Richard A. Tapia

On the formulation and theory of the Newton interior-point method for nonlinear programming

In this work, we first study in detail the formulation of the primal-dual interior-point method for linear programming. We show that, contrary to popular belief, it cannot be viewed as a damped Newton method applied to the Karush-Kuhn-Tucker conditions for the logarithmic barrier function problem. Next, we extend the formulation to general nonlinear programming, and then validate this extension by demonstrating that this algorithm can be implemented so that it is locally and Q-quadratically convergent under only the standard Newton method assumptions. We also establish a global convergence theory for this algorithm and include promising numerical experimentation.

Diagonalized multiplier methods and quasi-Newton methods for constrained optimization

Two approaches to quasi-Newton methods for constrained optimization problems inRn are presented. These approaches are based on a class of Lagrange multiplier approximation formulas used by the author in his previous work on Newtons method for constrained problems. The first approach is set in the framework of a diagonalized multiplier method. From this point of view, a new update rule for the Lagrange multipliers which depends on the particular quasi-Newton method employed is given. This update rule, in contrast to most other update rules, does not require exact minimization of the intermediate unconstrained problem. In fact, the optimal convergence rate is attained in the extreme case when only one step of a quasi-Newton method is taken on this intermediate problem. The second approach transforms the constrained optimization problem into an unconstrained problem of the same dimension.

Optimal Error Bounds for the Newton–Kantorovich Theorem

Best possible upper and lower bounds for the error in Newton’s method are established under the hypotheses of the Kantorovich theorem.

Nonparametric function estimation, modeling, and simulation

1. Historical Background 2. Some Approaches to Nonparametric Density Estimation 3. Maximum Likelihood Density Estimation 4. Maximum Penalized Likelihood Density Estimation 5. Discrete Maximum Penalized Likelihood Estimation 6. Nonparametric Density of Estimation in Higher Dimensions 7. Nonparametric Regression and Intensity Function Estimation 8. Model Building and Speculative Data Analysis Appendix I. An Introduction to Mathematical Optimization Theory Appendix II. Numerical Solution of Constrained Optimization Problems Appendix III. Optimization Algorithms for Noisy Problems Appendix IV. A Brief Primer in Simulation Index.

A quadratically convergent O n L -iteration algorithm for linear programming

AbstractRecently, Ye, Tapia and Zhang (1991) demonstrated that Mizuno—Todd—Yes predictor—corrector interior-point algorithm for linear programming maintains the O(

A study of indicators for identifying zero variables in interior-point methods

\sqrt n

The Gradient Projection Method under Mild Differentiability Conditions

L)-iteration complexity while exhibiting superlinear convergence of the duality gap to zero under the assumption that the iteration sequence converges, and quadratic convergence of the duality gap to zero under the assumption of nondegeneracy. In this paper we establish the quadratic convergence result without any assumption concerning the convergence of the iteration sequence or nondegeneracy. This surprising result, to our knowledge, is the first instance of a demonstration of polynomiality and superlinear (or quadratic) convergence for an interior-point algorithm which does not assume the convergence of the iteration sequence or nondegeneracy.

A Convergence Theory for a Class of Quasi-Newton Methods for Constrained Optimization

This paper presents a convergence rate analysis for interior point primal-dual linear programming algorithms. Conditions that guarantee Q-superlinear convergence are identified in two distinct theories. Both state that, under appropriate assumptions, Q-superlinear convergence is achieved by asymptotically taking the step to the boundary of the positive orthant and letting the barrier parameter approach zero at a rate that is superlinearly faster than the convergence of the duality gap to zero. The first theory makes no nondegeneracy assumption and explains why in recent numerical experimentation Q-superlinear convergence was always observed. The second theory requires the restrictive assumption of primal nondegeneracy. However, it gives the surprising result that Q-superlinear convergence can still be attained even if centering is not phased out, provided the iterates asymptotically approach the central path. The latter theory is extended to produce a satisfactory Q-quadratic convergence theory. It requir...