Geodesic optimization methods for multi-objective engineering design with conflicting performance requirements

Abstract

The aim of this research was to develop the geodesic minimax optimization framework which would be applicable for solving multi-objective engineering design problems where conflicting performance requirements are present. The proposed framework intended to improve the optimization accuracy, stability, Pareto coverage and worst-case performance through Riemannian manifolds representation of the decision space. The approach developed formulated several engineering objectives by using weighted minimax function, and applied the iterative optimization via Riemannian gradients and geodesics' exponential mappings. Numerical validation of the method was conducted via the use of Rosenbrock, Ackley and Rastrigin test functions as well as multi-criteria engineering design validation task. The proposed method was compared with the classical gradient descent technique, NSGA-II, particle swarm optimization and weighted-sum optimization. The average benchmark error achieved with the proposed geodesic minimax approach is 0.012 while gradient descent results in 0.079, NSGA-II gives 0.043 and particle swarm optimization generates 0.037. For the multi-criteria design problem validation, the proposed method provides 0.041 as objective function value, 96 Pareto solutions and 63 iteration convergences. Weighted-sum optimization, NSGA-II and particle swarm optimization give the objective value of 0.184, 0.126 and 0.097 correspondingly with 42, 68 and 75 Pareto solutions. The geodesic minimax framework provided superior accuracy, balanced multi-objective performance, Pareto coverage, and convergence efficiency. By integrating Riemannian geometry with minimax optimization, the proposed method offers a theoretically grounded and computationally effective approach for nonlinear engineering design problems with conflicting objectives.