A Review of Mathematical Optimization Methods and Related Algorithms in Management, Engineering, and Data Science

This study provides a comprehensive review of mathematical optimization methods and related algorithms in key domains including management, engineering, and data science. In today’s rapidly evolving world, mathematical optimization serves as a foundational tool that plays a pivotal role in advancing knowledge and technology, forming the core of complex decision-making processes. With the increasing volume and complexity of data, the demand for efficient algorithms and more precise problem-solving methodologies has become increasingly critical. The paper begins with an overview of the historical foundations and scientific developments of optimization theory, followed by a systematic examination of classical methods such as Linear Programming (LP) and Nonlinear Programming (NLP), along with associated algorithms including the Simplex method, Gradient Descent, Newton’s method, and Quasi-Newton methods. Given the challenges posed by real-world problems such as high dimensionality, complex constraints, and uncertainty the study further explores the development and application of metaheuristic approaches, including Genetic Algorithms (GA), Particle Swarm Optimization (PSO), and Ant Colony Optimization (ACO). Although these algorithms do not guarantee global optimality, they demonstrate substantial capability in identifying near-optimal solutions for highly complex and large-scale problems. The research also addresses practical implementation challenges, such as scalability and the selection of appropriate algorithms for specific problem structures. By proposing a novel classification framework and conducting a comprehensive comparative analysis, the paper evaluates the strengths and limitations of various optimization techniques across diverse application scenarios. Finally, emerging trends including the integration of optimization with machine learning and the development of hybrid and distributed algorithms are highlighted. The study emphasizes the critical importance of thoroughly understanding problem characteristics in order to select the most suitable optimization strategy. The findings provide a valuable practical guide for researchers and practitioners seeking to effectively apply optimization methods in management, engineering, and data science.

کلیدواژه ها:

Mathematical Optimization ، Metaheuristic Algorithms ، Linear and Nonlinear Programming ، Management ، engineering ، data science ، evolutionary algorithms