arXiv subjects

Alfredo Iusem

Publications and source records attributed to Alfredo Iusem.

8 recordsLinked to original sources

arXiv2025

On circumcentered direct methods for monotone variational inequality problems

Circumcentered techniques have been shown to significantly accelerate projection-based methods for convex feasibility problems. Motivated by this success, we propose two direct methods with circumcenter acceleration for solving variational inequality problems involving two classes of operators: paramonotone and monotone. Both schemes rely on approximate projections onto separating halfspaces, thereby avoiding computationally expensive exact projections. We establish convergence results for both methods and conduct numerical experiments, demonstrating that the proposed algorithms outperform classical methods, such as the extragradient algorithm, by orders of magnitude in terms of computational time, particularly when the feasible set is a complex intersection of convex sets.

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arXiv2025

Asymptotic Convergence Analysis of High-Order Proximal-Point Methods Beyond Sublinear Rates

This paper investigates the asymptotic convergence behavior of the high-order proximal-point algorithm (HiPPA) to global minimizers, extending existing analyses beyond sublinear convergence rates and complexity analysis. Specifically, we study the proximal operator of a proper lower semicontinuous function augmented with a $p$th-order regularization for $p>1$, and establish the convergence of HiPPA to a global minimizer with a particular focus on its convergence rate. To this end, we focus on minimizing functions in the class of uniformly quasiconvex functions, which includes strongly convex, uniformly convex, and strongly quasiconvex functions as special cases. Our analysis reveals the following convergence behaviors of HiPPA when the uniform quasiconvexity modulus $\phi$ admits a power function of degree $q$ as a lower bound, i.e., $\phi(t) \geq c t^q$ for some $c>0$, on an interval $\mathcal{I}$: (i) for $q\in (1,2)$ and $\mathcal{I}=[0,1)$, HiPPA exhibits a local linear rate for $p\in [q,2)$; (ii) HiPPA converges linearly when $p=2$, $q=2$, and also when $p=q>2$, provided that $\mathcal{I}=[0,\infty)$; (iii) for $q\geq 2$ and $\mathcal{I}=[0,\infty)$, HiPPA achieves a superlinear rate for $p>q$. Notably, to our knowledge, some of these results are novel, even in the context of strongly or uniformly convex functions, offering new insights into optimizing generalized convex problems.

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arXiv2023

A successive centralized circumcenter reflection method for the convex feasibility problem

In this paper we present the successive centralization of the circumcenter reflection scheme with several control sequences for solving the convex feasibility problem in Euclidean space. Assuming that a standard error bound holds, we prove the linear convergence of the method with the most violated constraint control sequence. Under additional smoothness assumptions, we prove the superlinear convergence. Numerical experiments confirm the efficiency of our method.

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arXiv2023

A finitely convergent circumcenter method for the Convex Feasibility Problem

In this paper, we present a variant of the circumcenter method for the Convex Feasibility Problem (CFP), ensuring finite convergence under a Slater assumption. The method replaces exact projections onto the convex sets with projections onto separating halfspaces, perturbed by positive exogenous parameters that decrease to zero along the iterations. If the perturbation parameters decrease slowly enough, such as the terms of a diverging series, finite convergence is achieved. To the best of our knowledge, this is the first circumcenter method for CFP that guarantees finite convergence.

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arXiv2021

Circumcentering approximate reflections for solving the convex feasibility problem

The circumcentered-reflection method (CRM) has been applied for solving convex feasibility problems. CRM iterates by computing a circumcenter upon a composition of reflections with respect to convex sets. Since reflections are based on exact projections, their computation might be costly. In this regard, we introduce the circumcentered approximate-reflection method (CARM), whose reflections rely on outer-approximate projections. The appeal of CARM is that, in rather general situations, the approximate projections we employ are available under low computational cost. We derive convergence of CARM and linear convergence under an error bound condition. We also present successful theoretical and numerical comparisons of CARM to the original CRM, to the classical method of alternating projections (MAP) and to a correspondent outer-approximate version of MAP, referred to as MAAP. Along with our results and numerical experiments, we present a couple of illustrative examples.

