Makoto Yamashita Inexact interior-point methods (IPMs) are a type of interior-point methods that inexactly solve the linear equation system for obtaining the search direction. On the other hand, arc-search IPMs approximate the central path with an ellipsoidal arc obtained by solving two linear equation systems in each iteration, while conventional line-search IPMs solve one linear system, therefore, … Read more
We study T-semidefinite programming (SDP) relaxation for constrained polynomial optimization problems (POPs). T-SDP relaxation for unconstrained POPs was introduced by Zheng, Huang and Hu in 2022. In this work, we propose a T-SDP relaxation for POPs with polynomial inequality constraints and show that the resulting T-SDP relaxation formulated with third-order tensors can be transformed into … Read more
We study exact matrix completion from partially available data with hidden connectivity patterns. Exact matrix completion was shown to be possible recently by Cosse and Demanet in 2021 with Lasserre’s relaxation using the trace of the variable matrix as the objective function with given data structured in a chain format. In this study, we introduce … Read more
An arc-search interior-point method is a type of interior-point method that approximates the central path by an ellipsoidal arc, and it can often reduce the number of iterations. In this work, to further reduce the number of iterations and the computation time for solving linear programming problems, we propose two arc-search interior-point methods using Nesterov’s … Read more
For nonconvex quadratically constrained quadratic programs (QCQPs), we first show that, under certain feasibility conditions, the standard semidefinite (SDP) relaxation is exact for QCQPs with bipartite graph structures. The exact optimal solutions are obtained by examining the dual SDP relaxation and the rank of the optimal solution of this dual SDP relaxation under strong duality. … Read more
Intensity modulated radiation therapy (IMRT) is one of radiation therapies for cancers, and it is considered to be effective for complicated shapes of tumors, since dose distributions from each irradiation can be modulated arbitrary. Fluence map optimization (FMO), which optimizes beam intensities with given beam angles, is often formulated as an optimization problem with dose … Read more
We study the exactness of the semidefinite programming (SDP) relaxation of quadratically constrained quadratic programs (QCQPs). With the aggregate sparsity matrix from the data matrices of a QCQP with $n$ variables, the rank and positive semidefiniteness of the matrix are examined. We prove that if the rank of the aggregate sparsity matrix is not less … Read more
Among many approaches to increase the computational efficiency of semidefinite programming (SDP) relaxation for quadratic constrained quadratic programming problems (QCQPs), exploiting the aggregate sparsity of the data matrices in the SDP by Fukuda et al. (2001) and second-order cone programming (SOCP) relaxation have been popular. In this paper, we exploit the aggregate sparsity of SOCP … Read more
In this paper, we propose an infeasible arc-search interior-point algorithm for solving nonlinear programming problems. Most algorithms based on interior-point methods are categorized as line search in the sense that they compute a next iterate on a straight line determined by a search direction which approximates the central path. The proposed arc-search interior-point algorithm uses … Read more
We extend the result on the spectral projected gradient method by Birgin et al in 2000 to a log-determinant semidefinite problem (SDP) with linear constraints and propose a spectral projected gradient method for the dual problem. Our method is based on alternate projections on the intersection of two convex sets, which first projects onto the … Read more