This book introduces the fundamental mathematical theory of optimization, focusing on the convergence properties of algorithms and their application to complex problems. It delves into the theoretical underpinnings of nonlinear optimization, exploring how algorithms converge to solutions, and extends these concepts to tackle combinatorial optimization challenges. The text highlights recent advancements, including global convergence theories, the Powell conjecture, semidefinite programming, and the use of relaxation techniques to approximate solutions for combinatorial problems.

The book is structured to serve as a textbook or reference for students in applied mathematics, operations research, and computer science. Its focus on theoretical aspects and recent developments suggests it is aimed at readers with a solid mathematical background seeking to understand the rigorous foundations of optimization techniques and their advanced applications.