Genetic Algorithm-Accelerated Optimization for Runge-Kutta Scheme Derivation in High-Dimensional Problems With Hard Constraints

Abstract

The numerical derivation of high-order Extended Stability Runge-Kutta (ERSK) integration schemes presents significant challenges due to the high-dimensional, non-convex, non-linear optimization problems with hard equality constraints. Traditional optimization techniques often struggle to converge on valid solutions efficiently. This paper proposes a hybrid approach that integrates Genetic Algorithm (GA) techniques with an interior-point optimizer to address these challenges. GA-generated values are used as starting points for the interior-point method, which refines candidate solutions to meet the stringent order conditions. Experimental results demonstrate a significant improvement in efficiency when GA-generated values are used. The average number of iterations required for convergence is reduced from 477 (with random initialization) to 143 when GA values are used. The median number of iterations drops from 280 to 105. The standard deviation is also reduced from 361 to 112, indicating more consistent performance. These reductions highlight the efficiency of the GA-plus-polish approach, which outperforms traditional random initialization in terms of convergence time and consistency. Despite some remaining variance, this hybrid method significantly reduces the computational cost of deriving stable Butcher tableaux, particularly in high-dimensional, non-convex optimization problems.

Keywords

• Genetic algorithms
• non-convex
• Runge-Kutta schemes
• optimization