Optimization Techniques in Applied Mathematics: From Physics Simulations to Real-World Problems

Abstract

Optimization techniques play a critical role in applied mathematics, enabling the effective and efficient solving of complex problems across various fields. This paper explores the application of optimization methods from theoretical frameworks to practical scenarios, particularly focusing on their utilization in physics simulations and real-world problems. We review classical optimization techniques, including linear programming and nonlinear optimization, as well as advanced methods such as evolutionary algorithms and machine learning-based approaches. By examining case studies and current applications, we highlight the effectiveness of these techniques in addressing real-world challenges, from engineering design to financial modeling. The insights provided offer a comprehensive understanding of how optimization contributes to advancing both theoretical research and practical problem-solving.