A Survey On Non – linear Optimization And Global Optimization Methods

Abstract

bjective functions and constraints are not linear. Non-linear optimization involves finding the best solution in the presence of non-linear relationships among the variables. These problems arise in various fields such as engineering design, economics, machine learning, and operational research. The primary challenge in non-linear optimization is the potential existence of multiple local optima, which can make it difficult to identify the global optimum. Global optimization methods are designed to overcome this challenge by searching for the global optimum, which is the best possible solution among all local optima. These methods can be broadly categorized into deterministic and stochastic approaches. Deterministic methods, such as branch and bound, interval analysis, and cutting-plane methods, systematically explore the search space and provide guarantees of finding the global optimum or bounds on the global optimum. These methods are often rigorous but can be computationally expensive, especially for high-dimensional problems. On the other hand, stochastic methods, such as genetic algorithms, simulated annealing, and particle swarm optimization, use probabilistic rules to explore the search space. These methods are inspired by natural processes and heuristics, offering flexibility and often being more efficient for large-scale problems. While they do not guarantee finding the global optimum, they are widely used due to their ability to escape local optima and explore the search space effectively. Hybrid methods that combine deterministic and stochastic approaches have also gained popularity. These methods leverage the strengths of both approaches, enhancing the robustness and efficiency of the optimization process. For example, a hybrid algorithm might use a stochastic method to explore the search space broadly and then apply a deterministic method to refine the search around promising areas. Recent advancements in computational power and algorithmic development have significantly improved the efficiency and applicability of non-linear and global optimization methods. Techniques such as machine learning-based optimization and parallel computing have enabled the handling of more complex and larger-scale optimization problems. Moreover, these advancements have facilitated the integration of optimization methods into real-time and adaptive systems, further broadening their applications.