arXiv subjects

Jan Harold Alcantara

Publications and source records attributed to Jan Harold Alcantara.

10 recordsLinked to original sources

arXiv2026

A Splitting Framework for Composite Semimonotone Inclusions

We introduce a general framework for composite inclusion problems with affine constraints, covering both monotone and semimonotone regimes. The central idea is a new interpretation of the constrained inclusion through an operator-vector pair that separates the implicit inclusion from the affine constraint: the former is handled through possibly preconditioned resolvent evaluations, while the latter is handled through an explicit forward step in an auxiliary variable. This yields a single abstract iteration applicable to multioperator inclusions, linearly coupled inclusions, and block-separable inclusions with affine constraints. The freedom in choosing the operator-vector pair enables the systematic construction of problem-adapted splitting algorithms, including new schemes for several important problem classes. Exploiting the orthogonal decomposition induced by the constraint subspace, we develop a unified and streamlined convergence analysis and establish weak and strong convergence guarantees under semimonotonicity assumptions. When specialized to multioperator inclusions, the framework permits general bounded linear operator coefficients, rather than only scalar coefficients, and therefore accommodates preconditioned resolvents. The resulting schemes recover several existing methods while extending them to previously uncovered regimes, and in several important cases, require weaker assumptions and admit provably larger admissible parameter ranges.

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arXiv2026

Accelerated projected gradient algorithms for sparsity constrained optimization problems

We consider the projected gradient algorithm for the nonconvex best subset selection problem that minimizes a given empirical loss function under an $\ell_0$-norm constraint. Through decomposing the feasible set of the given sparsity constraint as a finite union of linear subspaces, we present two acceleration schemes with global convergence guarantees, one by same-space extrapolation and the other by subspace identification. The former fully utilizes the problem structure to greatly accelerate the optimization speed with only negligible additional cost. The latter leads to a two-stage meta-algorithm that first uses classical projected gradient iterations to identify the correct subspace containing an optimal solution, and then switches to a highly-efficient smooth optimization method in the identified subspace to attain superlinear convergence. Experiments demonstrate that the proposed accelerated algorithms are magnitudes faster than their non-accelerated counterparts as well as the state of the art.

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arXiv2025

Douglas--Rachford algorithm for nonmonotone multioperator inclusion problems

The Douglas--Rachford algorithm is a classic splitting method for finding a zero of the sum of two maximal monotone operators. It has also been applied to settings that involve one weakly and one strongly monotone operator. In this work, we extend the Douglas--Rachford algorithm to address multioperator inclusion problems involving $m$ ($m\geq 2$) weakly and strongly monotone operators, reformulated as a two-operator inclusion in a product space. By selecting appropriate parameters, we establish the convergence of the algorithm to a fixed point, from which solutions can be extracted. Furthermore, we illustrate its applicability to sum-of-$m$-functions minimization problems characterized by weakly convex and strongly convex functions. For general nonconvex problems in finite-dimensional spaces, comprising Lipschitz continuously differentiable functions and a proper closed function, we provide global subsequential convergence guarantees.

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arXiv2025

A relaxed version of Ryu's three-operator splitting method for structured nonconvex optimization

In this work, we propose a modification of Ryu's splitting algorithm for minimizing the sum of three functions, where two of them are convex with Lipschitz continuous gradients, and the third is an arbitrary proper closed function that is not necessarily convex. The modification is essential to facilitate the convergence analysis, particularly in establishing a sufficient descent property for an associated envelope function. This envelope, tailored to the proposed method, is an extension of the well-known Moreau envelope. Notably, the original Ryu splitting algorithm is recovered as a limiting case of our proposal. The results show that the descent property holds as long as the stepsizes remain sufficiently small. Leveraging this result, we prove global subsequential convergence to critical points of the nonconvex objective.

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arXiv2025

Douglas--Rachford for multioperator comonotone inclusions with applications to multiblock optimization

We study the convergence of the adaptive Douglas--Rachford (aDR) algorithm for solving a multioperator inclusion problem involving the sum of maximally comonotone operators. To address such problems, we adopt a product space reformulation that accommodates nonconvex-valued operators, which is essential when dealing with comonotone mappings. We establish convergence of the aDR method under comonotonicity assumptions, subject to suitable conditions on the algorithm parameters and comonotonicity moduli of the operators. Our analysis leverages the Attouch--Théra duality framework, which allows us to study the convergence of the aDR algorithm via its application to the dual inclusion problem. As an application, we derive a multiblock ADMM-type algorithm for structured convex and nonconvex optimization problems by applying the aDR algorithm to the operator inclusion formulation of the KKT system. The resulting method extends to multiblock and nonconvex settings the classical duality between the Douglas--Rachford algorithm and the alternating direction method of multipliers in the convex two-block case. Moreover, we establish convergence guarantees for both the fully convex and strongly convex-weakly convex regimes.

