A Primal Nonlinear Fractional Programming Approach for Posynomial Geometric Programming

Abstract

Geometric programming (GP) is a well-established optimization framework widely used in engineering design and related areas for solving nonlinear optimization problems. Classical Classi approaches for solving GP problems typically rely on dual formulations, which may become restrictive when the degree of difficulty is high. In this paper, we propose a direct primal approach for solving constrained posynomial geometric programming problems by reformulating them as nonlinear fractional programming problems, without resorting to duality. The proposed method transforms the original GP into a ratio optimization problem and applies a parametrization technique based on the Dinkelbach method to obtain the optimal solution. This approach allows GP problems with nonzero degrees of difficulty to be handled in a systematic and computationally efficient manner. Theoretical results supporting the proposed formulation are presented, with several lemmas developed within the paper and standard results clearly distinguished from those recalled from existing literature. Numerical examples are provided to demonstrate the effectiveness and accuracy of the proposed approach. In addition, potential challenges associated with large-scale geometric programming problems, such as computational complexity and convergence issues, are briefly discussed.