arXiv subjects

Keita Kume

Publications and source records attributed to Keita Kume.

14 recordsLinked to original sources

arXiv2026

Asymptotic Analysis of Gradient Mapping-type Stationarity Measure for the Sum of Nonconvex Nonsmooth Functions and Applications to Proximal Gradient-type Algorithms

We propose a gradient mapping-type stationarity measure for the sum of two possibly nonsmooth nonconvex functions. Under suitable regularity assumptions, we show that Fréchet or proximal stationarity can be characterized through asymptotic vanishing of the proposed measure along some convergent sequence. We also establish analogous asymptotic results for a smoothing-based variant of the measure, which enables us to combine the proposed framework with smoothing techniques developed for nonsmooth optimization. As an application of our analysis of the stationarity measure, we provide an affirmative answer to an open question raised by [Olikier-Waldspurger, SIAM J. Optim., 2025] on whether every cluster point of a sequence generated by a proximal gradient method is a proximal stationary point under local Lipschitz smoothness of one component of the cost function. As a second application, we propose a proximal variable smoothing algorithm with a nonmonotone linesearch for minimizing the sum of two nonsmooth nonconvex functions under lower regularity of one component and prox-regularity of the other. For the proposed algorithm, we show that every cluster point of a subsequence such that the stationarity measure vanishes is a Fréchet stationary point.

math.OC↗

arXiv2026

A Proximal Variable Smoothing for Minimization of Nonlinearly Composite Nonsmooth Function -- Finite-Max Minimization and MIMO Applications

We propose a proximal variable smoothing algorithm for a nonsmooth optimization problem whose cost function is the sum of three functions including a weakly convex composite function. The proposed algorithm has a single-loop structure inspired by a proximal gradient-type method. More precisely, the proposed algorithm consists of two steps: (i) a gradient descent of a time-varying smoothed surrogate function designed partially with the Moreau envelope of the weakly convex function; (ii) an application of the proximity operator of the remaining function not covered by the smoothed surrogate function. For the proposed algorithm, we present a subsequential convergence guarantee in terms of a stationary point, and a convergence rate ${O}(ε^{-3})$ for achieving an $ε$-stationary point. Numerical experiments demonstrate the effectiveness of the proposed algorithm in two scenarios: (i) robust target localization and (ii) multiple-input-multiple-output (MIMO) signal detection.

math.OC↗

arXiv2026

A DC Composite Optimization via Variable Smoothing for Robust Phase Retrieval with Nonconvex Loss Functions

In this paper, we propose an optimization-based method for robust phase retrieval problem where the goal is to estimate an unknown signal from a quadratic measurement corrupted by outliers. To enhance the robustness of existing optimization models with the $\ell_1$ loss function, we propose a generalized model that can handle DC (Difference-of-Convex) loss functions beyond the $\ell_1$ loss. We view the cost function of the proposed model as a composition of a DC function with a smooth mapping, and develop a variable smoothing algorithm for minimizing such DC composite functions. At each step of our algorithm, we generate a smooth surrogate function by using the Moreau envelope of each (weakly) convex function in the DC function, and then perform the gradient descent update of the surrogate function. Unlike many existing algorithms for DC problems, the proposed algorithm does not require any inner loop. We also present a convergence analysis in terms of a DC composite critical point for the proposed algorithm. Our numerical experiment demonstrates that the proposed method with DC loss functions is more robust against outliers compared to existing methods with the $\ell_1$ loss.

math.OC↗

arXiv2026

A Variable Smoothing for Weakly Convex Composite Minimization with Manifold Constraint via Parametrization

In this paper, we address a manifold constrained nonsmooth optimization problem involving the composition of a weakly convex function and a smooth mapping under the availability of a parametrization of the manifold. To find a stationary point of the target problem, we propose a variable smoothing-type algorithm by combining the ideas of (i) translating the constrained problem into a Euclidean optimization problem with a parametrization of the constraint set; (ii) exploiting a sequence of smoothed surrogate functions, of the cost function, given with the Moreau envelope of a weakly convex function. The proposed algorithm produces a vector sequence by the gradient descent update of a smoothed surrogate function at each iteration. In a case where the proximity operator of the weakly convex function is available, the proposed algorithm does not require any iterative solver for subproblems therein. By leveraging tools in the variational analysis, we show the so-called {\em gradient consistency property}, which is a key ingredient for smoothing-type algorithms, of the smoothed surrogate function used in this paper. Based on the gradient consistency property, we also establish an asymptotic convergence analysis for the proposed algorithm regarding a stationary point. Numerical experiments demonstrate the efficacy of the proposed algorithm.

math.OC↗

arXiv2026

Minimization of Nonsmooth Weakly Convex Function over Prox-regular Set for Robust Low-rank Matrix Recovery

