Maicon Marques Alves

1. Consider the monotone inclusion problem (1) 0∈∑i=1n Ti(z), where each Ti:H⇉H (1≤i≤n) is a set-valued maximal monotone operator and H is a real Hilbert space endowed with inner product ⟨⋅ | ⋅⟩ an...

We study a bilevel variational inequality problem where the feasible set is itself the solution set of another variational inequality. Motivated by the difficulty of computing projections onto such sets, we consider a regularized extragradient method, as proposed by Samadi and Yousefian (2025), which operates over a simpler constraint set. Building on this framework, we introduce an inertial variant (called IneIREG) that incorporates momentum through extrapolation steps. We establish iteration-complexity bounds for the general (non-strongly monotone) case under both constant and diminishing regularization, and derive improved results under strong monotonicity assumptions. Our analysis extends and refines the results of the previous work by capturing both inertial and regularization effects within a unified framework. Preliminary numerical experiments are also presented to illustrate the behavior of the proposed method.

We propose and study a strongly convergent inexact inertial projective splitting (PS) algorithm for finding zeros of composite monotone inclusion problems involving the sum of finitely many maximal monotone operators. Strong convergence of the iterates is ensured by projections onto the intersection of appropriately defined half-spaces, even in the absence of inertial effects. We also establish iteration-complexity results for the proposed PS method, which likewise hold without requiring inertial terms. The algorithm includes two inertial sequences, controlled by parameters satisfying mild conditions, while preserving strong convergence and enabling iteration-complexity analysis. Furthermore, for more structured monotone inclusion problems, we derive two variants of the main algorithm that employ forward-backward and forward-backward-forward steps.

We propose and study the weak convergence of a projective splitting algorithm studied by A. Alotaibi, P. Combettes, and N. Shahzad [Solving coupled composite monotone inclusions by successive Fejér approximations of their Kuhn-Tucker set, SIAM J. Optim. 24/4 (2014) 2076–2095] and by J. Eckstein and B. F. Svaiter [A family of projective splitting methods for the sum of two maximal monotone operators, Math. Programming Ser. B 111/1-2 (2008) 173–199] for solving multi-term composite monotone inclusion problems involving the finite sum of n maximal monotone operators, each of which having an inner four-block structure: sum of maximal monotone, Lipschitz continuous, cocoercive and smooth differentiable operators. We show how to perform backward and half-forward steps [see L. M. Briceno Arias and D. Davis [Forward-backward-half forward algorithm for solving monotone inclusions, SIAM J. Optim. 28/4 (2018) 2839–2871] with respect to the maximal monotone and Lipschitz + cocoercive components, respectively, while performing proximal-Newton steps with respect to smooth differentiable blocks.

We present and study the iteration-complexity of a relative-error inexact proximal-Newton extragradient algorithm for solving smooth monotone variational inequality problems in real Hilbert spaces. We removed a search procedure from Monteiro and Svaiter (2012) by introducing a novel approach based on homotopy, which requires the resolution (at each iteration) of a single strongly monotone linear variational inequality. For a given tolerance [Formula: see text], our main algorithm exhibits pointwise [Formula: see text] and ergodic [Formula: see text] iteration-complexities. From a practical perspective, preliminary numerical experiments indicate that our main algorithm outperforms some previous proximal-Newton schemes.

We propose a new relative-error inexact version of the alternating direction method of multipliers (ADMM) for convex optimization. We prove the asymptotic convergence of our main algorithm as well as pointwise and ergodic iteration-complexities for residuals. We also justify the effectiveness of the proposed algorithm through some preliminary numerical experiments on regression problems.