Superlinear and Quadratic Convergence of Riemannian Interior Point Methods

We extend the classical primal-dual interior point algorithms from the Euclidean setting to the Riemannian one. Our method, named the Riemannian interior point (RIP) method, is for solving Riemannian constrained optimization problems. Under the standard assumptions in the Riemannian setting, we establish locally superlinear, quadratic convergence for the Newton version of RIP and locally linear, … Read more

A New Extension of Chubanov’s Method to Symmetric Cones

We propose a new variant of Chubanov’s method for solving the feasibility problem over the symmetric cone by extending Roos’s method (2018) for the feasibility problem over the nonnegative orthant. The proposed method considers a feasibility problem associated with a norm induced by the maximum eigenvalue of an element and uses a rescaling focusing on … Read more

Completely Positive Factorization by Riemannian Smoothing Method

Copositive optimization is a special case of convex conic programming, and it optimizes a linear function over the cone of all completely positive matrices under linear constraints. Copositive optimization provides powerful relaxations of NP-hard quadratic problems or combinatorial problems, but there are still many open problems regarding copositive or completely positive matrices. In this paper, … Read more

Evaluating approximations of the semidefinite cone with trace normalized distance

We evaluate the dual cone of the set of diagonally dominant matrices (resp., scaled diagonally dominant matrices), namely ${\cal DD}_n^*$ (resp., ${\cal SDD}_n^*$), as an approximation of the semidefinite cone. We prove that the norm normalized distance, proposed by Blekherman et al. (2022), between a set ${\cal S}$ and the semidefinite cone has the same … Read more

Centering ADMM for the Semidefinite Relaxation of the QAP

We propose a new method for solving the semidefinite (SD) relaxation of the quadratic assignment problem (QAP), called the Centering ADMM. The Centering ADMM is an alternating direction method of multipliers (ADMM) combining the centering steps used in the interior-point method. The first stage of the Centering ADMM updates the iterate so that it approaches … Read more

Polyhedral approximations of the semidefinite cone and their application

We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded … Read more

LP-based Tractable Subcones of the Semidefinite Plus Nonnegative Cone

The authors in a previous paper devised certain subcones of the semidefinite plus nonnegative cone and showed that satisfaction of the requirements for membership of those subcones can be detected by solving linear optimization problems (LPs) with $O(n)$ variables and $O(n^2)$ constraints. They also devised LP-based algorithms for testing copositivity using the subcones. In this … Read more

An LP-based Algorithm to Test Copositivity

A symmetric matrix is called copositive if it generates a quadratic form taking no negative values over the nonnegative orthant, and the linear optimization problem over the set of copositive matrices is called the copositive programming problem. Recently, many studies have been done on the copositive programming problem (see, for example, \cite{aDUR10, aBOMZE12}). Among others, … Read more

A Primal Barrier Function Phase I Algorithm for Nonsymmetric Conic Optimization Problems

We call a positive semidefinite matrix whose elements are nonnegative a doubly nonnegative matrix, and the set of those matrices the doubly nonnegative cone (DNN cone). The DNN cone is not symmetric but can be represented as the projection of a symmetric cone embedded in a higher dimension. In \cite{aYOSHISE10}, the authors demonstrated the efficiency … Read more

Complementarity Problems over Symmetric Cones: A Survey of Recent Developments in Several Aspects

The complementarity problem over a symmetric cone (that we call the Symmetric Cone Complementarity Problem, or the SCCP) has received much attention of researchers in the last decade. Most of studies done on the SCCP can be categorized into the three research themes, interior point methods for the SCCP, merit or smoothing function methods for … Read more