We propose a prox-regular-type low-rank constrained nonconvex nonsmooth optimization model for Robust Low-Rank Matrix Recovery (RLRMR), i.e., estimate problem of low-rank matrix from an observed signal corrupted by outliers. For RLRMR, the $\ell_{1}$-norm has been utilized as a convex loss to detect outliers as well as to keep tractability of optimization models. Nevertheless, the $\ell_{1}$-norm is not necessarily an ideal robust loss because the $\ell_{1}$-norm tends to overpenalize entries corrupted by outliers of large magnitude. In contrast, the proposed model can employ a weakly convex function as a more robust loss, against outliers, than the $\ell_{1}$-norm. For the proposed model, we present (i) a projected variable smoothing-type algorithm applicable for the minimization of a nonsmooth weakly convex function over a prox-regular set, and (ii) a convergence analysis of the proposed algorithm in terms of stationary point. Numerical experiments demonstrate the effectiveness of the proposed model compared with the existing models that employ the $\ell_{1}$-norm.

math.OC↗

arXiv2025

Linearly involved Generalized Moreau Enhanced Model with Non-quadratic Smooth Convex Data Fidelity Functions

In this paper, we introduce an overall convex model incorporating a nonconvex regularizer. The proposed model is designed by extending the least squares term in the constrained LiGME model [Yata Yamagishi Yamada 2022] to fairly general smooth convex functions for flexible utilization of non-quadratic data fidelity functions. Under an overall convexity condition for the proposed model, we present sufficient conditions for the existence of a minimizer of the proposed model and an inner-loop free algorithm with guaranteed convergence to a global minimizer of the proposed model. To demonstrate the effectiveness of the proposed model and algorithm, we conduct numerical experiments in scenarios of Poisson denoising problem and simultaneous declipping and denoising problem.

math.OC↗

arXiv2025

A convexity preserving nonconvex regularization for inverse problems under non-Gaussian noise

We propose a nonconvexly regularized convex model for linear regression problems under non-Gaussian noise. The cost function of the proposed model is designed with a possibly non-quadratic data fidelity term and a nonconvex regularizer via the generalized Moreau enhancement of a seed convex regularizer. We present sufficient conditions (i) for the cost function of the proposed model to be convex over the entire space, and (ii) for the existence of a minimizer of the proposed model. Under such conditions, we propose a proximal splitting type algorithm with guaranteed convergence to a global minimizer of the proposed model. As an application, we enhance nonconvexly a convex sparsity-promoting regularizer in a scenario of simultaneous declipping and denoising.

math.OC↗

arXiv2025

Variable smoothing algorithm for inner-loop-free DC composite optimizations

We propose a variable smoothing algorithm for minimizing a nonsmooth and nonconvex cost function. The cost function is the sum of a smooth function and a composition of a difference-of-convex (DC) function with a smooth mapping. At each step of our algorithm, we generate a smooth surrogate function by using the Moreau envelope of each weakly convex function in the DC function, and then perform the gradient descent update of the surrogate function. The proposed algorithm does not require any inner loop unlike many existing algorithms for DC problem. We also present a convergence analysis in terms of a DC critical point for the proposed algorithm as well as its application to robust phase retrieval.

math.OC↗

arXiv2025

Hierarchical Variational Inequality Problem for Noncooperative Game-Theoretic Selection of Generalized Nash Equilibrium

The equilibrium selection problem in the variational Generalized Nash Equilibrium Problem (v-GNEP) has been reported as an optimization problem defined over the solution set of v-GNEP, called in this paper the lower-level v-GNEP. However, to make such a selection fair for every player, we have to rely on an unrealistic assumption, that is, the availability of a trusted center that does not induce any bias among players. In this paper, to ensure fairness for every player even in the process of equilibrium selection, we propose a new equilibrium selection problem, named the upper-level v-GNEP. The proposed upper-level v-GNEP is formulated as a v-GNEP defined over the solution set of the lower-level v-GNEP. We also present an iterative algorithm, of guaranteed convergence to a solution of the upper-level v-GNEP, as an application of the hybrid steepest descent method to a fixed point set characterization of the solution of the lower-level v-GNEP. Numerical experiments illustrate the proposed equilibrium selection and algorithm.

math.OC↗

arXiv2025

An LiGME Regularizer of Designated Isolated Minimizers -- An Application to Discrete-Valued Signal Estimation

