arXiv subjects

Yasushi Narushima

Publications and source records attributed to Yasushi Narushima.

3 recordsLinked to original sources

arXiv2026

Proximal Nonlinear Conjugate Gradient Methods for Composite Optimization

The nonlinear conjugate gradient methods are known to be an effective approach for standard unconstrained optimization problems especially for large-scale problems.This paper proposes a proximal nonlinear conjugate gradient method, which extends the nonlinear conjugate gradient methods to composite objective functions, namely, the sum of a smooth nonconvex function and a nonsmooth convex function, and its extension to the case where the nonsmooth function is weakly convex. The proposed method uses the \textit{forward-backward residual} which is defined by using the proximal mapping instead of the gradient and determines the search direction based on the three-term Hestenes-Stiefel (HS) formula. We establish the global convergence of the proposed algorithm under standard assumptions for both convex and weakly convex nonsmooth functions, and analyze its convergence rate in terms of stationarity.In addition, when the smooth term is strongly convex, we prove that the generated sequence converges to the optimal solution and establish its convergence rate. Finally, numerical experiments show that the proposed method is stable and achieves better performance than existing methods in both convex and nonconvex settings.

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arXiv2026

Convergence Analysis via ODE Approach for Convex Optimization with Linear Equality Constraints

This paper studies the continuous-time dynamics of primal-dual algorithms for linearly constrained convex optimization problems and provides a quantitative convergence analysis using the Lyapunov functions. With the growing prevalence of sparse and low-rank models, large-scale problems involving nonsmooth objective functions have become increasingly important. Our approach addresses nonsmooth and nonstrong convex objective functions, which is particularly effective in extending classical accelerated methods to broader large-scale optimization problems. Building upon the ordinary differential equation (ODE) approach inspired by the recent work on Nesterov's acceleration methods, we extend the analysis to an ODE associated with an optimization problem with linear equality constraints. Moreover, by imposing a geometric condition analogous to the Kurdyka--$£$ojasiewicz (K$£$) property} on the objective function, we derive convergence rates that depend explicitly on the local geometry and establish the $O(1/t^2)$ local convergence rate. For the algorithmic construction, a numerical scheme is derived by discretizing the proposed ODE. Furthermore, we investigate the influence of algorithm parameters and provide insights into their optimal selection. Finally, preliminary numerical experiments are provided to validate the consistency with the theoretical results.

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arXiv2023

Inexact proximal DC Newton-type method for nonconvex composite functions

We consider a class of difference-of-convex (DC) optimization problems where the objective function is the sum of a smooth function and a possible nonsmooth DC function. The application of proximal DC algorithms to address this problem class is well-known. In this paper, we combine a proximal DC algorithm with an inexact proximal Newton-type method to propose an inexact proximal DC Newton-type method. We demonstrate global convergence properties of the proposed method. In addition, we give a memoryless quasi-Newton matrix for scaled proximal mappings and consider a two-dimensional system of semi-smooth equations that arise in calculating scaled proximal mappings. To efficiently obtain the scaled proximal mappings, we adopt a semi-smooth Newton method to inexactly solve the system. Finally, we present some numerical experiments to investigate the efficiency of the proposed method, showing that the proposed method outperforms existing methods.

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