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Now showing items 1-10 of 17

#### On the Quadratic Convergence of the Singular Newton's Method

(1992-12)

The purpose of this essay is to describe a situation that we have found particularly exciting in our recent work in interior-point methods for linear programming. To our surprise, we have seen considerable theory developed ...

#### Interior-Point Algorithms for Semidefinite Programming Based on A Nonlinear Programming Formulation

(1999-12)

Recently, the authors of this paper introduced a nonlinear transformation to convert the positive definiteness constraint on an n × n matrix function of a certain form into the positivity constraint on n scalar variables ...

#### A Fast Newton's Algorithm for Entropy Maximization in Phase Determination

(1999-05)

A long-standing problem in X-ray crystallography, known as the phase problem, is to determine the phases for a large set of complex variables, called the structure factors of the crystal, given their magnitudes obtained ...

#### On Convergence of Minimization Methods: Attraction, Repulsion and Selection

(1999-03)

In this paper, we introduce a rather straightforward but fundamental observation concerning the convergence of the general iteration process. x^(k+1) = x^k - alpha(x^k) [B(x^k)]^(-1) gradf(x^k) for minimizing a function ...

#### An Interior-Point Method with Polynomial Complexity and Superlinear Convergence for Linear Complementarity Problems

(1991-07)

For linear programming, a primal-dual interior-point algorithm was recently constructed by Zhang and Tapia that achieves both polynomial complexity and Q-superlinear convergence (Q-quadratic in the nondegenerate case). In ...

#### Properties of A Class of Preconditioners for Weighted Least Squares Problems

(1999-04)

A sequence of weighted linear least squares problems arises from interior-point methods for linear programming where the changes from one problem to the next are the weights and the right hand side. One approach for solving ...

#### The Mehrotra Predictor-Corrector Interior-Point Method as a Perturbed Composite Newton Method

(1990-07)

The simplified Newton method reduces the work required by Newton's method per iteration by reusing the initial Jacobian matrix. However, fast convergence is sacrificed. The level-m composite Newton method attempts to balance ...

#### The Bayesian Statistical Approach to the Phase Problem in Protein X-ray Crystallography

(1999-04)

We review a Bayesian statistical approach to the phase problem in protein X-ray crystallography. We discuss the mathematical foundations and the computational issues. The introduction to the theory and the algorithms does ...

#### Solving Semidefinite Programs via Nonlinear Programming, Part I: Transformations and Derivatives

(1999-09)

In this paper, we introduce transformations that convert a large class of linear and/or nonlinear semidefinite programming (SDP) problems into nonlinear optimization problems over "orthants" of the form (R^n)++ × R^N, ...

#### An Interior-Point Algorithm for the Maximum-Volume Ellipsoid Problem

(1998-06)

In this report, we consider the problem on finding the maximum-volume ellipsoid inscribing a given full-dimensional polytope in R^n defined by a finite set of affine inequalities. We present several formulations for the ...