دانلود کتاب تحدب و کاربردهای آن در بهینه سازی گسسته و پیوسته

This book is about convex analysis and geometry, as well as their use in mathematical optimization. Mathematical optimization plays a central role in almost all mathematical and computational disciplines, and its importance in the age of data science, machine learning, and artificial intelligence cannot be overemphasized. Almost every problem in these areas has a mathematical optimization problem at its core. This book focuses primarily on convex optimization (both discrete and continuous) and its mathematical foundations. The reach of convex optimization methods and, more generally, the underlying concepts from convex analysis and geometry in business, engineering, and scientific applications is also significant. The simple idea behind convexity – that the “value” of an average of multiple potential solutions is at least as good as the average “value” of these solutions – has the ability to model an amazing variety of phenomena. There are enough textbooks on convexity and convex optimization to occupy an entire section of an academic (virtual) library. There are two reasons why I believe this book provides something new and valuable, particularly from the perspective of designing a course that introduces these ideas rigorously. First, I feel that previous textbooks and monographs focus either on the continuous/analytic aspects of convexity [42, 44, 45, 57, 59, 114, 131, 140, 141, 170, 183, 199, 210, 220] or on the discrete/combinatorial aspects of convexity [24, 116, 124, 128, 234], and no one single book surveys in a unified way the main ideas in both of these aspects (theory and applications in optimization).1 Second, there have been several recent developments, especially in the use of convexity in discrete optimization, that appear only in specialized research papers or surveys. These developments have now reached a level of maturity that can be presented in a textbook, with clean and elegant proofs available for the major results. I am convinced that the time is right for a textbook that gives a unified perspective on continuous and discrete optimization via the lens of convexity, and presents some of the recent advances in optimization that have not yet made it to the texts on convexity. The book is divided into two parts. The first part focuses on the mathematical foundations of convex analysis and geometry, starting with a study of convex sets, followed by an exposition of convex functions, and ending with an introduction to the geometry of numbers. The second part focuses on optimization, presenting discrete and continuous ideas in a common framework. Here are some topics covered in this textbook that, to the best of my knowledge, are not contained in any previous textbook on convex analysis, geometry, or optimization.