arXiv subjects

V. Vetrivel

Publications and source records attributed to V. Vetrivel.

12 recordsLinked to original sources

arXiv2026

Two Adaptive Accelerated Golden Ratio Primal--Dual Algorithms With an Application to Poisson Imaging Problem

This paper revisits the adaptive extended golden-ratio primal--dual algorithm (aEGRPDA) proposed by Soe et al. (2026) for structured convex optimisation problems involving a differentiable term that is only locally smooth. We prove that the artificial upper bound imposed on the primal step-size in aEGRPDA is redundant, since the adaptive rule itself keeps the step-sizes bounded above. As a consequence, the ergodic $\mathcal O(1/N)$ estimates for the objective residual and feasibility violation, where $N\ge1$ denotes the number of iterations, are independent of this hyperparameter. Consequently, the resulting adaptive golden-ratio primal--dual method, therefore, requires neither a step-size cap, nor a linesearch procedure, nor a known global Lipschitz constant. We establish linear convergence of the algorithm when both the primal and dual functions are strongly convex. Furthermore, we develop two accelerated variants, in addition to the local smoothness assumption: one for the case where the nonsmooth primal component is strongly convex, and another for the case where the differentiable term is globally strongly convex. For these accelerated methods, we prove an ergodic $\mathcal O(1/N^2)$ convergence rate. Preliminary numerical experiments on a Poisson imaging problem illustrate the efficiency and robustness of the proposed approaches.

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arXiv2026

Proximal Gradient Methods for Unconstrained Set Optimization Problems with Set-Valued Maps of Finite Cardinality

This work presents two different types of proximal gradient methods, with line search and without line search, for solving unconstrained set-valued optimization problems under the lower set-less ordering relation induced by a solid cone that is convex, pointed, and closed. The objective mapping of the problem involves finitely many functions, with each one being the sum of a continuously differentiable function and a convex function that is proper and closed. We present an approach to characterize weakly minimal points of the problem with the help of weakly efficient points of a family of vector optimization problems. Thereafter, we establish a stationarity condition along with its connection with weakly minimal points of the problem under study. Based on the stationary condition, the concept of a descent direction at a non-stationary point is discussed. In view of the line search-based method, we formulate an Armijo-type line search condition and establish the existence of such a step-size. For the proposed methods, global convergence is established under mild assumptions. The convergence analysis of the proximal gradient method with line search provides a theoretical advancement over the convergence results previously established for the steepest descent method in set-valued optimization problems. In addition, we analyze the computational complexity of the proposed methods and show that both methods achieve a convergence rate of $\mathcal{O}(1/\sqrt{k})$. Numerical results are reported to test the performance of the methods in practice.

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arXiv2026

The Golden Ratio Proximal ADMM with Norm Independent Step-Sizes for Separable Convex Optimization

In this work, we propose two step-size strategies for the Golden ratio proximal ADMM (GrpADMM) to solve linearly constrained separable convex optimization problems. Both strategies eliminate explicit operator norm estimates by relying on inexpensive local information computed at the current iterate and requiring no backtracking. However, the key difference is that the second step-size strategy allows recovery from poor initial steps and can increase from iteration to iteration. Under standard assumptions, we establish global convergence of the generated iterates and derive sublinear convergence rates for both algorithms. We also obtain pointwise convergence rate results for the iterates of the algorithms. In addition, we show that the first proposed step-size rule for GrpADMM reduces to the fixed-step-size counterpart when the initial step-size is chosen below a certain threshold. Preliminary numerical experiments demonstrate the practical adaptability and effectiveness of the proposed approaches.

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arXiv2026

On the Convexity of the Solution Set of Linear Complementarity Problem over Tensor Spaces

This paper investigates the convexity of the solution set of the linear complementarity problems over tensor spaces (TLCPs). We introduce the notion of a $T$-column sufficient tensor and study its properties and relationships with several structured tensors. An equivalent condition for the convexity of the solution set of the $\mathrm{TLCP}$ is established. In addition, sufficient conditions for uniqueness and for feasibility implying solvability are derived.

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arXiv2026

On Generalized Forward-Reflected-Backward Method for Monotone Inclusion Problems

We study the generalized forward-reflected-backward (GFRB) method, an extension of the forward-reflected-backward (FRB) scheme due to Malitsky and Tam, for solving monotone inclusion problems in real Hilbert spaces. We first analyze GFRB equipped with a non-decreasing step-size rule that does not require prior knowledge of the Lipschitz constant of the operator involved. We then present two illustrative examples: in the first, we show that the convergence rate of GFRB is bounded from below by that of FRB, and in the second, we obtain an improved convergence rate for GFRB via an appropriate choice of initial parameters. In the sequel, we propose an extended primal-dual twice-reflected (PDTR) algorithm and show that it can be recovered from GFRB under suitable metric selections. Finally, we validate the proposed approach on several state-of-the-art problems and demonstrate better numerical performance compared to the existing ones.

