A large number of mathematical models in many diverse areas of science and engineering have lead to the formulation of optimization problems where the best solution (globally optimal) is needed. This book covers a small subset of important topics in global optimization with emphasis on theoretical developments and scientific applications.
This book provides a comprehensive, modern introduction to convex optimization, a field that is becoming increasingly important in applied mathematics, economics and finance, engineering, and computer science, notably in data science and machine learning. Written by a leading expert in the field, this book includes recent advances in the algorithmic theory of convex optimization, naturally complementing the existing literature. It contains a unified and rigorous presentation of the acceleration techniques for minimization schemes of first- and second-order. It provides readers with a full treatment of the smoothing technique, which has tremendously extended the abilities of gradient-type methods. Several powerful approaches in structural optimization, including optimization in relative scale and polynomial-time interior-point methods, are also discussed in detail. Researchers in theoretical optimization as well as professionals working on optimization problems will find this book very useful. It presents many successful examples of how to develop very fast specialized minimization algorithms. Based on the author’s lectures, it can naturally serve as the basis for introductory and advanced courses in convex optimization for students in engineering, economics, computer science and mathematics.
Significant research activity has occurred in the area of global optimization in recent years. Many new theoretical, algorithmic, and computational contributions have resulted. Despite the major importance of test problems for researchers, there has been a lack of representative nonconvex test problems for constrained global optimization algorithms. This book is motivated by the scarcity of global optimization test problems and represents the first systematic collection of test problems for evaluating and testing constrained global optimization algorithms. This collection includes problems arising in a variety of engineering applications, and test problems from published computational reports.
Due to the general complementary convex structure underlying most nonconvex optimization problems encountered in applications, convex analysis plays an essential role in the development of global optimization methods. This book develops a coherent and rigorous theory of deterministic global optimization from this point of view. Part I constitutes an introduction to convex analysis, with an emphasis on concepts, properties and results particularly needed for global optimization, including those pertaining to the complementary convex structure. Part II presents the foundation and application of global search principles such as partitioning and cutting, outer and inner approximation, and decomposition to general global optimization problems and to problems with a low-rank nonconvex structure as well as quadratic problems. Much new material is offered, aside from a rigorous mathematical development. Audience: The book is written as a text for graduate students in engineering, mathematics, operations research, computer science and other disciplines dealing with optimization theory. It is also addressed to all scientists in various fields who are interested in mathematical optimization.
This book provides a unified and insightful treatment of deterministic global optimization. It introduces theoretical and algorithmic advances that address the computation and characterization of global optima, determine valid lower and upper bounds on the global minima and maxima, and enclose all solutions of nonlinear constrained systems of equations. Among its special features, the book: Introduces the fundamentals of deterministic global optimization; Provides a thorough treatment of decomposition-based global optimization approaches for biconvex and bilinear problems; Covers global optimization methods for generalized geometric programming problems Presents in-depth global optimization algorithms for general twice continuously differentiable nonlinear problems; Provides a detailed treatment of global optimization methods for mixed-integer nonlinear problems; Develops global optimization approaches for the enclosure of all solutions of nonlinear constrained systems of equations; Includes many important applications from process design, synthesis, control, and operations, phase equilibrium, design under uncertainty, parameter estimation, azeotrope prediction, structure prediction in clusters and molecules, protein folding, and peptide docking. Audience: This book can be used as a textbook in graduate-level courses and as a desk reference for researchers in all branches of engineering and applied science, applied mathematics, industrial engineering, operations research, computer science, economics, computational chemistry and molecular biology.
This book treats the subject of global optimization with minimal restrictions on the behavior on the objective functions. In particular, optimal conditions were developed for a class of noncontinuous functions characterized by their having level sets that are robust. The integration-based approach contrasts with existing approaches which require some degree of convexity or differentiability of the objective function. Some computational results on a personal computer are presented.
Here is a book devoted to well-structured and thus efficiently solvable convex optimization problems, with emphasis on conic quadratic and semidefinite programming. The authors present the basic theory underlying these problems as well as their numerous applications in engineering, including synthesis of filters, Lyapunov stability analysis, and structural design. The authors also discuss the complexity issues and provide an overview of the basic theory of state-of-the-art polynomial time interior point methods for linear, conic quadratic, and semidefinite programming. The book's focus on well-structured convex problems in conic form allows for unified theoretical and algorithmical treatment of a wide spectrum of important optimization problems arising in applications.
This textbook is an introduction to global optimization, which treats mathematical facts stringently on the one hand, but also motivates them in great detail and illustrates them with 80 figures. The book is therefore not only aimed at mathematicians, but also at natural scientists, engineers and economists who want to understand and apply mathematically sound methods in their field. With almost two hundred pages, the book provides enough choices to use it as a basis for differently designed lectures on global optimization. The detailed treatment of the global solvability of optimization problems under application-relevant conditions sets a new accent that enriches the stock of previous textbooks on optimization. Using the theory and algorithms of smooth convex optimization, the book illustrates that the global solution of a class of optimization problems frequently encountered in practice is efficiently possible, while for the more difficult-to-handle non-convex problems itdevelops in detail the ideas of branch-and-bound methods. This book is the English translation of the 2nd edition of “Grundzüge der Globalen Optimierung” (Springer, 2021) written in German. The translation was done with the help of artificial intelligence. A subsequent revision was performed by the author to further refine the work and to ensure that the translation is appropriate concerning content and scientific correctness. It may, however, read stylistically different from a conventional translation.
Employing a closed set-theoretic foundation for interval computations, Global Optimization Using Interval Analysis simplifies algorithm construction and increases generality of interval arithmetic. This Second Edition contains an up-to-date discussion of interval methods for solving systems of nonlinear equations and global optimization problems. It expands and improves various aspects of its forerunner and features significant new discussions, such as those on the use of consistency methods to enhance algorithm performance. Provided algorithms are guaranteed to find and bound all solutions to these problems despite bounded errors in data, in approximations, and from use of rounded arithmetic.
