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
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 interpolation procedure of Gomory and Johnson (1972), which generates cutting planes for general integer programs from facets of master cyclic group polyhedra. This idea has recently been re-considered by Evans (2002) and Gomory, Johnson and Evans (2003). We compare inequalities generated by this procedure with mixed-integer rounding (MIR) based inequalities discussed in … Read more
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
We describe a graph coloring problem associated with the determination of mathematical derivatives. The coloring instances are obtained as intersection graphs of row partitioned sparse derivative matrices. The size of the graph is dependent on the partition and can be varied between the number of columns and the number of nonzero entries. If solved exactly … 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
We present a primal-dual interior-point algorithm with a filter line-search method for nonlinear programming. Local and global convergence properties of this method were analyzed in previous work. Here we provide a comprehensive description of the algorithm, including the feasibility restoration phase for the filter method, second-order corrections, and inertia correction of the KKT matrix. Heuristics … Read more
We describe FORTRAN subroutines for network flow optimization using an interior point network flow algorithm. We provide FORTRAN and C language drivers, as well as C language functions that, together with the subroutines, make up PDNET (Portugal, Resende, Veiga, and Júdice, 2000). The algorithm is described in detail and its implementation is outlined. Usage of … 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