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arXiv2025

The Golden Ratio Primal-Dual Algorithm with Two New Stepsize Rules for Convex-Concave Saddle Point Problems

In this paper, we present two stepsize strategies for the extended Golden Ratio primal-dual algorithm (E-GRPDA) designed to address structured convex optimization problems in finite-dimensional real Hilbert spaces. The first rule features a non-increasing primal stepsize that remains bounded below by a positive constant and is updated adaptively at each iteration, eliminating the need to compute the Lipschitz constant of the gradient of the function and the norm of the operator, without using backtracking. The second stepsize rule is adaptive, adjusting based on the local smoothness of the smooth component function and the norm of the operator involved. In other words, we present an adaptive version of the E-GRPDA algorithm. We prove that E-GRPDA achieves an ergodic sublinear convergence rate with both stepsize rules, based on the function-value residual and constraint violation rather than on the so-called primal-dual gap function. Additionally, we establish an R-linear convergence rate for E-GRPDA with the first stepsize rule, under standard assumptions and with appropriately chosen parameters. Through numerical experiments on various convex optimization problems, we demonstrate the effectiveness of our approaches and compare their performance to existing ones.

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arXiv2025

On the Finiteness Property of the Polynomial Complementarity Problem

This paper explores the finiteness of the solution set of the polynomial complementarity problem (PCP). To achieve this goal, we introduce two new classes of structured tensor tuples, namely the nondegenerate tensor tuple and the strong nondegenerate tensor tuple, as a generalization of nondegenerate tensors, and discuss their properties and interconnections. We investigate the finiteness of the solution set of the PCP in the context of these structured tensor tuples and establish a sufficient condition that guarantees a finite solution set. As a consequence, we establish a result related to the finiteness of the solution set of tensor complementarity problems.

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arXiv2025

Weak convergence of projection algorithm with momentum terms and new step size rule for quasimonotone variational inequalities

This article analyses the simple projection method proposed by Izuchukwu et al. [8, Algorithm 3.2] for solving variational inequality problems by incorporating momentum terms. A new step size strategy is also introduced, in which the step size sequence increases after a finite number of iterations. Under the assumptions that the underlying operator is quasimonotone and Lipschitz continuous, we establish weak convergence of the proposed method. The effectiveness and efficiency of the algorithm are demonstrated through numerical experiments and are compared with existing approaches from the literature. Finally, we apply the proposed algorithm to a signal recovery problem.

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arXiv2025

Extended Horizontal Tensor Complementarity Problems

In this paper, we study the nonemptiness, compactness, uniqueness, and finiteness of the solution set of a new type of nonlinear complementarity problem, namely the extended horizontal tensor complementarity problem (EHTCP). We introduce several classes of structured tensors and discuss the interconnections among these tensors. Consequently, we study the properties of the solution set of the EHTCP with the help of degree theory.

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arXiv2025

Modified Bregman Golden Ratio Algorithm for Mixed Variational Inequality Problems

In this article, we provide a modification to the Bregman Golden Ratio Algorithm (B-GRAAL). We analyze the B-GRAAL algorithm with a new step size rule, where the step size increases after a certain number of iterations and does not require prior knowledge of the global Lipschitz constant of the cost operator. Under suitable assumptions, we establish the global iterate convergence as well as the R-linear rate of convergence of the modified algorithm. The numerical performance of the proposed approach is validated for the matrix game problem and the sparse logistic regression problem in machine learning.

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arXiv2024

Forward-backward-forward dynamics for bilevel equilibrium problem

We introduce a forward-backward-forward (FBF) algorithm for solving bilevel equilibrium problem associated with bifunctions on a real Hilbert space. This modifies the forward-backward algorithm by relaxing cocoercivity with monotone and Lipschitzness. Further, we present the FBF dynamical system and investigate the generated trajectory's existence, uniqueness and weak convergence. We illustrate the proposed method for equilibrium problem under saddle point constraint.

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arXiv2023

Well-posedness for the split equilibrium problem

We extend the concept of well-posedness to the split equilibrium problem and establish Furi-Vignoli-type characterizations for the well-posedness. We prove that the well-posedness of the split equilibrium problem is equivalent to the existence and uniqueness of its solution under certain assumptions on the bifunctions involved. We also characterize the generalized well-posedness of the split equilibrium problem via the Kuratowski measure of noncompactness. We illustrate our theoretical results by several examples.

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