Global Optimization of Polynomial Functions and Applications
Author: Jiawang Nie
Publisher:
Published: 2006
Total Pages: 324
ISBN-13:
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Global Optimization
Author: Marco Locatelli
Publisher: SIAM
Published: 2013-10-16
Total Pages: 439
ISBN-13: 1611972671
DOWNLOAD EBOOKThis volume contains a thorough overview of the rapidly growing field of global optimization, with chapters on key topics such as complexity, heuristic methods, derivation of lower bounds for minimization problems, and branch-and-bound methods and convergence. The final chapter offers both benchmark test problems and applications of global optimization, such as finding the conformation of a molecule or planning an optimal trajectory for interplanetary space travel. An appendix provides fundamental information on convex and concave functions. Intended for Ph.D. students, researchers, and practitioners looking for advanced solution methods to difficult optimization problems. It can be used as a supplementary text in an advanced graduate-level seminar.
Moments, Positive Polynomials and Their Applications
Author: Jean-Bernard Lasserre
Publisher: World Scientific
Published: 2010
Total Pages: 384
ISBN-13: 1848164467
DOWNLOAD EBOOK1. The generalized moment problem. 1.1. Formulations. 1.2. Duality theory. 1.3. Computational complexity. 1.4. Summary. 1.5. Exercises. 1.6. Notes and sources -- 2. Positive polynomials. 2.1. Sum of squares representations and semi-definite optimization. 2.2. Nonnegative versus s.o.s. polynomials. 2.3. Representation theorems : univariate case. 2.4. Representation theorems : mutivariate case. 2.5. Polynomials positive on a compact basic semi-algebraic set. 2.6. Polynomials nonnegative on real varieties. 2.7. Representations with sparsity properties. 2.8. Representation of convex polynomials. 2.9. Summary. 2.10. Exercises. 2.11. Notes and sources -- 3. Moments. 3.1. The one-dimensional moment problem. 3.2. The multi-dimensional moment problem. 3.3. The K-moment problem. 3.4. Moment conditions for bounded density. 3.5. Summary. 3.6. Exercises. 3.7. Notes and sources -- 4. Algorithms for moment problems. 4.1. The overall approach. 4.2. Semidefinite relaxations. 4.3. Extraction of solutions. 4.4. Linear relaxations. 4.5. Extensions. 4.6. Exploiting sparsity. 4.7. Summary. 4.8. Exercises. 4.9. Notes and sources. 4.10. Proofs -- 5. Global optimization over polynomials. 5.1. The primal and dual perspectives. 5.2. Unconstrained polynomial optimization. 5.3. Constrained polynomial optimization : semidefinite relaxations. 5.4. Linear programming relaxations. 5.5. Global optimality conditions. 5.6. Convex polynomial programs. 5.7. Discrete optimization. 5.8. Global minimization of a rational function. 5.9. Exploiting symmetry. 5.10. Summary. 5.11. Exercises. 5.12. Notes and sources -- 6. Systems of polynomial equations. 6.1. Introduction. 6.2. Finding a real solution to systems of polynomial equations. 6.3. Finding all complex and/or all real solutions : a unified treatment. 6.4. Summary. 6.5. Exercises. 6.6. Notes and sources -- 7. Applications in probability. 7.1. Upper bounds on measures with moment conditions. 7.2. Measuring basic semi-algebraic sets. 7.3. Measures with given marginals. 7.4. Summary. 7.5. Exercises. 7.6. Notes and sources -- 8. Markov chains applications. 8.1. Bounds on invariant measures. 8.2. Evaluation of ergodic criteria. 8.3. Summary. 8.4. Exercises. 8.5. Notes and sources -- 9. Application in mathematical finance. 9.1. Option pricing with moment information. 9.2. Option pricing with a dynamic model. 9.3. Summary. 9.4. Notes and sources -- 10. Application in control. 10.1. Introduction. 10.2. Weak formulation of optimal control problems. 10.3. Semidefinite relaxations for the OCP. 10.4. Summary. 10.5. Notes and sources -- 11. Convex envelope and representation of convex sets. 11.1. The convex envelope of a rational function. 11.2. Semidefinite representation of convex sets. 11.3. Algebraic certificates of convexity. 11.4. Summary. 11.5. Exercises. 11.6. Notes and sources -- 12. Multivariate integration 12.1. Integration of a rational function. 12.2. Integration of exponentials of polynomials. 12.3. Maximum entropy estimation. 12.4. Summary. 12.5. Exercises. 12.6. Notes and sources -- 13. Min-max problems and Nash equilibria. 13.1. Robust polynomial optimization. 13.2. Minimizing the sup of finitely many rational cunctions. 13.3. Application to Nash equilibria. 13.4. Exercises. 13.5. Notes and sources -- 14. Bounds on linear PDE. 14.1. Linear partial differential equations. 14.2. Notes and sources
Recent Advances in Global Optimization
Author: Christodoulos A. Floudas
Publisher: Princeton University Press
Published: 2014-07-14
Total Pages: 644
ISBN-13: 1400862523
