arXiv subjects

Masaru Ito

Publications and source records attributed to Masaru Ito.

14 recordsLinked to original sources

arXiv2026

A proximal gradient method with adaptive backtracking for weakly smooth multiobjective optimization

In this paper, we propose a proximal gradient method with adaptive linesearch for multiobjective optimization problems whose objective functions are weakly smooth, i.e., they have H\"older continuous gradients. The proposed method is parameter-free as we do not require prior knowledge of parameters related to the weak smoothness of the objective function; the proposed linesearch finds an appropriate step-size that adapts to the weak smoothness. The complexity guarantee analyzed in this paper for the non-convex case is compatible with related works and our algorithm accepts coercer stationarity measure compared to existing methods. We also establish a novel complexity result for the convex case which improves the one in non-convex case.

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arXiv2025

Faces of homogeneous cones and applications to homogeneous chordality

A convex cone $\mathcal{K}$ is said to be homogeneous if its group of automorphisms acts transitively on its relative interior. Important examples of homogeneous cones include symmetric cones and cones of positive semidefinite (PSD) matrices that follow a sparsity pattern given by a homogeneous chordal graph. Our goal in this paper is to elucidate the facial structure of homogeneous cones and make it as transparent as the faces of the PSD matrices. We prove that each face of a homogeneous cone $\mathcal{K}$ is mapped by an automorphism of $\mathcal{K}$ to one of its finitely many so-called principal faces. Furthermore, constructing such an automorphism can be done algorithmically by making use of a generalized Cholesky decomposition. Among other consequences, we give a proof that homogeneous cones are projectionally exposed, which strengthens the previous best result that they are amenable. Using our results, we will carefully analyze the facial structure of cones of PSD matrices satisfying homogeneous chordality and discuss consequences for the corresponding family of PSD completion problems.

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arXiv2025

Simple linesearch-free first-order methods for nonconvex optimization

This paper presents an auto-conditioned proximal gradient method for nonconvex optimization. The method determines the stepsize using an estimation of local curvature and does not require any prior knowledge of problem parameters and any linesearch procedures. Its convergence analysis is carried out in a simple manner without assuming the convexity, unlike previous studies. We also provide convergence analysis in the presence of the Kurdyka--Łojasiewicz property, adaptivity to the weak smoothness, and the extension to the Bregman proximal gradient method. Furthermore, the auto-conditioned stepsize strategy is also applied to the conditional gradient (Frank--Wolfe) method and the Riemannian gradient method.

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arXiv2025

Minimal hyperbolic polynomials and ranks of homogeneous cones

The starting point of this paper is the computation of minimal hyperbolic polynomials of duals of cones arising from chordal sparsity patterns. From that, we investigate the relation between ranks of homogeneous cones and their minimal polynomials. Along the way, we answer in the negative a question posed in an earlier paper and show examples of homogeneous cones that cannot be realized as rank-one generated (ROG) hyperbolicity cones.

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arXiv2025

Proximal gradient-type method with generalized distance and convergence analysis without global descent lemma

We consider solving nonconvex composite optimization problems in which the sum of a smooth function and a nonsmooth function is minimized. Many of convergence analyses of proximal gradient-type methods rely on global descent property between the smooth term and its proximal term. On the other hand, the ability to efficiently solve the subproblem depends on the compatibility between the nonsmooth term and the proximal term. Selecting an appropriate proximal term by considering both factors simultaneously is generally difficult. We overcome this issue by providing convergence analyses for proximal gradient-type methods with general proximal terms, without requiring global descent property of the smooth term. As a byproduct, new convergence results of the interior gradient methods for conic optimization are also provided.

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arXiv2024

Eigenvalue programming beyond matrices

In this paper we analyze and solve eigenvalue programs, which consist of the task of minimizing a function subject to constraints on the "eigenvalues" of the decision variable. Here, by making use of the FTvN systems framework introduced by Gowda, we interpret "eigenvalues" in a broad fashion going beyond the usual eigenvalues of matrices. This allows us to shed new light on classical problems such as inverse eigenvalue problems and also leads to new applications. In particular, after analyzing and developing a simple projected gradient algorithm for general eigenvalue programs, we show that eigenvalue programs can be used to express what we call vanishing quadratic constraints. A vanishing quadratic constraint requires that a given system of convex quadratic inequalities be satisfied and at least a certain number of those inequalities must be tight. As a particular case, this includes the problem of finding a point $x$ in the intersection of $m$ ellipsoids in such a way that $x$ is also in the boundary of at least $\ell$ of the ellipsoids, for some fixed $\ell > 0$. At the end, we also present some numerical experiments.

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arXiv2023

A Parameter-Free Conditional Gradient Method for Composite Minimization under Hölder Condition

In this paper we consider a composite optimization problem that minimizes the sum of a weakly smooth function and a convex function with either a bounded domain or a uniformly convex structure. In particular, we first present a parameter-dependent conditional gradient method for this problem, whose step sizes require prior knowledge of the parameters associated with the Hölder continuity of the gradient of the weakly smooth function, and establish its rate of convergence. Given that these parameters could be unknown or known but possibly conservative, such a method may suffer from implementation issue or slow convergence. We therefore propose a parameter-free conditional gradient method whose step size is determined by using a constructive local quadratic upper approximation and an adaptive line search scheme, without using any problem parameter. We show that this method achieves the same rate of convergence as the parameter-dependent conditional gradient method. Preliminary experiments are also conducted and illustrate the superior performance of the parameter-free conditional gradient method over the methods with some other step size rules.

