An Analytical Study of Optimization Techniques Using Linear and Nonlinear Programming Models

Optimization plays a vital role in solving real-world engineering, management, and scientific problems. This paper presents an analytical study of linear and nonlinear programming techniques used for optimal decision-making under constraints. The study focuses on mathematical formulation, solution procedures, and comparative performance of classical optimization methods such as the simplex method, graphical method, Karush–Kuhn–Tucker (KKT) conditions, and gradient-based techniques. Several numerical case studies are included to demonstrate the applicability of these methods in production planning, cost minimization, and resource allocation problems. The results show that linear programming is efficient for proportional systems, whereas nonlinear programming provides greater flexibility for complex engineering systems. The study concludes that optimization techniques significantly enhance system performance and decision accuracy in multidisciplinary applications.