We study the algebraic and facial structures of hyperbolic programs, and examine natural relaxations of hyperbolic programs, the relaxations themselves being hyperbolic programs. Citation TR 1406, School of Operations Research, Cornell University, Ithaca, NY 14853, U.S., 3/04 Article Download View Hyperbolic Programs, and Their Derivative Relaxations
We consider here the problem of minimizing a polynomial function on $\oR^n$. The problem is known to be hard even for degree $4$. Therefore approximation algorithms are of interest. Lasserre \cite{lasserre:2001} and Parrilo \cite{Pa02a} have proposed approximating the minimum of the original problem using a hierarchy of lower bounds obtained via semidefinite programming relaxations. We … Read more
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
Let $I=(g_1,…, g_n)$ be a zero-dimensional ideal of $ \R[x_1,…,x_n]$ such that its associated set $G$ of polynomial equations $g_i(x)=0$ for all $i=1,…,n$, is in triangular form. By introducing multivariate Newton sums we provide a numerical characterization of polynomials in the radical ideal of $I$. We also provide a necessary and sufficient (numerical) condition for … Read more
In this paper we consider the proximal point method with Bregman distance applied to linear programming problems, and study the dual sequence obtained from the optimal multipliers of the linear constraints of each subproblem. We establish the convergence of this dual sequence, as well as convergence rate results for the primal sequence, for a suitable … Read more
We present a unifying framework to establish a lower-bound on the number of semidefinite programming based, lift-and-project iterations (rank) for computing the convex hull of the feasible solutions of various combinatorial optimization problems. This framework is based on the maps which are commutative with the lift-and-project operators. Some special commutative maps were originally observed by … Read more
In this paper we propose a new class of primal-dual path-following interior point algorithms for solving monotone linear complementarity problems. At each iteration, the method would select a target on the central path with a large update from the current iterate, and then the Newton method is used to get the search directions, followed by … Read more
The goal of this paper is to formulate and solve structural optimization problems with constraints on the global stability of the structure. The stability constraint is based on the linear buckling phenomenon. We formulate the problem as a nonconvex semidefinite programming problem and introduce an algorithm based on the Augmented Lagrangian method combined with the … Read more
Interior-point methods (IPMs), originally conceived in the context of linear programming have found a variety of applications in integer programming, and combinatorial optimization. This survey presents an up to date account of IPMs in solving NP-hard combinatorial optimization problems to optimality, and also in developing approximation algorithms for some of them. The surveyed approaches include … Read more
A classical theorem by Block and Levin says that certain variants of the relaxation method for solving systems of linear inequalities produce bounded sequences of intermediate solutions even when running on inconsistent input data. Using a new approach, we prove a more general version of this result and answer an old open problem of quantifying … Read more