arXiv subjects
Publications and source records attributed to Stephen A. Vavasis.
We present a gradient-based algorithm for unconstrained minimization derived from iterated linear change of basis. The new method is equivalent to linear conjugate gradient in the case of a quadratic objective function. In the case of exact line search it is a secant method. In practice, it performs comparably to BFGS and DFP and is sometimes more robust.
We propose an algorithm based on Newton's method and subdivision for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the intersections between a line and a surface, which has applications in graphics and computer-aided geometric design. The algorithm can operate on polynomials represented in any basis that satisfies a few conditions. The power basis, the Bernstein basis, and the first-kind Chebyshev basis are among those compatible with the algorithm. The main novelty of our algorithm is an analysis showing that its running is bounded only in terms of the condition number of the polynomial's zeros and a constant depending on the polynomial basis.
Nonnegative matrix factorization (NMF) has become a prominent technique for the analysis of image databases, text databases and other information retrieval and clustering applications. In this report, we define an exact version of NMF. Then we establish several results about exact NMF: (1) that it is equivalent to a problem in polyhedral combinatorics; (2) that it is NP-hard; and (3) that a polynomial-time local search heuristic exists.
We conjecture that the roots of a degree-n univariate complex polynomial are located in a union of n-1 annuli, each of which is centered at a root of the derivative and whose radii depend on higher derivatives. We prove the conjecture for the cases of degrees 2 and 3, and we report on tests with randomly generated polynomials of higher degree. We state two other closely related conjectures concerning Newton's method. If true, these conjectures imply the existence of a simple, rapidly convergent algorithm for finding all roots of a polynomial.
We propose a new two-dimensional meshing algorithm called PINW able to generate meshes that accurately approximate the distance between any two domain points by paths composed only of cell edges. This technique is based on an extension of pinwheel tilings proposed by Radin and Conway. We prove that the algorithm produces triangles of bounded aspect ratio. This kind of mesh would be useful in cohesive interface finite element modeling when the crack propagation pathis an outcome of a simulation process.
In this paper, we show a way to exploit sparsity in the problem data in a primal-dual potential reduction method for solving a class of semidefinite programs. When the problem data is sparse, the dual variable is also sparse, but the primal one is not. To avoid working with the dense primal variable, we apply Fukuda et al.'s theory of partial matrix completion and work with partial matrices instead. The other place in the algorithm where sparsity should be exploited is in the computation of the search direction, where the gradient and the Hessian-matrix product of the primal and dual barrier functions must be computed in every iteration. By using an idea from automatic differentiation in backward mode, both the gradient and the Hessian-matrix product can be computed in time proportional to the time needed to compute the barrier functions of sparse variables itself. Moreover, the high space complexity that is normally associated with the use of automatic differentiation in backward mode can be avoided in this case. In addition, we suggest a technique to efficiently compute the determinant of the positive definite matrix completion that is required to compute primal search directions. The method of obtaining one of the primal search directions that minimizes the number of the evaluations of the determinant of the positive definite completion is also proposed. We then implement the algorithm and test it on the problem of finding the maximum cut of a graph.