Kenneth O. Kortanek

An infeasible interior-point algorithm for solving primal and dual geometric programs

In this paper an algorithm is presented for solving the classical posynomial geometric programming dual pair of problems simultaneously. The approach is by means of a primal-dual infeasible algorithm developed simultaneously for (i) the dual geometric program after logarithmic transformation of its objective function, and (ii) its Lagrangian dual program. Under rather general assumptions, the mechanism defines a primal-dual infeasible path from a specially constructed, perturbed Karush-Kuhn-Tucker system.Subfeasible solutions, as described by Duffin in 1956, are generated for each program whose primal and dual objective function values converge to the respective primal and dual program values. The basic technique is one of a predictor-corrector type involving Newton’s method applied to the perturbed KKT system, coupled with effective techniques for choosing iterate directions and step lengths. We also discuss implementation issues and some sparse matrix factorizations that take advantage of the very special structure of the Hessian matrix of the logarithmically transformed dual objective function. Our computational results on 19 of the most challenging GP problems found in the literature are encouraging. The performance indicates that the algorithm is effective regardless of thedegree of difficulty, which is a generally accepted measure in geometric programming.

On some efficient interior point methods for nonlinear convex programming

Abstract We introduce two interior point algorithms for minimizing a convex function subject to linear constraints. Our algorithms require the solution of a nonlinear system of equations at each step. We show that if sufficiently good approximations to the solutions of the nonlinear systems can be found, then the primal-dual gap becomes less that e in O( n |lne|) steps, where n is the number of variables.

A second order affine scaling algorithm for the geometric programming dual with logarithmic barrier

We present an interior point algorithm for solving the dual geometric programming problem, which avoids nondifferentiability at the boundary, yet uses singular Hessian information. The algorithm generates three sequences which converge, respectively, to optimal solutions for the primal and dual geometric programs, and to a Lagrangian dual solution of the dual geometric program. The sequences are connected by a robust procedure for converting duai GP solutions to primai GP solutions, and error bounds are given. Extensive computational experience is reported including solutions to GP problems having largest known degree of difficulty.

Maximum likelihood estimates with order restrictions on probabilities and odds ratios: A geometric programming approach

The problem of assigning cell probabilities to maximize a multinomial likelihood with order restrictions on the probabilies and/or restrictions on the local odds ratios is modeled as a posynomial geometric program (GP), a class of nonlinear optimization problems with a well-developed duality theory and collection of algorithms. (Local odds ratios provide a measure of association between categorical random variables.) A constrained multinomial MLE example from the literature is solved, and the quality of the solution is compared with that obtained by the iterative method of El Barmi and Dykstra, which is based upon Fenchel duality. Exploiting the proximity of the GP model of MLE problems to linear programming (LP) problems, we also describe as an alternative, in the absence of special-purpose GP software, an easily implemented successive LP approximation method for solving this class of MLE problems using one of the readily available LP solvers.

Near boundary behavior of primal-dual potential reduction algorithms for linear programming

AbstractThis paper is concerned with selection of theρ-parameter in the primal—dual potential reduction algorithm for linear programming. Chosen from [n +

OBSERVATIONS ON INFEASIBILITY DETECTORS FOR CLASSIFYING CONIC CONVEX PROGRAMS

\sqrt n

On a Compound Duality Classification for Geometric Programming

, ∞), the level ofρ determines the relative importance placed on the centering vs. the Newton directions. Intuitively, it would seem that as the iterate drifts away from the central path towards the boundary of the positive orthant,ρ must be set close ton +