A multivariate polynomial $p(x)=p(x_1,…,x_n)$ is sos-convex if its Hessian $H(x)$ can be factored as $H(x)= M^T(x) M(x)$ with a possibly nonsquare polynomial matrix $M(x)$. It is easy to see that sos-convexity is a sufficient condition for convexity of $p(x)$. Moreover, the problem of deciding sos-convexity of a polynomial can be cast as the feasibility of … Read more
Semidefinite programming (SDP) is one of the most active areas in mathematical programming, due to varied applications and the availability of interior point algorithms. In this paper we propose a new pre-processing technique for SDP instances that exhibit algebraic symmetry. We present computational results to show that the solution times of certain SDP instances may … Read more
This paper explores generalizations of the Goemans-Williamson randomization technique. It establishes a simple equivalence of binary linear feasibility problems and max-cut problems and presents an analysis of the semidefinite max-cut relaxation for the case of a single linear equation. Numerical examples for feasible random binary problems indicate that the randomization technique is efficient when the … Read more
The kissing number in n-dimensional Euclidean space is the maximal number of non-overlapping unit spheres which simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing number based on semidefinite programming. This paper is a report on high accuracy calculations of these upper bounds for … Read more
The goal of this paper is to formulate and solve free material optimization problems with constraints on the smallest eigenfrequency of the optimal structure. A natural formulation of this problem as linear semidefinite program turns out to be numerically intractable. As an alternative, we propose a new approach, which is based on a nonlinear semidefinite … Read more
Inspired by a question of Lov\’asz, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lov{\’a}sz’s theta body of the graph. … Read more
A basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities. The sparsity, characterized with a chordal graph structure, can be detected in the variable matrix or in a linear or nonlinear matrix-inequality constraint of the problem. We classify the sparsity in two … Read more
Given a subset of all the pair-wise distances between a set of points in a fixed dimension, and possibly the positions of few of the points (called anchors), can we estimate the (relative) positions of all the unknown points (in the given dimension) accurately? This problem is known as the Euclidean Distance Geometry or Graph … Read more
Given a connected graph $G=(N,E)$ with node weights $s\in\R^N_+$ and nonnegative edge lengths, we study the following embedding problem related to an eigenvalue optimization problem over the second smallest eigenvalue of the (scaled) Laplacian of $G$: Find $v_i\in\R^{|N|}$, $i\in N$ so that distances between adjacent nodes do not exceed prescribed edge lengths, the weighted barycenter … Read more
Option price data is often used to infer risk-neutral densities for future prices of an underlying asset. Given the prices of a set of options on the same underlying asset with different strikes and maturities, we propose a nonparametric approach for estimating the evolution of the risk-neutral density in time. Our method uses bicubic splines … Read more