arXiv subjects

Hayato Waki

Publications and source records attributed to Hayato Waki.

16 recordsLinked to original sources

arXiv2026

Induced Norm Analysis of Linear Systems for Nonnegative Input Signals

This paper is concerned with the analysis of the $L_p\ (p\in[1,\infty), p=\infty)$ induced norms of continuous-time linear systems where input signals are restricted to be nonnegative. This norm is referred to as the $L_{p+}$ induced norm in this paper. It has been shown recently that the $L_{2+}$ induced norm is effective for the stability analysis of nonlinear feedback systems where the nonlinearity returns only nonnegative signals. However, the exact computation of the $L_{2+}$ induced norm is essentially difficult. To get around this difficulty, in the first part of this paper, we provide a copositive-programming-based method for the upper bound computation by capturing the nonnegativity of the input signals by copositive multipliers. Then, in the second part of the paper, we derive uniform lower bounds of the $L_{p+}\ (p\in[1,\infty), p=\infty)$ induced norms with respect to the standard $L_{p}$ induced norms that are valid for all linear systems including infinite-dimensional ones. For each linear system, we finally derive a computation method of the lower bounds of the $L_{2+}$ induced norm that are larger than (or equal to) the uniform one. The effectiveness of the upper/lower bound computation methods are fully illustrated by numerical examples.

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arXiv2025

Expansions of the Characteristic Polynomial of a Perturbed Positive Semidefinite Matrix and Convergence Analysis of Alternating Projections

We observe that the characteristic polynomial of a linearly perturbed semidefinite matrix can be used to give the convergence rate of alternating projections for the positive semidefinite cone and a line. As a consequence, we show that such alternating projections converge at $O(k^{-1/2})$, independently of the singularity degree. A sufficient condition for the linear convergence is also obtained. Our method directly analyzes the defining equation for an alternating projection sequence and does not use error bounds.

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arXiv2025

Analytic Formulas for Alternating Projection Sequences for the Positive Semidefinite Cone and an Application to Convergence Analysis

We derive analytic formulas for the alternating projection method applied to the cone $\mathbb{S}^n_+$ of positive semidefinite matrices and an affine subspace. More precisely, we find recursive relations on parameters representing a sequence constructed by the alternating projection method. By applying these formulas, we analyze the alternating projection method in detail and show that the upper bound given by the singularity degree is actually tight when the alternating projection method is applied to $\mathbb{S}^3_+$ and a $3$-plane whose intersection is a singleton with singularity degree $2$.

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arXiv2024

$L_{2+}$ Induced Norm Analysis of Continuous-Time LTI Systems Using Positive Filters and Copositive Programming

This paper is concerned with the analysis of the $L_{2}$ induced norm of continuous-time LTI systems where the input signals are restricted to be nonnegative. This induced norm is referred to as the $L_{2+}$ induced norm in this paper. It has been shown very recently that the $L_{2+}$ induced norm is particularly useful for the stability analysis of nonlinear feedback systems constructed from linear systems and static nonlinearities where the nonlinear elements only provide nonnegative signals. For the upper bound computation of the $L_{2+}$ induced norm, an approach with copositive programming has also been proposed. It is nonetheless true that this approach becomes effective only for multi-input systems, and for single-input systems this approach does not bring any improvement over the trivial upper bound, the standard $L_2$ norm. To overcome this difficulty, we newly introduce positive filters to increase the number of positive signals. This enables us to enlarge the size of the copositive multipliers so that we can obtain better (smaller) upper bounds with copositive programming.

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arXiv2024

Convergence rate of alternating projection method for the intersection of an affine subspace and the second-order cone

We study the convergence rate of the alternating projection method (APM) applied to the intersection of an affine subspace and the second-order cone. We show that when they intersect non-transversally, the convergence rate is $O(k^{-1/2})$, where $k$ is the number of iterations of the APM. In particular, when the intersection is not at the origin or forms a half-line with the origin as the endpoint, the obtained convergence rate can be exact because a lower bound of the convergence rate is evaluated. These results coincide with the worst-case convergence rate obtained from the error bound discussed in [Borwein et al., SIOPT, 2014] and [Drusvyatskiy et al., Math. Prog., 2017]. Moreover, we consider the convergence rate of the APM for the intersection of an affine subspace and the product of two second-order cones. We provide an example that the worst-case convergence rate of the APM is better than the rate expected from the error bound for the example.

