Approaches to Nonlinear Programming Problems: Taylor Series Expansion, RBF Surrogate Modeling, DOE-Based Dimensionality Reduction, and Adaptive Domain Splitting

<title>Abstract</title> <p>Solving nonlinear constrained optimization problems efficiently remains a challenge. This paper describes a four-component method developed to address these difficulties: Design of Experiments (DOE) screening using L9 and L81 orthogonal arrays to eliminate statistically non-significant variables before optimization begins; adaptive domain splitting into 4–8 gradient-guided subregions to handle multimodality; Taylor series expansion (1st–3rd order) and Radial Basis Function (RBF) surrogate modeling for local and global function approximation respectively; and a hybrid Genetic Algorithm–Sequential Quadratic Programming (GA–SQP) solver. The methodology was evaluated on 20 benchmark nonlinear programming functions spanning low- (1–3 variables), medium- (4–6 variables), and high-dimensional (7–12 variables) problems, as well as 15 practical engineering case studies. DOE-based achieved 95 and 99% confidence in variable selection, reducing problem dimensionality by up to 75% in some cases. RBF surrogates achieved convergence on all 20 benchmark functions, reducing computational time by 70–80% compared to Taylor-series approaches. Adaptive domain splitting reduced function evaluations by 30–35% on multimodal problems. Taken as a whole, the results make a clear case that pre-screening variables with DOE, approximating the reduced function with RBF surrogates, and refining solutions with GA–SQP can cut both computational effort and convergence time substantially without sacrificing accuracy.</p>