We discuss in detail an additive structure of cones of trigonometric polynomials nonnegative on the union of finite number of pairwise disjoint segments of the unit circle. We derive new descriptions of these cones in terms of semidefinite constraints. We explain the results of M. Krein and A. Nudelman providing a description of dual cones … Read more
We study the extension of a column generation technique to eigenvalue optimization. In our approach we utilize the method of analytic center to obtain the query points at each iteration. A restricted master problem in the primal space is formed corresponding to the relaxed dual problem. At each step of the algorithm, an oracle is … Read more
We establish polynomial-time convergence of infeasible-interior-point methods for conic programs over symmetric cones using a wide neighborhood of the central path. The convergence is shown for a commutative family of search directions used in Schmieta and Alizadeh. These conic programs include linear and semidefinite programs. This extends the work of Rangarajan and Todd, which established … Read more
The notion of weighted centers is essential in V-space interior-point algorithms for linear programming. Although there were some successes in generalizing this notion to semidefinite programming via weighted center equations, we still do not have a generalization that preserves two important properties — 1) each choice of weights uniquely determines a pair of primal-dual weighted … Read more
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