Homogeneous optimization strategies for nonlinear engineering design problems with complex decision variables

Abstract

Minimax optimization problems with coupled and complex decision variables pose serious challenges due to strong coupling among decision variables, irregular optimization landscape, sensitivity to scaling factors, and inability to achieve convergence guarantees. Existing minimax optimization techniques suffer from low convergence rates, dependency on parameter choice, and insufficient theoretical understanding of their performance under nonlinear conditions. In this paper, a homogeneous transformation-based optimization technique is proposed for better performance in nonlinear minimax optimization problems with a scale-invariant mathematical model. The proposed framework implements normalization of the decision variables space via homogeneity while keeping the optimization direction regardless of changes in decision variables scaling factors. In order to apply the developed homogeneous technique, a homogeneous minimax problem formulation based on homogeneous functions theory is introduced, and its stability and convergence characteristics are rigorously analyzed through Lyapunov theoretical analysis. The efficiency of the proposed method is assessed using nonlinear design benchmark problems, where the framework shows better convergence speed, lower objective function value, and improved optimization stability in comparison to gradient, penalty, and genetic optimization methods. Obtained results prove the ability of the homogeneous optimization approach to improve minimax optimization problem performance in the case of complex nonlinear design problem formulations with complicated decision variables.