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Optimization on Low Rank Nonconvex Structures

2013-12-01 Mathematics
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Release : 2013-12-01
Genre : Mathematics
Kind : eBook
Book Rating : 984/5 ( reviews)

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Book Synopsis Optimization on Low Rank Nonconvex Structures by : Hiroshi Konno

Download or read book Optimization on Low Rank Nonconvex Structures written by Hiroshi Konno. This book was released on 2013-12-01. Available in PDF, EPUB and Kindle. Book excerpt: Global optimization is one of the fastest developing fields in mathematical optimization. In fact, an increasing number of remarkably efficient deterministic algorithms have been proposed in the last ten years for solving several classes of large scale specially structured problems encountered in such areas as chemical engineering, financial engineering, location and network optimization, production and inventory control, engineering design, computational geometry, and multi-objective and multi-level optimization. These new developments motivated the authors to write a new book devoted to global optimization problems with special structures. Most of these problems, though highly nonconvex, can be characterized by the property that they reduce to convex minimization problems when some of the variables are fixed. A number of recently developed algorithms have been proved surprisingly efficient for handling typical classes of problems exhibiting such structures, namely low rank nonconvex structures. Audience: The book will serve as a fundamental reference book for all those who are interested in mathematical optimization.

Nonconvex Optimization for Low-rank Matrix Related Problems

2020
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Release : 2020
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Book Synopsis Nonconvex Optimization for Low-rank Matrix Related Problems by : Zhenzhen Li

Download or read book Nonconvex Optimization for Low-rank Matrix Related Problems written by Zhenzhen Li. This book was released on 2020. Available in PDF, EPUB and Kindle. Book excerpt:

Non-convex Optimization for Machine Learning

2017-12-04 Machine learning
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Release : 2017-12-04
Genre : Machine learning
Kind : eBook
Book Rating : 683/5 ( reviews)

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Book Synopsis Non-convex Optimization for Machine Learning by : Prateek Jain

Download or read book Non-convex Optimization for Machine Learning written by Prateek Jain. This book was released on 2017-12-04. Available in PDF, EPUB and Kindle. Book excerpt: Non-convex Optimization for Machine Learning takes an in-depth look at the basics of non-convex optimization with applications to machine learning. It introduces the rich literature in this area, as well as equips the reader with the tools and techniques needed to apply and analyze simple but powerful procedures for non-convex problems. Non-convex Optimization for Machine Learning is as self-contained as possible while not losing focus of the main topic of non-convex optimization techniques. The monograph initiates the discussion with entire chapters devoted to presenting a tutorial-like treatment of basic concepts in convex analysis and optimization, as well as their non-convex counterparts. The monograph concludes with a look at four interesting applications in the areas of machine learning and signal processing, and exploring how the non-convex optimization techniques introduced earlier can be used to solve these problems. The monograph also contains, for each of the topics discussed, exercises and figures designed to engage the reader, as well as extensive bibliographic notes pointing towards classical works and recent advances. Non-convex Optimization for Machine Learning can be used for a semester-length course on the basics of non-convex optimization with applications to machine learning. On the other hand, it is also possible to cherry pick individual portions, such the chapter on sparse recovery, or the EM algorithm, for inclusion in a broader course. Several courses such as those in machine learning, optimization, and signal processing may benefit from the inclusion of such topics.

Approximation and Complexity in Numerical Optimization

2013-06-29 Technology & Engineering
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Release : 2013-06-29
Genre : Technology & Engineering
Kind : eBook
Book Rating : 450/5 ( reviews)

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Book Synopsis Approximation and Complexity in Numerical Optimization by : Panos M. Pardalos

Download or read book Approximation and Complexity in Numerical Optimization written by Panos M. Pardalos. This book was released on 2013-06-29. Available in PDF, EPUB and Kindle. Book excerpt: There has been much recent progress in approximation algorithms for nonconvex continuous and discrete problems from both a theoretical and a practical perspective. In discrete (or combinatorial) optimization many approaches have been developed recently that link the discrete universe to the continuous universe through geomet ric, analytic, and algebraic techniques. Such techniques include global optimization formulations, semidefinite programming, and spectral theory. As a result new ap proximate algorithms have been discovered and many new computational approaches have been developed. Similarly, for many continuous nonconvex optimization prob lems, new approximate algorithms have been developed based on semidefinite pro gramming and new randomization techniques. On the other hand, computational complexity, originating from the interactions between computer science and numeri cal optimization, is one of the major theories that have revolutionized the approach to solving optimization problems and to analyzing their intrinsic difficulty. The main focus of complexity is the study of whether existing algorithms are efficient for the solution of problems, and which problems are likely to be tractable. The quest for developing efficient algorithms leads also to elegant general approaches for solving optimization problems, and reveals surprising connections among problems and their solutions. A conference on Approximation and Complexity in Numerical Optimization: Con tinuous and Discrete Problems was held during February 28 to March 2, 1999 at the Center for Applied Optimization of the University of Florida.

A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems

2013-04-17 Mathematics
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Release : 2013-04-17
Genre : Mathematics
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Book Rating : 882/5 ( reviews)

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Book Synopsis A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems by : Hanif D. Sherali

Download or read book A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems written by Hanif D. Sherali. This book was released on 2013-04-17. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with the theory and applications of the Reformulation- Linearization/Convexification Technique (RL T) for solving nonconvex optimization problems. A unified treatment of discrete and continuous nonconvex programming problems is presented using this approach. In essence, the bridge between these two types of nonconvexities is made via a polynomial representation of discrete constraints. For example, the binariness on a 0-1 variable x . can be equivalently J expressed as the polynomial constraint x . (1-x . ) = 0. The motivation for this book is J J the role of tight linear/convex programming representations or relaxations in solving such discrete and continuous nonconvex programming problems. The principal thrust is to commence with a model that affords a useful representation and structure, and then to further strengthen this representation through automatic reformulation and constraint generation techniques. As mentioned above, the focal point of this book is the development and application of RL T for use as an automatic reformulation procedure, and also, to generate strong valid inequalities. The RLT operates in two phases. In the Reformulation Phase, certain types of additional implied polynomial constraints, that include the aforementioned constraints in the case of binary variables, are appended to the problem. The resulting problem is subsequently linearized, except that certain convex constraints are sometimes retained in XV particular special cases, in the Linearization/Convexijication Phase. This is done via the definition of suitable new variables to replace each distinct variable-product term. The higher dimensional representation yields a linear (or convex) programming relaxation.

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