For a regularized least squares estimation of discrete-valued signals, we propose a Linearly involved Generalized Moreau Enhanced (LiGME) regularizer, as a nonconvex regularizer, of designated isolated minimizers. The proposed regularizer is designed as a Generalized Moreau Enhancement (GME) of the so-called sum-of-absolute-values (SOAV) convex regularizer. Every candidate vector in the discrete-valued set is aimed to be assigned to an isolated local minimizer of the proposed regularizer while the overall convexity of the regularized least squares model is maintained. Moreover, a global minimizer of the proposed model can be approximated iteratively by using a variant of the constrained LiGME (cLiGME) algorithm. To enhance the accuracy of the proposed estimation, we also propose a pair of simple modifications, called respectively an iterative reweighting and a generalized superiorization. Numerical experiments demonstrate the effectiveness of the proposed model and algorithms in a scenario of multiple-input multiple-output (MIMO) signal detection.

eess.SP↗

arXiv2025

Hierarchical Nash Equilibrium over Variational Equilibria via Fixed-point Set Expression of Quasi-nonexpansive Operator

The equilibrium selection problem in the generalized Nash equilibrium problem (GNEP) has recently been studied as an optimization problem, defined over the set of all variational equilibria achievable through a lower-level non-cooperative game among players. However, to make such a selection fair for every player, we have to rely on an unrealistic assumption, that is, the availability of a trusted center that does not induce any bias for every player. In this paper, we study a new equilibrium selection problem, named the hierarchical Nash equilibrium problem (HNEP), and propose an iterative algorithm for solving the HNEP. The HNEP is designed to ensure a fair selection without assuming any trusted center. More precisely, the HNEP is the GNEP for an upper-level non-cooperative game defined over the set of all variational equilibria of the lower-level non-cooperative game. The proposed algorithm for the HNEP is established by applying the hybrid steepest descent method to a variational inequality defined over the fixed point set of a quasi-nonexpansive operator. Numerical experiments show the effectiveness of the proposed equilibrium selection problem and its algorithmic solution.

math.OC↗

arXiv2025

A Proximal Variable Smoothing for Nonsmooth Minimization Involving Weakly Convex Composite with MIMO Application

We propose a proximal variable smoothing algorithm for nonsmooth optimization problem with sum of three functions involving weakly convex composite function. The proposed algorithm is designed as a time-varying forward-backward splitting algorithm with two steps: (i) a time-varying forward step with the gradient of a smoothed surrogate function, designed with the Moreau envelope, of the sum of two functions; (ii) the backward step with a proximity operator of the remaining function. For the proposed algorithm, we present a convergence analysis in terms of a stationary point by using a newly smoothed surrogate stationarity measure. As an application of the target problem, we also present a formulation of multiple-input-multiple-output (MIMO) signal detection with phase-shift keying. Numerical experiments demonstrate the efficacy of the proposed formulation and algorithm.

math.OC↗

arXiv2024

A variable smoothing for Nonconvexly constrained nonsmooth optimization with application to sparse spectral clustering

We propose a variable smoothing algorithm for solving nonconvexly constrained nonsmooth optimization problems. The target problem has two issues that need to be addressed: (i) the nonconvex constraint and (ii) the nonsmooth term. To handle the nonconvex constraint, we translate the target problem into an unconstrained problem by parameterizing the nonconvex constraint in terms of a Euclidean space. We show that under a certain condition, these problems are equivalent in view of finding a stationary point. To find a stationary point of the parameterized problem, the proposed algorithm performs the gradient descent update for the smoothed version of the parameterized problem with replacement of the nonsmooth function by the Moreau envelope, inspired by a variable smoothing algorithm [Böhm-Wright, J. Optim. Theory Appl., 2021] specialized for unconstrained nonsmooth optimization. We also present a convergence analysis of the proposed algorithm as well as its application to a nonconvex reformulation of the sparse spectral clustering.

math.OC↗

arXiv2023

Adaptive Localized Cayley Parametrization for Optimization over Stiefel Manifold

We present an adaptive parametrization strategy for optimization problems over the Stiefel manifold by using generalized Cayley transforms to utilize powerful Euclidean optimization algorithms efficiently. The generalized Cayley transform can translate an open dense subset of the Stiefel manifold into a vector space, and the open dense subset is determined according to a tunable parameter called a center point. With the generalized Cayley transform, we recently proposed the naive Cayley parametrization, which reformulates the optimization problem over the Stiefel manifold as that over the vector space. Although this reformulation enables us to transplant powerful Euclidean optimization algorithms, their convergences may become slow by a poor choice of center points. To avoid such a slow convergence, in this paper, we propose to estimate adaptively 'good' center points so that the reformulated problem can be solved faster. We also present a unified convergence analysis, regarding the gradient, in cases where fairly standard Euclidean optimization algorithms are employed in the proposed adaptive parametrization strategy. Numerical experiments demonstrate that (i) the proposed strategy succeeds in escaping from the slow convergence observed in the naive Cayley parametrization strategy; (ii) the proposed strategy outperforms the standard strategy which employs a retraction.

math.OC↗