arXiv subjects

Akatsuki Nishioka

Publications and source records attributed to Akatsuki Nishioka.

9 recordsLinked to original sources

arXiv2026

Pseudo-concave optimization of the first eigenvalue of elliptic operators with application to topology optimization by homogenization

We study optimization problems for the first eigenvalue of a linear elliptic operator. As applications, we consider homogenized two-phase optimal design problems, also known as topology optimization problems, for conductivity and simplified elasticity settings. Under suitable assumptions, we prove that the first eigenvalue is pseudo-concave with respect to the density-like parameter. This pseudo-concavity implies that every stationary point of the corresponding maximization problem is a global maximizer. Also, for a certain pseudo-concave minimization problem in the conductivity setting, a classical $0$-$1$ minimizer exists. Finally, we present simple numerical experiments illustrating the theoretical results.

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arXiv2026

A class of nonconvex semidefinite programming in which every KKT point is globally optimal

We consider a special class of nonconvex semidefinite programming problems and show that every point satisfying the Karush--Kuhn--Tucker (KKT) conditions is globally optimal despite nonconvexity. This property is related to pseudoconvex optimization and fractional programming. We also present several applications to robust fractional programming and generalized eigenvalue optimization appearing in topology optimization, network control, finance, etc.

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arXiv2026

An Improved Lower Bound for the Three-Dimensional Blaschke--Lebesgue Problem from Spectral and Dual Perspectives

The Blaschke--Lebesgue problem asks for convex bodies of minimum volume among all convex bodies of prescribed constant width. In the plane, the minimizer is the Reuleaux triangle, whereas the corresponding three-dimensional problem remains open and is also known as Meissner's conjecture. In this paper, we establish the lower bound $(4π/33)d^3 \simeq 0.380799\,d^3$ for the volume of any three-dimensional convex body of constant width d. This improves upon Chakerian's lower bound, approximately $0.364916\,d^3$, although it remains below the volume of the conjectured minimizers, Meissner's tetrahedra, whose volume is approximately $0.419860\,d^3$. The proof is based on a support-function formulation, spectral estimates via spherical harmonics, and Bochner's formula. We also show that the resulting lower bound can be interpreted as a Lagrange dual bound for the associated concave quadratic minimization problem. This dual viewpoint suggests possible routes toward sharper lower bounds.

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arXiv2026

Revisiting Invex Functions: Explicit Kernel Constructions and Characterizations

An invex function generalizes a convex function in the sense that every stationary point is a global minimizer. Recently, invex functions and their subclasses have attracted attention in signal processing and machine learning. However, verifying invexity is often difficult because its definition involves an unknown function called a kernel function. This paper studies kernel functions associated with invex functions, which have received relatively limited attention in the literature. In particular, we develop several methods for constructing explicit kernel functions and establish a characterization of pseudoconvexity in terms of kernel functions. These results provide constructive tools for proving invexity of new functions and for clarifying their structural properties. We also present examples of nonsmooth, non-pseudoconvex invex functions arising in signal processing.

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arXiv2024

Variational analysis of unbounded and discontinuous generalized eigenvalue functions with application to topology optimization

The maximum (or minimum) generalized eigenvalue of symmetric positive semidefinite matrices that depend on optimization variables often appears as objective or constraint functions in structural topology optimization when we consider robustness, vibration, and buckling. It can be an unbounded or discontinuous function where matrices become singular (where a topological change of the structural design occurs). Based on variational analysis, we redefine the maximum (and minimum) generalized eigenvalue function as an extended real-valued function and propose a real-valued continuous approximation of it. Then, we show that the proposed approximation epi-converges to the original redefined function, which justifies solving problems with the approximation instead. We consider two specific topology optimization problems: robust compliance optimization and eigenfrequency optimization and conduct simple numerical experiments.

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arXiv2024

A unified Euler--Lagrange system for analyzing continuous-time accelerated gradient methods

This paper presents an Euler--Lagrange system for a continuous-time model of the accelerated gradient methods in smooth convex optimization and proposes an associated Lyapunov-function-based convergence analysis framework. Recently, ordinary differential equations (ODEs) with dumping terms have been developed to intuitively interpret the accelerated gradient methods, and the design of unified model describing the various individual ODE models have been examined. In existing reports, the Lagrangian, which results in the Euler-Lagrange equation, and the Lyapunov function for the convergence analysis have been separately proposed for each ODE. This paper proposes a unified Euler--Lagrange system and its Lyapunov function to cover the existing various models. In the convergence analysis using the Lyapunov function, a condition that parameters in the Lagrangian and Lyapunov function must satisfy is derived, and a parameter design for improving the convergence rate naturally results in the mysterious dumping coefficients. Especially, a symmetric Bregman divergence can lead to a relaxed condition of the parameters and a resulting improved convergence rate. As an application of this study, a slight modification in the Lyapunov function establishes the similar convergence proof for ODEs with smooth approximation in nondifferentiable objective function minimization.

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arXiv2023

A feasible smoothing accelerated projected gradient method for nonsmooth convex optimization

Smoothing accelerated gradient methods achieve faster convergence rates than that of the subgradient method for some nonsmooth convex optimization problems. However, Nesterov's extrapolation may require gradients at infeasible points, and thus they cannot be applied to some structural optimization problems. We introduce a variant of smoothing accelerated projected gradient methods where every variable is feasible. The $O(k^{-1}\log k)$ convergence rate is obtained using the Lyapunov function. We conduct a numerical experiment on the robust compliance optimization of a truss structure.

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arXiv2023

Convergence Rate Analysis of Continuous- and Discrete-Time Smoothing Gradient Algorithms

This paper addresses the gradient flow -- the continuous-time representation of the gradient method -- with the smooth approximation of a non-differentiable objective function and presents convergence analysis framework. Similar to the gradient method, the gradient flow is inapplicable to the non-differentiable function minimization; therefore, this paper addresses the smoothing gradient method, which exploits a decreasing smoothing parameter sequence in the smooth approximation. The convergence analysis is presented using conventional Lyapunov-function-based techniques, and a Lyapunov function applicable to both strongly convex and non-strongly convex objective functions is provided by taking into consideration the effect of the smooth approximation. Based on the equivalence of the stepsize in the smoothing gradient method and the discretization step in the forward Euler scheme for the numerical integration of the smoothing gradient flow, the sample values of the exact solution of the smoothing gradient flow are compared with the state variable of the smoothing gradient method, and the equivalence of the convergence rates is shown.

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arXiv2023

On a minimization problem of the maximum generalized eigenvalue: properties and algorithms

We study properties and algorithms of a minimization problem of the maximum generalized eigenvalue of symmetric-matrix-valued affine functions, which is nonsmooth and quasiconvex, and has application to eigenfrequency optimization of truss structures. We derive an explicit formula of the Clarke subdifferential of the maximum generalized eigenvalue and prove the maximum generalized eigenvalue is a pseudoconvex function, which is a subclass of a quasiconvex function, under suitable assumptions. Then, we consider smoothing methods to solve the problem. We introduce a smooth approximation of the maximum generalized eigenvalue and prove the convergence rate of the smoothing projected gradient method to a global optimal solution in the considered problem. Also, some heuristic techniques to reduce the computational costs, acceleration and inexact smoothing, are proposed and evaluated by numerical experiments.

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