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arXiv2023

Automorphisms of rank-one generated hyperbolicity cones and their derivative relaxations

A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank one, where the rank is computed with respect to the underlying hyperbolic polynomial. This is a natural class of hyperbolicity cones which are strictly more general than the ROG spectrahedral cones. In this work, we present a study of the automorphisms of ROG hyperbolicity cones and their derivative relaxations. One of our main results states that the automorphisms of the derivative relaxations are exactly the automorphisms of the original cone fixing a certain direction. As an application, we completely determine the automorphisms of the derivative relaxations of the nonnegative orthant and of the cone of positive semidefinite matrices. More generally, we also prove relations between the automorphisms of a spectral cone and the underlying permutation-invariant set, which might be of independent interest.

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arXiv2020

Nearly optimal first-order methods for convex optimization under gradient norm measure: An adaptive regularization approach

In the development of first-order methods for smooth (resp., composite) convex optimization problems, where smooth functions with Lipschitz continuous gradients are minimized, the gradient (resp., gradient mapping) norm becomes a fundamental optimality measure. Under this measure, a fixed iteration algorithm with the optimal iteration complexity is known, while determining this number of iteration to obtain a desired accuracy requires the prior knowledge of the distance from the initial point to the optimal solution set. In this paper, we report an adaptive regularization approach, which attains the nearly optimal iteration complexity without knowing the distance to the optimal solution set. To obtain further faster convergence adaptively, we secondly apply this approach to construct a first-order method that is adaptive to the Hölderian error bound condition (or equivalently, the Łojasiewicz gradient property), which covers moderately wide classes of applications. The proposed method attains nearly optimal iteration complexity with respect to the gradient mapping norm.

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arXiv2018

The automorphism group and the non-self-duality of $p$-cones

In this paper, we determine the automorphism group of the $p$-cones ($p\neq 2$) in dimension greater than two. In particular, we show that the automorphism group of those $p$-cones are the positive scalar multiples of the generalized permutation matrices that fix the main axis of the cone. Next, we take a look at a problem related to the duality theory of the $p$-cones. Under the Euclidean inner product it is well-known that a $p$-cone is self-dual only when $p=2$. However, it was not known whether it is possible to construct an inner product depending on $p$ which makes the $p$-cone self-dual. Our results shows that no matter which inner product is considered, a $p$-cone will never become self-dual unless $p=2$ or the dimension is less than three.

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arXiv2017

The $p$-cones in dimension $n \geq 3$ are not homogeneous when $p\neq 2$

Using the T-algebra machinery we show that, up to linear isomorphism, the only strictly convex homogeneous cones in $\Re^n$ with $n \geq 3$ are the 2-cones, also known as Lorentz cones or second order cones. In particular, this shows that the p-cones are not homogeneous when $p\neq 2$, $1 < p <\infty$ and $n\geq 3$, thus answering a problem proposed by Gowda and Trott.

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arXiv2016

A bound on the Carathéodory number

The Carathéodory number k(K) of a pointed closed convex cone K is the minimum among all the k for which every element of K can be written as a nonnegative linear combination of at most k elements belonging to extreme rays. Carathéodory's Theorem gives the bound k(K) <= dim (K). In this work we observe that this bound can be sharpened to k(K) <= l-1, where l is the length of the longest chain of nonempty faces contained in K, thus tying the Carathéodory number with a key quantity that appears in the analysis of facial reduction algorithms. We show that this bound is tight for several families of cones, which include symmetric cones and the so-called smooth cones. We also give a family of examples showing that this bound can also fail to be sharp. In addition, we furnish a new proof of a result by Güler and Tunçel which states that the Carathéodory number of a symmetric cone is equal to its rank. Finally, we connect our discussion to the notion of cp-rank for completely positive matrices.

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arXiv2015

A Family of Subgradient-Based Methods for Convex Optimization Problems in a Unifying Framework

We propose a new family of subgradient- and gradient-based methods which converges with optimal complexity for convex optimization problems whose feasible region is simple enough. This includes cases where the objective function is non-smooth, smooth, have composite/saddle structure, or are given by an inexact oracle model. We unified the way of constructing the subproblems which are necessary to be solved at each iteration of these methods. This permitted us to analyze the convergence of these methods in a unified way compared to previous results which required different approaches for each method/algorithm. Our contribution rely on two well-known methods in non-smooth convex optimization: the mirror-descent method by Nemirovski-Yudin and the dual-averaging method by Nesterov. Therefore, our family of methods includes them and many other methods as particular cases. For instance, the proposed family of classical gradient methods and its accelerations generalize Devolder et al.'s, Nesterov's primal/dual gradient methods, and Tseng's accelerated proximal gradient methods. Also our family of methods can partially become special cases of other universal methods, too. As an additional contribution, the novel extended mirror-descent method removes the compactness assumption of the feasible region and the fixation of the total number of iterations which is required by the original mirror-descent method in order to attain the optimal complexity.

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arXiv2015

New results on subgradient methods for strongly convex optimization problems with a unified analysis

We develop subgradient- and gradient-based methods for minimizing strongly convex functions under a notion which generalizes the standard Euclidean strong convexity. We propose a unifying framework for subgradient methods which yields two kinds of methods, namely, the Proximal Gradient Method (PGM) and the Conditional Gradient Method (CGM), unifying several existing methods. The unifying framework provides tools to analyze the convergence of PGMs and CGMs for non-smooth, (weakly) smooth, and further for structured problems such as the inexact oracle models. The proposed subgradient methods yield optimal PGMs for several classes of problems and yield optimal and nearly optimal CGMs for smooth and weakly smooth problems, respectively.

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