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Interior Point And Outer Approximation Methods For Conic Optimization
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Book Synopsis Interior Point and Outer Approximation Methods for Conic Optimization by : Christopher Daniel Lang Coey
Download or read book Interior Point and Outer Approximation Methods for Conic Optimization written by Christopher Daniel Lang Coey and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Any convex optimization problem may be represented as a conic problem that minimizes a linear function over the intersection of an affine subspace with a convex cone. An advantage of representing convex problems in conic form is that, under certain regularity conditions, a conic problem has a simple and easily checkable certificate of optimality, primal infeasibility, or dual infeasibility. As a natural generalization of linear programming duality, conic duality allows us to design powerful algorithms for continuous and mixed-integer convex optimization. The main goal of this thesis is to improve the generality and practical performance of (i) interior point methods for continuous conic problems and (ii) outer approximation methods for mixed-integer conic problems. We implement our algorithms in extensible open source solvers accessible through the convenient modeling language JuMP. From around 50 applied examples, we formulate continuous and mixed-integer problems over two dozen different convex cone types, many of which are new. Our extensive computational experiments with these examples explore which algorithmic features and what types of equivalent conic formulations lead to the best performance.
Book Synopsis A Mathematical View of Interior-point Methods in Convex Optimization by : James Renegar
Download or read book A Mathematical View of Interior-point Methods in Convex Optimization written by James Renegar and published by SIAM. This book was released on 2001-01-01 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Interior-point Polynomial Algorithms in Convex Programming by : Yurii Nesterov
Download or read book Interior-point Polynomial Algorithms in Convex Programming written by Yurii Nesterov and published by SIAM. This book was released on 1994-01-01 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: Specialists working in the areas of optimization, mathematical programming, or control theory will find this book invaluable for studying interior-point methods for linear and quadratic programming, polynomial-time methods for nonlinear convex programming, and efficient computational methods for control problems and variational inequalities. A background in linear algebra and mathematical programming is necessary to understand the book. The detailed proofs and lack of "numerical examples" might suggest that the book is of limited value to the reader interested in the practical aspects of convex optimization, but nothing could be further from the truth. An entire chapter is devoted to potential reduction methods precisely because of their great efficiency in practice.
Book Synopsis Interior Point Techniques in Optimization by : B. Jansen
Download or read book Interior Point Techniques in Optimization written by B. Jansen and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: Operations research and mathematical programming would not be as advanced today without the many advances in interior point methods during the last decade. These methods can now solve very efficiently and robustly large scale linear, nonlinear and combinatorial optimization problems that arise in various practical applications. The main ideas underlying interior point methods have influenced virtually all areas of mathematical programming including: analyzing and solving linear and nonlinear programming problems, sensitivity analysis, complexity analysis, the analysis of Newton's method, decomposition methods, polynomial approximation for combinatorial problems etc. This book covers the implications of interior techniques for the entire field of mathematical programming, bringing together many results in a uniform and coherent way. For the topics mentioned above the book provides theoretical as well as computational results, explains the intuition behind the main ideas, gives examples as well as proofs, and contains an extensive up-to-date bibliography. Audience: The book is intended for students, researchers and practitioners with a background in operations research, mathematics, mathematical programming, or statistics.
Book Synopsis Full-Newton Step Interior-point Methods for Conic Optimization by : Hossein Mansouri
Download or read book Full-Newton Step Interior-point Methods for Conic Optimization written by Hossein Mansouri and published by . This book was released on 2008 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Modeling and Optimization: Theory and Applications by : Luis F. Zuluaga
Download or read book Modeling and Optimization: Theory and Applications written by Luis F. Zuluaga and published by Springer Science & Business Media. This book was released on 2013-11-22 with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains a selection of contributions that were presented at the Modeling and Optimization: Theory and Applications Conference (MOPTA) held at Lehigh University in Bethlehem, Pennsylvania, USA on July 30-August 1, 2012. The conference brought together a diverse group of researchers and practitioners, working on both theoretical and practical aspects of continuous or discrete optimization. Topics presented included algorithms for solving convex, network, mixed-integer, nonlinear, and global optimization problems, and addressed the application of optimization techniques in finance, logistics, health, and other important fields. The contributions contained in this volume represent a sample of these topics and applications and illustrate the broad diversity of ideas discussed at the meeting.