DOWNLOAD EBOOKThis book will present the papers delivered at the first U.S. conference devoted exclusively to global optimization and will thus provide valuable insights into the significant research on the topic that has been emerging during recent years. Held at Princeton University in May 1991, the conference brought together an interdisciplinary group of the most active developers of algorithms for global optimization in order to focus the attention of the mathematical programming community on the unsolved problems and diverse applications of this field. The main subjects addressed at the conference were advances in deterministic and stochastic methods for global optimization, parallel algorithms for global optimization problems, and applications of global optimization. Although global optimization is primarily a mathematical problem, it is relevant to several other disciplines, including computer science, applied mathematics, physical chemistry, molecular biology, statistics, physics, engineering, operations research, communication theory, and economics. Global optimization problems originate from a wide variety of mathematical models of real-world systems. Some of its applications are allocation and location problems and VLSI and data-base design problems. Originally published in 1991. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Handbook of Global Optimization
Author: Panos M. Pardalos
Publisher: Springer Science & Business Media
Published: 2013-04-18
Total Pages: 571
ISBN-13: 1475753624
DOWNLOAD EBOOKIn 1995 the Handbook of Global Optimization (first volume), edited by R. Horst, and P.M. Pardalos, was published. This second volume of the Handbook of Global Optimization is comprised of chapters dealing with modern approaches to global optimization, including different types of heuristics. Topics covered in the handbook include various metaheuristics, such as simulated annealing, genetic algorithms, neural networks, taboo search, shake-and-bake methods, and deformation methods. In addition, the book contains chapters on new exact stochastic and deterministic approaches to continuous and mixed-integer global optimization, such as stochastic adaptive search, two-phase methods, branch-and-bound methods with new relaxation and branching strategies, algorithms based on local optimization, and dynamical search. Finally, the book contains chapters on experimental analysis of algorithms and software, test problems, and applications.
Global Optimality Conditions and Optimization Methods for Polynomial Programming Problems and Their Applications
Author: Jing Tian
Publisher:
Published: 2014
Total Pages: 478
ISBN-13:
DOWNLOAD EBOOK"The polynomial programming problem which has a polynomial objective function, either with no constraints or with polynomial constraints occurs frequently in engineering design, investment science, control theory, network distribution, signal processing and location-allocation contexts. Moreover, the polynomial programming problem is known to be Nondeterministic Polynomial-time hard (NP-hard). The polynomial programming problem has attracted a lot of attention, including quadratic, cubic, homogenous or normal quartic programming problems as special cases. Existing methods for solving polynomial programming problems include algebraic methods and various convex relaxation methods. Especially, among these methods, semidefinite programming (SDP) and sum of squares (SOS) relaxations are very popular. Theoretically, SDP and SOS relaxation methods are very powerful and successful in solving the general polynomial programming problem with a compact feasible region. However, the solvability in practice depends on the size or the degree of the polynomial programming problem and the required accuracy. Hence, solving large scale SDP problems still remains a computational challenge. It is well-known that traditional local optimization methods are designed based on necessary local optimality conditions, i.e., Karush-Kuhn-Tucker (KKT) conditions. Motivated by this, some researchers proposed a necessary global optimality condition for a quadratic programming problem and designed a new local optimization method according to the necessary global optimality condition. In this thesis, we try to apply this idea to cubic and quatic programming problems, and further to general unconstrained and constrained polynomial programming problems. For these polynomial programming problems, we will investigate necessary global optimality conditions and design new local optimization methods according to these conditions. These necessary global optimality conditions are generally stronger than KKT conditions. Hence, the obtained new local minimizers by using the new local optimization methods may improve some KKT points. Our ultimate aim is to design global optimization methods for these polynomial programming problems. We notice that the filled function method is one of the well-known and practical auxiliary function methods used to achieve a global minimizer. In this thesis, we design global optimization methods by combining the new proposed local optimization methods and some auxiliary functions. The numerical examples illustrate the efficiency and stability of the optimization methods. Finally, we discuss some applications for solving some sensor network localization problems and systems of polynomial equations. It is worth mentioning that we apply the idea and the results for polynomial programming problems to nonlinear programming problems (NLP). We provide an optimality condition and design new local optimization methods according to the optimality condition and design global optimization methods for the problem (NLP) by combining the new local optimization methods and an auxiliary function. In order to test the performance of the global optimization methods, we compare them with two other heuristic methods. The results demonstrate our methods outperform the two other algorithms.." -- Abstract.
