinearly to this point, which is a KKT pair, and eventually identifies the active stratum. The rate is quadratic if the problem data are additionally of class $LC^2$ near the solution.

math.OC↗

arXiv2026

Convergence Rate Analysis of SOAP with Arbitrary Orthogonal Projection Matrices

In this short note, we establish, for the first time, the convergence rate of SOAP, an efficient and popular matrix-based optimizer for training deep neural networks. Our analysis extends to a more general variant of SOAP that admits arbitrary orthogonal projection matrices and requires only that these matrices be conditionally independent of the current stochastic gradient at each iteration. For example, they may be constructed from information available up to the preceding step.

math.OC↗