Analysis of Copositive Optimization Based Linear Programming Bounds on Standard Quadratic Optimization

The problem of minimizing a quadratic form over the unit simplex, referred to as a standard quadratic optimization problem, admits an exact reformulation as a linear optimization problem over the convex cone of completely positive matrices. This computationally intractable cone can be approximated from the inside and from the outside by two sequences of nested … Read more

Numerical Optimization of Eigenvalues of Hermitian Matrix Functions

The eigenvalues of a Hermitian matrix function that depends on one parameter analytically can be ordered so that each eigenvalue is an analytic function of the parameter. Ordering these analytic eigenvalues from the largest to the smallest yields continuous and piece-wise analytic functions. For multi-variate Hermitian matrix functions that depend on $d$ parameters analytically, the … Read more

On the Accuracy of Uniform Polyhedral Approximations of the Copositive Cone

We consider linear optimization problems over the cone of copositive matrices. Such conic optimization problems, called {\em copositive programs}, arise from the reformulation of a wide variety of difficult optimization problems. We propose a hierarchy of increasingly better outer polyhedral approximations to the copositive cone. We establish that the sequence of approximations is exact in … Read more

A Linearly Convergent Linear-Time First-Order Algorithm for Support Vector Classification with a Core Set Result

We present a simple, first-order approximation algorithm for the support vector classification problem. Given a pair of linearly separable data sets and $\epsilon \in (0,1)$, the proposed algorithm computes a separating hyperplane whose margin is within a factor of $(1-\epsilon)$ of that of the maximum-margin separating hyperplane. We discuss how our algorithm can be extended … Read more

Identification and Elimination of Interior Points for the Minimum Enclosing Ball Problem

Given $\cA := \{a^1,\ldots,a^m\} \subset \R^n$, we consider the problem of reducing the input set for the computation of the minimum enclosing ball of $\cA$. In this note, given an approximate solution to the minimum enclosing ball problem, we propose a simple procedure to identify and eliminate points in $\cA$ that are guaranteed to lie … Read more

An Algorithm and a Core Set Result for the Weighted Euclidean One-Center Problem

Given ${\cal A} := \{a^1,\ldots,a^m\} \subset \R^n$ with corresponding positive weights ${\cal W} := \{\omega_1,\ldots,\omega_m\}$, the weighted Euclidean one-center problem, which is a generalization of the minimum enclosing ball problem, involves the computation of a point $c_{\cal A} \in \R^n$ that minimizes the maximum weighted Euclidean distance from $c_{\cal A}$ to each point in ${\cal … Read more

Two Algorithms for the Minimum Enclosing Ball Problem

Given $\cA := \{a^1,\ldots,a^m\} \subset \R^n$ and $\eps > 0$, we propose and analyze two algorithms for the problem of computing a $(1 + \eps)$-approximation to the radius of the minimum enclosing ball of $\cA$. The first algorithm is closely related to the Frank-Wolfe algorithm with a proper initialization applied to the dual formulation of … Read more

Computing Minimum Volume Enclosing Axis-Aligned Ellipsoids

Given a set of points $\cS = \{x^1,\ldots,x^m\} \subset \R^n$ and $\eps > 0$, we propose and analyze an algorithm for the problem of computing a $(1 + \eps)$-approximation to the the minimum volume axis-aligned ellipsoid enclosing $\cS$. We establish that our algorithm is polynomial for fixed $\eps$. In addition, the algorithm returns a small … Read more

Implementation of Warm-Start Strategies in Interior-Point Methods for Linear Programming in Fixed Dimension

We implement several warm-start strategies in interior-point methods for linear programming (LP). We study the situation in which both the original LP instance and the perturbed one have exactly the same dimensions. We consider different types of perturbations of data components of the original instance and different sizes of each type of perturbation. We modify … Read more

On Khachiyan’s Algorithm for the Computation of Minimum Volume Enclosing Ellipsoids

Given $\cA := \{a^1,\ldots,a^m\} \subset \R^d$ whose affine hull is $\R^d$, we study the problems of computing an approximate rounding of the convex hull of $\cA$ and an approximation to the minimum volume enclosing ellipsoid of $\cA$. In the case of centrally symmetric sets, we first establish that Khachiyan’s barycentric coordinate descent (BCD) method is … Read more