Moment and Polynomial Optimization
Author: Jiawang Nie
Publisher: SIAM
Published: 2023-06-15
Total Pages: 484
ISBN-13: 1611977606
DOWNLOAD EBOOKMoment and polynomial optimization is an active research field used to solve difficult questions in many areas, including global optimization, tensor computation, saddle points, Nash equilibrium, and bilevel programs, and it has many applications. The author synthesizes current research and applications, providing a systematic introduction to theory and methods, a comprehensive approach for extracting optimizers and solving truncated moment problems, and a creative methodology for using optimality conditions to construct tight Moment-SOS relaxations. This book is intended for applied mathematicians, engineers, and researchers entering the field. It can be used as a textbook for graduate students in courses on convex optimization, polynomial optimization, and matrix and tensor optimization.
Genericity In Polynomial Optimization
Author: Tien Son Pham
Publisher: World Scientific
Published: 2016-12-22
Total Pages: 261
ISBN-13: 1786342235
DOWNLOAD EBOOKIn full generality, minimizing a polynomial function over a closed semi-algebraic set requires complex mathematical equations. This book explains recent developments from singularity theory and semi-algebraic geometry for studying polynomial optimization problems. Classes of generic problems are defined in a simple and elegant manner by using only the two basic (and relatively simple) notions of Newton polyhedron and non-degeneracy conditions associated with a given polynomial optimization problem. These conditions are well known in singularity theory, however, they are rarely considered within the optimization community.Explanations focus on critical points and tangencies of polynomial optimization, Hölderian error bounds for polynomial systems, Frank-Wolfe-type theorem for polynomial programs and well-posedness in polynomial optimization. It then goes on to look at optimization for the different types of polynomials. Through this text graduate students, PhD students and researchers of mathematics will be provided with the knowledge necessary to use semi-algebraic geometry in optimization.
Introduction to Global Optimization
Author: R. Horst
Publisher: Springer Science & Business Media
Published: 2000-12-31
Total Pages: 376
ISBN-13: 9780792367567
DOWNLOAD EBOOKA textbook for an undergraduate course in mathematical programming for students with a knowledge of elementary real analysis, linear algebra, and classical linear programming (simple techniques). Focuses on the computation and characterization of global optima of nonlinear functions, rather than the locally optimal solutions addressed by most books on optimization. Incorporates the theoretical, algorithmic, and computational advances of the past three decades that help solve globally multi-extreme problems in the mathematical modeling of real world systems. Annotation copyright by Book News, Inc., Portland, OR
Handbook of Global Optimization
Author: R. Horst
Publisher: Springer Science & Business Media
Published: 2013-12-11
Total Pages: 891
ISBN-13: 1461520258
DOWNLOAD EBOOKGlobal optimization is concerned with the computation and characterization of global optima of nonlinear functions. During the past three decades the field of global optimization has been growing at a rapid pace, and the number of publications on all aspects of global optimization has been increasing steadily. Many applications, as well as new theoretical, algorithmic, and computational contributions have resulted. The Handbook of Global Optimization is the first comprehensive book to cover recent developments in global optimization. Each contribution in the Handbook is essentially expository in nature, but scholarly in its treatment. The chapters cover optimality conditions, complexity results, concave minimization, DC programming, general quadratic programming, nonlinear complementarity, minimax problems, multiplicative programming, Lipschitz optimization, fractional programming, network problems, trajectory methods, homotopy methods, interval methods, and stochastic approaches. The Handbook of Global Optimization is addressed to researchers in mathematical programming, as well as all scientists who use optimization methods to model and solve problems.
