Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.34975

GRAAL for variational inequalities revisited: sharp parameter range and adaptive step sizes

Santanu Soe·Matthew K. Tam·V. Vetrivel
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper revisits the Golden Ratio Algorithm (GRAAL) of Malitsky (Math. Program. 184, 383--410, 2020) for monotone variational inequalities. The original analysis of GRAAL restricts the convex-combination parameter $c$ to $1<c\leqφ$, where $φ=(1+\sqrt5)/2$ is the golden ratio. Through a new Lyapunov analysis, we show that this range of $c$ can be extended to $1<c<3$. Further, a two-dimensional linear counterexample shows that this parameter range is sharp. In particular, for every $c\geq3$ and every positive constant step size, the iterates of GRAAL can be unbounded. Building on this analysis, we propose a simpler version of the adaptive GRAAL (aGRAAL) that requires neither a line search nor a prescribed upper bound on the step sizes, and demonstrate its convergence for locally Lipschitz cost operators over the same parameter range. For both methods, we establish an $O(1/N)$ ergodic gap bound and, under a residual-based error bound condition, prove the R-linear convergence of the adaptive iterates. Finally, numerical experiments are conducted to compare the proposed strategies with existing ones.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Santanu Soe, Matthew K. Tam, V. Vetrivel. 2026-10-26. GRAAL for variational inequalities revisited: sharp parameter range and adaptive step sizes. https://arxiv.org/abs/2610.34975

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

KEEP EXPLORING

Related papers

arXiv2026

Mini-Batch Stochastic Conjugate Gradient Algorithms with Minimal Variance

Stochastic gradient descent (SGD) methods have gained widespread popularity for solving large-scale optimization problems. However, the inherent variance in SGD often leads to slow convergence rates. We introduce a family of unbiased stochastic gradient estimators that encompasses existing estimators from the literature and identify a gradient estimator that not only maintains unbiasedness but also achieves minimal variance. Compared with the existing estimator used in SGD algorithms, the proposed estimator demonstrates a significant reduction in variance. By utilizing this stochastic gradient estimator to approximate the full gradient, we propose two mini-batch stochastic conjugate gradient algorithms with minimal variance. Under the assumptions of strong convexity and smoothness on the objective function, we prove that the two algorithms achieve linear convergence rates. Numerical experiments validate the effectiveness of the proposed gradient estimator in reducing variance and demonstrate that the two stochastic conjugate gradient algorithms exhibit accelerated convergence rates and enhanced stability.

math.OC↗
arXiv2026

Analysis of Primal-Dual Langevin Algorithms

We analyze a recently proposed class of algorithms for the problem of sampling from probability distributions $μ^\ast$ in $\mathbb{R}^d$ with a Lebesgue density of the form $μ^\ast(x) \propto \exp(-f(Kx)-g(x))$, where $K$ is a linear operator and $f,g$ convex and non-smooth. The method is a generalization of the primal-dual hybrid gradient optimization algorithm to a sampling scheme. We give the iteration's continuous time limit, a stochastic differential equation in the joint primal-dual variable, and its mean field limit Fokker-Planck equation. Under mild conditions, the scheme converges to a unique stationary state in continuous and discrete time. Contrary to purely primal overdamped Langevin diffusion, the stationary state in continuous time does not have $μ^\ast$ as its primal marginal. Thus, further analysis is carried out to bound the bias induced by the partial dualization, and potentially correct for it in the diffusion. Time discretizations of the diffusion lead to implementable algorithms, but, as is typical in Langevin Monte Carlo methods, introduce further bias. We prove bounds for these discretization errors, which allow to give convergence results relating the produced samples to the target. We demonstrate our findings numerically first on small-scale examples in which we can exactly verify the theoretical results, and subsequently on typical examples of larger scale from Bayesian imaging inverse problems.

math.OC↗
arXiv2026

An Efficient Parallel Solver for Linear-Equality-Constrained LQR Problems

We present a parallel solver for finite-horizon linear-quadratic regulator (LQR) problems with arbitrary stagewise linear equality constraints, using linear work and logarithmic depth in the horizon for fixed local dimensions. The solver uses a two-part reduction that separates feasibility from optimization. First, an associative suffix scan computes the states from which each remaining horizon is feasible. Second, we use these feasible-continuation sets to eliminate the constraints independently at each stage, producing an unconstrained LQR problem with smaller or equal, possibly stage-varying dimensions. A parallel unconstrained-LQR solver completes the primal solution. Finally, we recover valid multipliers from the original KKT stationarity equations using a logarithmic-depth contraction-expansion procedure, even when the multipliers are not unique. Sequential and parallel implementations are validated against dense KKT solutions and original residuals. We release a C++/CUDA implementation with JAX bindings under the MIT license.

math.OC↗