1 The Role of Mathematical Optimization in Advanced AI Applications

This chapter explores the significance of mathematical optimization across diverse fields, focusing on its critical role in artificial intelligence (AI) applications. Optimization techniques are essential for finding the best solutions while considering constraints, and they are widely applied in AI to improve decision-making and performance. We discuss key optimization methods, including linear programming, gradient descent, and evolutionary algorithms, demonstrating their applications in machine learning, robotics, and natural language processing. Practical examples illustrate how these techniques optimize model training, clustering, and policy development in reinforcement learning. Additionally, the chapter examines the emerging role of quantum computing in solving complex optimization problems and highlights ethical considerations in AI development.