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arXiv2023

Exact Convergence Rates of Alternating Projections for Nontransversal Intersections

We consider the convergence rate of the alternating projection method for the nontransversal intersection of a semialgebraic set and a linear subspace. For such an intersection, the convergence rate is known as sublinear in the worst case. We study the exact convergence rate for a given semialgebraic set and an initial point, and investigate when the convergence rate is linear or sublinear. As a consequence, we show that the exact rates are expressed by multiplicities of the defining polynomials of the semialgebraic set, or related power series in the case that the linear subspace is a line, and we also decide the convergence rate for given data by using elimination theory. Our methods are also applied to give upper bounds for the case that the linear subspace has the dimension more than one. The upper bounds are shown to be tight by obtaining exact convergence rates for a specific semialgebraic set, which depend on the initial points.

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arXiv2022

Tractable semidefinite bounds of positive maximal singular values

We focus on computing certified upper bounds for the positive maximal singular value (PMSV) of a given matrix. The PMSV problem boils down to maximizing a quadratic polynomial on the intersection of the unit sphere and the nonnegative orthant. We provide a hierarchy of tractable semidefinite relaxations to approximate the value of the latter polynomial optimization problem as closely as desired. This hierarchy is based on an extension of Pólya's representation theorem. Doing so, positive polynomials can be decomposed as weighted sums of squares of $s$-nomials, where $s$ can be a priori fixed ($s=1$ corresponds to monomials, $s=2$ corresponds to binomials, etc.). This in turn allows us to control the size of the resulting semidefinite relaxations.

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arXiv2022

Stability Analysis of Recurrent Neural Networks by IQC with Copositive Mutipliers

This paper is concerned with the stability analysis of the recurrent neural networks (RNNs) by means of the integral quadratic constraint (IQC) framework. The rectified linear unit (ReLU) is typically employed as the activation function of the RNN, and the ReLU has specific nonnegativity properties regarding its input and output signals. Therefore, it is effective if we can derive IQC-based stability conditions with multipliers taking care of such nonnegativity properties. However, such nonnegativity (linear) properties are hardly captured by the existing multipliers defined on the positive semidefinite cone. To get around this difficulty, we loosen the standard positive semidefinite cone to the copositive cone, and employ copositive multipliers to capture the nonnegativity properties. We show that, within the framework of the IQC, we can employ copositive multipliers (or their inner approximation) together with existing multipliers such as Zames-Falb multipliers and polytopic bounding multipliers, and this directly enables us to ensure that the introduction of the copositive multipliers leads to better (no more conservative) results. We finally illustrate the effectiveness of the IQC-based stability conditions with the copositive multipliers by numerical examples.

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arXiv2020

$l_2$ Induced Norm Analysis of Discrete-Time LTI Systems for Nonnegative Input Signals and Its Application to Stability Analysis of Recurrent Neural Networks

In this paper, we focus on the "positive" $l_2$ induced norm of discrete-time linear time-invariant systems where the input signals are restricted to be nonnegative. To cope with the nonnegativity of the input signals, we employ copositive programming as the mathematical tool for the analysis. Then, by applying an inner approximation to the copositive cone, we derive numerically tractable semidefinite programming problems for the upper and lower bound computation of the "positive" $l_2$ induced norm. This norm is typically useful for the stability analysis of feedback systems constructed from an LTI system and nonlinearities where the nonlinear elements provide only nonnegative signals. As a concrete example, we illustrate the usefulness of the "positive" $l_2$ induced norm for the stability analysis of recurrent neural networks with activation functions being rectified linear units.

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arXiv2020

Perturbation Analysis of Singular Semidefinite Programs and Its Applications to Control Problems

We consider sensitivity of a semidefinite program under perturbations in the case that the primal problem is strictly feasible and the dual problem is weakly feasible. When the coefficient matrices are perturbed, the optimal values can change discontinuously as explained in concrete examples. We show that the optimal value of such a semidefinite program changes continuously under conditions involving the behavior of the minimal faces of the perturbed dual problems. In addition, we determine what kinds of perturbations keep the minimal faces invariant, by using the reducing certificates, which are produced in facial reduction. Our results allow us to classify the behavior of the minimal face of a semidefinite program obtained from a control problem.

