Approximation Hierarchies for Copositive Cone over Symmetric Cone and Their Comparison

We first provide an inner-approximation hierarchy described by a sum-of-squares (SOS) constraint for the copositive (COP) cone over a general symmetric cone. The hierarchy is a generalization of that proposed by Parrilo (2000) for the usual COP cone (over a nonnegative orthant). We also discuss its dual. Second, we characterize the COP cone over a … Read more

Generalization of Doubly Nonnegative Cone: Focusing on Inner-Approximation for Generalized Copositive Cone

We aim to provide better relaxation for generalized completely positive (copositive) programming. We first develop an inner-approximation hierarchy for the generalized copositive cone over a symmetric cone. Exploiting this hierarchy as well as the existing hierarchy proposed by Zuluaga et al. (SIAM J Optim 16(4):1076–1091, 2006), we then propose two (NN and ZVP) generalized doubly … Read more

Mixed Integer Quadratic Optimization Formulations for Eliminating Multicollinearity Based on Variance Inflation Factor

The variance inflation factor, VIF, is the most frequently used indicator for detecting multicollinearity in multiple linear regression models. This paper proposes two mixed integer quadratic optimization formulations for selecting the best subset of explanatory variables under upper-bound constraints on VIF of selected variables. Computational results illustrate the effectiveness of our optimization formulations based on … Read more

Best subset selection for eliminating multicollinearity

This paper proposes a method for eliminating multicollinearity from linear regression models. Specifically, we select the best subset of explanatory variables subject to the upper bound on the condition number of the correlation matrix of selected variables. We first develop a cutting plane algorithm that, to approximate the condition number constraint, iteratively appends valid inequalities … Read more

Fast implementation for semidefinite programs with positive matrix completion

Solving semidefinite programs (SDP) in a short time is the key to managing various mathematical optimization problems in practical time. The matrix-completion primal-dual interior-point method (MC-PDIPM) extracts a structural sparsity of input SDP by factorizing the variable matrices, and it shrinks the computation time. In this paper, we propose a new factorization based on the … Read more

Parallel solver for semidefinite programming problem having sparse Schur complement matrix

SemiDefinite Programming (SDP) problem is one of the most central problems in mathematical programming. SDP provides a practical computation framework for many research fields. Some applications, however, require solving large-scale SDPs whose size exceeds the capacity of a single processor in terms of computational time and available memory. SDPARA (SemiDefinite Programming Algorithm paRAllel version) developed … Read more

A high-performance software package for semidefinite programs: SDPA 7

The SDPA (SemiDefinite Programming Algorithm) Project launched in 1995 has been known to provide high-performance packages for solving large-scale Semidefinite Programs (SDPs). SDPA Ver. 6 solves efficiently large-scale dense SDPs, however, it required much computation time compared with other software packages, especially when the Schur complement matrix is sparse. SDPA Ver. 7 is now completely … Read more

An Extension of a Minimax Approach to Multiple Classification

When the mean vectors and the covariance matrices of two classes are available in a binary classification problem, Lanckriet et al.\ \cite{mpm} propose a minimax approach for finding a linear classifier which minimizes the worst-case (maximum) misclassification probability. We extend the minimax approach to a multiple classification problem, where the number $m$ of classes could … Read more

Parallel Primal-Dual Interior-Point Methods for SemiDefinite Programs

The Semidefinite Program (SDP) is a fundamental problem in mathematical programming. It covers a wide range of applications, such as combinatorial optimization, control theory, polynomial optimization, and quantum chemistry. Solving extremely large-scale SDPs which could not be solved before is a significant work to open up a new vista of future applications of SDPs. Our … Read more

Preprocessing sparse semidefinite programs via matrix completion

Considering that preprocessing is an important phase in linear programming, it should be systematically more incorporated in semidefinite programming solvers. The conversion method proposed by the authors (SIAM Journal on Optimization, vol.~11, pp.~647–674, 2000, and Mathematical Programming, Series B, vol.~95, pp.~303–327, 2003) is a preprocessing of sparse semidefinite programs based on matrix completion. This article … Read more