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arXiv2017

Variance-based stochastic extragradient methods with line search for stochastic variational inequalities

A dynamic sampled stochastic approximated (DS-SA) extragradient method for stochastic variational inequalities (SVI) is proposed that is \emph{robust} with respect to an unknown Lipschitz constant $L$. To the best of our knowledge, it is the first provably convergent \emph{robust} SA \emph{method with variance reduction}, either for SVIs or stochastic optimization, assuming just an unbiased stochastic oracle in a large sample regime. This widens the applicability and improves, up to constants, the desired efficient acceleration of previous variance reduction methods, all of which still assume knowledge of $L$ (and, hence, are not robust against its estimate). Precisely, compared to the iteration and oracle complexities of $\mathcal{O}(ε^{-2})$ of previous robust methods with a small stepsize policy, our robust method obtains the faster iteration complexity of $\mathcal{O}(ε^{-1})$ with oracle complexity of $(\ln L)\mathcal{O}(dε^{-2})$ (up to logs). This matches, up to constants, the sample complexity of the sample average approximation estimator which does not assume additional problem information (such as $L$). Differently from previous robust methods for ill-conditioned problems, we allow an unbounded feasible set and an oracle with multiplicative noise (MN) whose variance is not necessarily uniformly bounded. These properties are seen in our complexity estimates which depend only on $L$ and local second or forth moments at solutions. The robustness and variance reduction properties of our DS-SA line search scheme come at the expense of nonmartingale-like dependencies (NMD) due to the needed inner statistical estimation of a lower bound for $L$. In order to handle a NMD and a MN, our proofs rely on a novel localization argument based on empirical process theory. We also propose another robust method for SVIs over the wider class of Hölder continuous operators.

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arXiv2017

Incremental constraint projection methods for monotone stochastic variational inequalities

We consider stochastic variational inequalities with monotone operators defined as the expected value of a random operator. We assume the feasible set is the intersection of a large family of convex sets. We propose a method that combines stochastic approximation with incremental constraint projections meaning that at each iteration, a step similar to some variant of a deterministic projection method is taken after the random operator is sampled and a component of the intersection defining the feasible set is chosen at random. Such sequential scheme is well suited for applications involving large data sets, online optimization and distributed learning. First, we assume that the variational inequality is weak-sharp. We provide asymptotic convergence, feasibility rate of $O(1/k)$ in terms of the mean squared distance to the feasible set and solvability rate of $O(1/\sqrt{k})$ (up to first order logarithmic terms) in terms of the mean distance to the solution set for a bounded or unbounded feasible set. Then, we assume just monotonicity of the operator and introduce an explicit iterative Tykhonov regularization to the method. We consider Cartesian variational inequalities so as to encompass the distributed solution of stochastic Nash games or multi-agent optimization problems under a limited coordination. We provide asymptotic convergence, feasibility rate of $O(1/k)$ in terms of the mean squared distance to the feasible set and, in the case of a compact set, we provide a near-optimal solvability convergence rate of $O\left(\frac{k^δ\ln k}{\sqrt{k}}\right)$ in terms of the mean dual gap-function of the SVI for arbitrarily small $δ>0$.

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arXiv2017

Extragradient method with variance reduction for stochastic variational inequalities

We propose an extragradient method with stepsizes bounded away from zero for stochastic variational inequalities requiring only pseudo-monotonicity. We provide convergence and complexity analysis, allowing for an unbounded feasible set, unbounded operator, non-uniform variance of the oracle and, also, we do not require any regularization. Alongside the stochastic approximation procedure, we iteratively reduce the variance of the stochastic error. Our method attains the optimal oracle complexity $\mathcal{O}(1/ε^2)$ (up to a logarithmic term) and a faster rate $\mathcal{O}(1/K)$ in terms of the mean (quadratic) natural residual and the D-gap function, where $K$ is the number of iterations required for a given tolerance $ε>0$. Such convergence rate represents an acceleration with respect to the stochastic error. The generated sequence also enjoys a new feature: the sequence is bounded in $L^p$ if the stochastic error has finite $p$-moment. Explicit estimates for the convergence rate, the oracle complexity and the $p$-moments are given depending on problem parameters and distance of the initial iterate to the solution set. Moreover, sharper constants are possible if the variance is uniform over the solution set or the feasible set. Our results provide new classes of stochastic variational inequalities for which a convergence rate of $\mathcal{O}(1/K)$ holds in terms of the mean-squared distance to the solution set. Our analysis includes the distributed solution of pseudo-monotone Cartesian variational inequalities under partial coordination of parameters between users of a network.

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