Optimization with Constraints: Theory and Applications

This lecture introduces the concepts of optimization with constraints, focusing on the theoretical foundations and practical applications. The instructor begins by reviewing optimization without constraints, emphasizing the importance of finding a minimum point in a defined space. The discussion progresses to optimization with constraints, where the search for optimal solutions is limited to a specific domain, referred to as Omega. The instructor explains the role of inequality and equality constraints, providing examples to illustrate these concepts. Key formulas and theorems, including the Karush-Kuhn-Tucker (KKT) conditions, are presented as essential tools for solving constrained optimization problems. The lecture also covers numerical methods for addressing these problems and highlights the significance of understanding the underlying principles for practical applications, such as control problems in engineering. Throughout the session, the instructor emphasizes the need to master these concepts for successful problem-solving in optimization scenarios.