Kim-Chuan Toh

SDPT3 — A Matlab software package for semidefinite programming, Version 1.3

This invention relates to stabilizing compositions; to the process for preparing a novel halogen-containing polymer; and to polymers stabilized against the deteriorative effect of heat which comprises a vinyl chloride or vinylidene chloride homopolymer or copolymer and a stabilizing amount of: as a first stabilizer an organotin halide exhibiting the formula RSnX3 wherein R is a hydrocarbon and X is halogen and as a second stabilizer a sulfur-containing organotin compound exhibiting two direct carbon to tin bonds and two direct sulfur to tin bonds.

Solving semidefinite-quadratic-linear programs using SDPT3

Abstract. This paper discusses computational experiments with linear optimization problems involving semidefinite, quadratic, and linear cone constraints (SQLPs). Many test problems of this type are solved using a new release of SDPT3, a Matlab implementation of infeasible primal-dual path-following algorithms. The software developed by the authors uses Mehrotra-type predictor-corrector variants of interior-point methods and two types of search directions: the HKM and NT directions. A discussion of implementation details is provided and computational results on problems from the SDPLIB and DIMACS Challenge collections are reported.

On the Nesterov--Todd Direction in Semidefinite Programming

We study different choices of search direction for primal-dual interior-point methods for semidefinite programming problems. One particular choice we consider comes from a specialization of a class of algorithms developed by Nesterov and Todd for certain convex programming problems. We discuss how the search directions for the Nesterov--Todd (NT) method can be computed efficiently and demonstrate how they can be viewed as Newton directions. This last observation also leads to convenient computation of accelerated steps, using the Mehrotra predictor-corrector approach, in the NT framework. We also provide an analytical and numerical comparison of several methods using different search directions, and suggest that the method using the NT direction is more robust than alternative methods.

On the Implementation and Usage of SDPT3 – A Matlab Software Package for Semidefinite-Quadratic-Linear Programming, Version 4.0

This software is designed to solve primal and dual semidefinite-quadratic-linear conic programming problems (known as SQLP problems) whose constraint conic is a product of semidefinite conics, second-order conics, nonnegative orthants and Euclidean spaces, and whose objective function is the sum of linear functions and log-barrier terms associated with the constraint conics. This includes the special case of determinant maximization problems with linear matrix inequalities. It employs an infeasible primal-dual predictor-corrector path-following method, with either the HKM or the NT search direction. The basic code is written in Matlab, but key subroutines in C are incorporated via Mex files. Routines are provided to read in problems in either SDPA or SeDuMi format. Sparsity and block diagonal structure are exploited. We also exploit low-rank structures in the constraint matrices associated with the semidefinite blocks if such structures are explicitly given. To help the users in using our software, we also include some examples to illustrate the coding of problem data for our solver. Various techniques to improve the efficiency and robustness of the main solver are incorporated. For example, step-lengths associated with semidefinite conics are calculated via the Lanczos method. The current version also implements algorithms for solving a 3-parameter homogeneous self-dual model of the primal and dual SQLP problems. Routines are also provided to determine whether the primal and dual feasible regions of a given SQLP have empty interiors. Numerical experiments show that this general-purpose code can solve more than 80% of a total of about 430 test problems to an accuracy of at least 10 − 6 in relative duality gap and infeasibilities.

An implementable proximal point algorithmic framework for nuclear norm minimization

The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In this paper, we study inexact proximal point algorithms in the primal, dual and primal-dual forms for solving the nuclear norm minimization with linear equality and second order cone constraints. We design efficient implementations of these algorithms and present comprehensive convergence results. In particular, we investigate the performance of our proposed algorithms in which the inner sub-problems are approximately solved by the gradient projection method or the accelerated proximal gradient method. Our numerical results for solving randomly generated matrix completion problems and real matrix completion problems show that our algorithms perform favorably in comparison to several recently proposed state-of-the-art algorithms. Interestingly, our proposed algorithms are connected with other algorithms that have been studied in the literature.

A coordinate gradient descent method for l 1-regularized convex minimization

In applications such as signal processing and statistics, many problems involve finding sparse solutions to under-determined linear systems of equations. These problems can be formulated as a structured nonsmooth optimization problems, i.e., the problem of minimizing ℓ1-regularized linear least squares problems. In this paper, we propose a block coordinate gradient descent method (abbreviated as CGD) to solve the more general ℓ1-regularized convex minimization problems, i.e., the problem of minimizing an ℓ1-regularized convex smooth function. We establish a Q-linear convergence rate for our method when the coordinate block is chosen by a Gauss-Southwell-type rule to ensure sufficient descent. We propose efficient implementations of the CGD method and report numerical results for solving large-scale ℓ1-regularized linear least squares problems arising in compressed sensing and image deconvolution as well as large-scale ℓ1-regularized logistic regression problems for feature selection in data classification. Comparison with several state-of-the-art algorithms specifically designed for solving large-scale ℓ1-regularized linear least squares or logistic regression problems suggests that an efficiently implemented CGD method may outperform these algorithms despite the fact that the CGD method is not specifically designed just to solve these special classes of problems.

From Potential Theory to Matrix Iterations in Six Steps

The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, Bi-CGSTAB, and so on) is reviewed. For a computation of this kind, an estimated asymptotic convergence factor

Calculation of pseudospectra by the Arnoldi iteration

\rho \le 1

A Distributed SDP Approach for Large-Scale Noisy Anchor-Free Graph Realization with Applications to Molecular Conformation

can be derived by solving a problem of potential theory or conformal mapping. Six approximations are involved in relating the actual computation to this scalar estimate. These six approximations are discussed in a systematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming.

An inexact interior point method for L1-regularized sparse covariance selection

The Arnoldi iteration, usually viewed as a method for calculating eigenvalues, can also be used to estimate pseudospectra. This possibility may be of practical importance, because in applications involving highly nonnormal matrices or operators, such as hydrodynamic stability, pseudospectra may be physically more significant than spectra.