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arXiv2025

Theoretical smoothing frameworks for nonsmooth simple bilevel problems

Bilevel programming has recently received a great deal of attention due to its abundant applications in many areas. The optimal value function approach provides a useful reformulation of the bilevel problem, but its utility is often limited due to the nonsmoothness of the value function even in cases when the associated lower-level function is smooth. In this paper, we present two smoothing strategies for the value function associated with lower-level functions that are not necessarily smooth but are Lipschitz continuous. The first method employs quadratic regularization for partially convex lower-level functions, while the second utilizes entropic regularization for general lower-level objective functions. Meanwhile, the property known as gradient consistency is crucial in ensuring that a designed smoothing algorithm is globally subsequentially convergent to stationary points of the value function reformulation. With this motivation, we prove that the proposed smooth approximations satisfy the gradient consistent property under certain conditions on the lower-level function.

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arXiv2025

A four-operator splitting algorithm for nonconvex and nonsmooth optimization

In this work, we address a class of nonconvex nonsmooth optimization problems where the objective function is the sum of two smooth functions (one of which is proximable) and two nonsmooth functions (one proper, closed and proximable, and the other continuous and weakly concave). We introduce a new splitting algorithm that extends the Davis-Yin splitting (DYS) algorithm to handle such four-term nonconvex nonsmooth problems. We prove that with appropriately chosen stepsizes, our algorithm exhibits global subsequential convergence to stationary points with a stationarity measure converging at a global rate of $1/T$, where $T$ is the number of iterations. When specialized to the setting of the DYS algorithm, our results allow for larger stepsizes compared to existing bounds in the literature. Experimental results demonstrate the practical applicability and effectiveness of our proposed algorithm.

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arXiv2023

Global convergence and acceleration of projection methods for feasibility problems involving union convex sets

We prove global convergence of classical projection algorithms for feasibility problems involving union convex sets, which refer to sets expressible as the union of a finite number of closed convex sets. We present a unified strategy for analyzing global convergence by means of studying fixed-point iterations of a set-valued operator that is the union of a finite number of compact-valued upper semicontinuous maps. Such a generalized framework permits the analysis of a class of proximal algorithms for minimizing the sum of a piecewise smooth function and the difference between pointwise minimum of finitely many weakly convex functions and a piecewise smooth convex function. When realized on two-set feasibility problems, this algorithm class recovers alternating projections and averaged projections as special cases, and thus we obtain global convergence criterion for these projection algorithms. Using these general results, we derive sufficient conditions to guarantee global convergence for several projection algorithms for solving the sparse affine feasibility problem and a feasibility reformulation of the linear complementarity problem. Notably, we obtain global convergence of both the alternating and the averaged projection methods to the solution set for linear complementarity problems involving $P$-matrices. By leveraging the structures of the classes of problems we consider, we also propose acceleration algorithms with guaranteed global convergence. Numerical results further exemplify that the proposed acceleration schemes significantly improve upon their non-accelerated counterparts in efficiency.

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arXiv2023

Unified Smoothing Approach for Best Hyperparameter Selection Problem Using a Bilevel Optimization Strategy

Strongly motivated from use in various fields including machine learning, the methodology of sparse optimization has been developed intensively so far. Especially, the recent advance of algorithms for solving problems with nonsmooth regularizers is remarkable. However, those algorithms suppose that weight parameters of regularizers, called hyperparameters hereafter, are pre-fixed, and it is a crucial matter how the best hyperparameter should be selected. In this paper, we focus on the hyperparameter selection of regularizers related to the $\ell_p$ function with $0<p\le 1$ and apply a bilevel programming strategy, wherein we need to solve a bilevel problem, whose lower-level problem is nonsmooth, possibly nonconvex and non-Lipschitz. Recently, for solving a bilevel problem for hyperparameter selection of the pure $\ell_p\ (0<p \le 1)$ regularizer Okuno et al. discovered new necessary optimality conditions, called SB(scaled bilevel)-KKT conditions, and further proposed a smoothing-type algorithm using a certain smoothing function. However, this optimality measure is loose in the sense that there could be many points that satisfy the SB-KKT conditions. In this work, we propose new bilevel KKT conditions, which are new necessary optimality conditions tighter than the ones proposed by Okuno et al. Moreover, we propose a unified smoothing approach using smoothing functions that belong to the Chen-Mangasarian class, and then prove that generated iteration points accumulate at \alert{bilevel KKT points under milder constraint qualifications. Another contribution is that our approach and analysis are applicable to a wider class of regularizers. Numerical comparisons demonstrate which smoothing functions work well for hyperparameter optimization via bilevel optimization approach.

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arXiv2021

Method of Alternating Projection for the Absolute Value Equation

A novel approach for solving the general absolute value equation $Ax+B|x| = c$ where $A,B\in \mathbb{R}^{m\times n}$ and $c\in \mathbb{R}^m$ is presented. We reformulate the equation as a feasibility problem which we solve via the method of alternating projections (MAP). The fixed points set of the alternating projections map is characterized under nondegeneracy conditions on $A$ and $B$. Furthermore, we prove linear convergence of the algorithm. Unlike most of the existing approaches in the literature, the algorithm presented here is capable of handling problems with $m\neq n$, both theoretically and numerically.

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