Fractional programming.

Abstract

Fractional programming (FP) involves optimization problems where the objective function or constraints are expressed as ratios of two functions, typically of the form f(x) g(x), with f(x) and g(x) being continuous functions. The research in fractional programming encompasses both linear and nonlinear forms, offering valuable models in fields like economics, engineering, and management. Linear fractional programming (LFP) is characterized by linear objective functions and linear constraints, while nonlinear fractional programming (NFP) extends the concept to nonlinear functions, posing more complex optimization challenges. The primary objective of this work is to review efficient methodologies for solving both linear and nonlinear fractional programming problems, including the use of optimization techniques like the Charnes-Cooper transformation for LFP and more advanced algorithms such as iterative methods for NFP. We employ a combination of mathematical modeling and computational techniques, applying existing algorithms and proposing novel improvements in terms of convergence rates and computational complexity. Through theoretical analysis and numerical experiments, the study highlights the efficiency and practical applicability of the proposed methods for solving real-world problems. The results demonstrate significant improvements in the solution quality and computational efficiency when compared to traditional methods. The methodologies presented offer a valuable framework for tackling both linear and nonlinear fractional optimization problems, with implications for various applied fields such as resource allocation, decision-making, and financial modeling. In summary, the work contributes to advancing the general survey of linear and nonlinear fractional programming.