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arXiv2019

Reduction of SISO H-infinity Output Feedback Control Problem

We consider the linear matrix inequality (LMI) problem of $H_\infty$ output feedback control problem for a generalized plant whose control input, measured output, disturbance input, and controlled output are scalar. We provide an explicit form of the optimal value. This form is the unification of some results in the literature of $H_\infty$ performance limitation analysis. To obtain the form of the optimal value, we focus on the non-uniqueness of perpendicular matrices, which appear in the LMI problem. We use the null vectors of invariant zeros associated with the dynamical system for the expression of the perpendicular matrices. This expression enables us to reduce and simplify the LMI problem. Our approach uses some well-known fundamental tools, e.g., the Schur complement, Lyapunov equation, Sylvester equation, and matrix completion. We use these techniques for the simplification of the LMI problem. Also, we investigate the structure of dual feasible solutions and reduce the size of the dual. This reduction is called a facial reduction in the literature of convex optimization.

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arXiv2016

Minimization of Akaike's Information Criterion in Linear Regression Analysis via Mixed Integer Nonlinear Program

Akaike's information criterion (AIC) is a measure of the quality of a statistical model for a given set of data. We can determine the best statistical model for a particular data set by the minimization of the AIC. Since we need to evaluate exponentially many candidates of the model by the minimization of the AIC, the minimization is unreasonable. Instead, stepwise methods, which are local search algorithms, are commonly used to find a better statistical model though it may not be the best. We propose a branch and bound search algorithm for a mixed integer nonlinear programming formulation of the AIC minimization by Miyashiro and Takano (2015). More concretely, we propose methods to find lower and upper bounds, and branching rules for this minimization. We then combine them with SCIP, which is a mathematical optimization software and a branch-and-bound framework. We show that the proposed method can provide the best statistical model based on AIC for small-sized or medium-sized benchmark data sets in UCI Machine Learning Repository. Furthermore, we show that this method finds good quality solutions for large-sized benchmark data sets.

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arXiv2016

Boundary modeling in model-based calibration for automotive engines via the vertex representation of the convex hulls

When using the convex hull approach in the boundary modeling process, Model-Based Calibration (MBC) software suites -- such as Model-Based Calibration Toolbox from MathWorks -- can be computationally intensive depending on the amount of data modeled. The reason for this is that the half-space representation of the convex hull is used. We discuss here another representation of the convex hull, the vertex representation, which proves capable to reduce the computational cost. Numerical comparisons in this article are executed in MATLAB by using MBC Toolbox commands, and show that for certain conditions, the vertex representation outperforms the half-space representation.

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arXiv2016

Application of Facial Reduction to $H_\infty$ State Feedback Control Problem

One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from $H_\infty$ control problems. We discuss the reason from the viewpoint of optimization, and provide necessary and sufficient conditions for LMI problem and its dual not to be strongly feasible. Moreover, we interpret them in terms of control system. In this analysis, facial reduction, which was proposed by Borwein and Wolkowicz, plays an important role. We show that a necessary and sufficient condition closely related to the existence of invariant zeros in the closed left-half plane in the system, and present a way to remove the numerical difficulty with the null vectors associated with invariant zeros in the closed left-half plane. Numerical results show that the numerical stability is improved by applying it.

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arXiv2014

A Perturbed Sums of Squares Theorem for Polynomial Optimization and its Applications

We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by $(f^\ast - ε)$ and from above by $f^\ast + ε(n+1)$, where $f^\ast$ is the optimal value of the POP. We propose new SDP relaxations for POP based on modifications of existing sums-of-squares representation theorems. An advantage of our SDP relaxations is that in many cases they are of considerably smaller dimension than those originally proposed by Lasserre. We present some applications and the results of our computational experiments.

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arXiv2009

Real ideal and the duality of semidefinite programming for polynomial optimization

We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial optimization problems with polynomial equality constraints can be modified equivalently so that the associated semidefinite programming relaxation problems have no duality gap. Elementary proofs for some criteria on reality of ideals are also given.

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