There is a large number of implementational choices to be made for the primal-dual interior point method in the context of mixed semidefinite and second order cone optimization. This paper presents such implementational issues in a unified framework, and compares the choices made by different research groups. This is also the first paper to provide … Read more
We present a practical algorithm for computing the minimum volume n-dimensional ellipsoid that must contain m given points a_1, …, a_m \in R^n. This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it … Read more
This paper presents smoothing heuristics for an NP-hard combinatorial problem based on Lagrangian relaxation. We formulate the Lagrangian dual for this nonconvex quadratic problem and propose eigenvalue nonsmooth unconstrained optimization to solve the dual problem with bundle or subgradient methods. Derived heuristics are considered to obtain good primal solutions through pathfollowing methods using a projected … Read more
The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that … Read more
It was recently shown that, unlike in linear optimization, the central path in semidefinite optimization (SDO) does not converge to the analytic center of the optimal set in general. In this paper we analyze the limiting behavior of the central path to explain this unexpected phenomenon. This is done by deriving a new necessary and … Read more
In linear programming, the central path is known to converge to the analytic center of the set of optimal solutions. Recently, it has been shown that this is not necessarily true for linear semidefinite programming in the absence of strict complementarity. The present paper deals with the formulation of a convex problem whose solution defines … Read more
The class of POPs (Polynomial Optimization Problems) over cones covers a wide range of optimization problems such as $0$-$1$ integer linear and quadratic programs, nonconvex quadratic programs and bilinear matrix inequalities. This paper presents a new framework for convex relaxation of POPs over cones in terms of linear optimization problems over cones. It provides a … Read more
$T$-algebras are non-associative algebras defined by Vinberg in the early 1960’s for the purpose of studying homogeneous cones. Vinberg defined a cone $K(\mathcal A)$ for each $T$-algebra $\mathcal A$ and proved that every homogeneous cone is isomorphic to one such $K(\mathcal A)$. We relate each $T$-algebra $\mathcal A$ with a space of linear operators in … Read more
We build upon the work of Fukuda et al.\ \cite{FuKoMuNa01} and Nakata et al.\ \cite{NaFuFuKoMu01}, in which the theory of partial positive semidefinite matrices has been applied to the semidefinite programming (SDP) problem as a technique for exploiting sparsity in the data. In contrast to their work, which improves an existing algorithm that is based … Read more
DSDP4 is an implementation of the dual-scaling algorithm for semidefinite program ming. New features in this version include a Lanczos procedure for determining the step size, more precise primal solutions, a parallel solver, and improved performance on the standard test suites. Citation ANL/MCS-TM-255; Mathematics and Computer Science Division; Argonne National Laboratory; Argonne, IL; March 2002 … Read more