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 the degree of difficulty, which is a generally accepted measure in geometric programming.