Book Synopsis Chordal Sparsity in Interior-point Methods for Conic Optimization by : Marin Skovgaard Andersen
Download or read book Chordal Sparsity in Interior-point Methods for Conic Optimization written by Marin Skovgaard Andersen and published by . This book was released on 2011 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Implementation of Interior Point Methods for Second Order Conic Optimization by : Bixiang Wang
Download or read book Implementation of Interior Point Methods for Second Order Conic Optimization written by Bixiang Wang and published by . This book was released on 2003 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Lectures on Modern Convex Optimization by : Aharon Ben-Tal
Download or read book Lectures on Modern Convex Optimization written by Aharon Ben-Tal and published by SIAM. This book was released on 2001-01-01 with total page 500 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Fast and Parallelizable Numerical Algorithms for Large Scale Conic Optimization Problems by : Muhammad Adil
Download or read book Fast and Parallelizable Numerical Algorithms for Large Scale Conic Optimization Problems written by Muhammad Adil and published by . This book was released on 2022 with total page 77 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many real world problems from various application areas such as engineering, finance and operation research can be cast as optimization problems. Generally, the goal is to optimize an objective function under a set of constraints. Traditionally, convex optimization problems are solved by an interior point method (IPM). Interior point methods proved to achieve high accuracy for moderate size problems. However, the computation cost of iterations of these iterative algorithms grows non-linearly with the dimension of the problem. Although interior-point methods are robust and theoretically sound, they do not scale well for very large conic optimization programs. Computational cost, memory issues, and incompatibility with distributed platforms are among the major impediments for interior point methods in solving large-scale and practical conic optimization programs. The rapid growth of problem size in application areas such as power systems, finance, signal processing and machine learning motivated researchers to develop computationally efficient optimization solvers. In recent years, first orders methods have received a particular attention for solving large convex optimization problems. Various optimization solvers based on first order numerical algorithms have been developed in the past decade. Although first order methods provide low accuracy solutions, but inexpensive iterations and low computational cost makes them attractive mathematical tools for handling large-scale problems. One of the major shortcomings of first order methods to achieve a higher accuracy is their slow tail convergence behavior. The first part of this work is an attempt to remedy the problem of slow convergence for first-order numerical algorithms by proposing an adaptive conditioning heuristic policy. First, a parallelizable numerical algorithm is proposed that is capable of dealing with large-scale conic programs on distributed platforms such as graphics processing unit (GPU) with orders-of-magnitude time improvement. The mathematical proof for global convergence of proposed numerical algorithm is provided. In the past decade, several preconditioning methods have been applied to improve the condition number and convergence of first order methods. Diagonal preconditioning and matrix equilibration techniques are most commonly used for this purpose. In contrary to the existing techniques, in this work, it is argued that the condition number of the problem data is not a reliable predictor of convergence speed. In light of this observation, an adaptive conditioning heuristic is proposed which enables higher accuracy compared to other first-order numerical algorithms. A wide range of experiments are conducted on a variety of large-scale linear programming and second-order cone programming problems to demonstrate the scalability and computational advantages of the proposed algorithm compared to commercial and open-source solvers. Solving the linear system is probably the most computationally expensive part in first order methods. The existing methods rely on direct and indirect methods for solving the linear systems of equations. Direct methods rely on factorization techniques which usually destroy the sparsity structure of original sparse problems and hence become computationally prohibitive. Alternatively, indirect methods are iterative and various preconditioning variants of indirect or iterative methods have been studied in the literature to improve accuracy, but again the preconditioners do not necessarily retain the sparsity patterns of original problems. In the second part of this work, a matrix-free first order approach is proposed for solving large-scales parse conic optimization problems. This method is based on an easy-to-compute decomposition of large sparse matrices into two factors. The proposed numerical algorithm is based on matrix-free decomposition and alternating direction method of multipliers. The iterations of the designed algorithm are computationally cheap, highly parallelizable and enjoy closed form solutions. The algorithm can easily be implemented on distributed platforms such as graphics processing units with orders of-magnitude time improvements. The performance of the proposed algorithm is demonstrated on a variety of conic problems and the performance gain is compared with competing first-order solvers.
Book Synopsis Handbook on Semidefinite, Conic and Polynomial Optimization by : Miguel F. Anjos
Download or read book Handbook on Semidefinite, Conic and Polynomial Optimization written by Miguel F. Anjos and published by Springer Science & Business Media. This book was released on 2011-11-19 with total page 955 pages. Available in PDF, EPUB and Kindle. Book excerpt: Semidefinite and conic optimization is a major and thriving research area within the optimization community. Although semidefinite optimization has been studied (under different names) since at least the 1940s, its importance grew immensely during the 1990s after polynomial-time interior-point methods for linear optimization were extended to solve semidefinite optimization problems. Since the beginning of the 21st century, not only has research into semidefinite and conic optimization continued unabated, but also a fruitful interaction has developed with algebraic geometry through the close connections between semidefinite matrices and polynomial optimization. This has brought about important new results and led to an even higher level of research activity. This Handbook on Semidefinite, Conic and Polynomial Optimization provides the reader with a snapshot of the state-of-the-art in the growing and mutually enriching areas of semidefinite optimization, conic optimization, and polynomial optimization. It contains a compendium of the recent research activity that has taken place in these thrilling areas, and will appeal to doctoral students, young graduates, and experienced researchers alike. The Handbook’s thirty-one chapters are organized into four parts: Theory, covering significant theoretical developments as well as the interactions between conic optimization and polynomial optimization; Algorithms, documenting the directions of current algorithmic development; Software, providing an overview of the state-of-the-art; Applications, dealing with the application areas where semidefinite and conic optimization has made a significant impact in recent years.
Book Synopsis Convex Optimization by : Stephen P. Boyd
Download or read book Convex Optimization written by Stephen P. Boyd and published by Cambridge University Press. This book was released on 2004-03-08 with total page 744 pages. Available in PDF, EPUB and Kindle. Book excerpt: Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.
Book Synopsis Handbook on Semidefinite, Conic and Polynomial Optimization by : Miguel F. Anjos
Download or read book Handbook on Semidefinite, Conic and Polynomial Optimization written by Miguel F. Anjos and published by Springer. This book was released on 2011-11-18 with total page 957 pages. Available in PDF, EPUB and Kindle. Book excerpt: Semidefinite and conic optimization is a major and thriving research area within the optimization community. Although semidefinite optimization has been studied (under different names) since at least the 1940s, its importance grew immensely during the 1990s after polynomial-time interior-point methods for linear optimization were extended to solve semidefinite optimization problems. Since the beginning of the 21st century, not only has research into semidefinite and conic optimization continued unabated, but also a fruitful interaction has developed with algebraic geometry through the close connections between semidefinite matrices and polynomial optimization. This has brought about important new results and led to an even higher level of research activity. This Handbook on Semidefinite, Conic and Polynomial Optimization provides the reader with a snapshot of the state-of-the-art in the growing and mutually enriching areas of semidefinite optimization, conic optimization, and polynomial optimization. It contains a compendium of the recent research activity that has taken place in these thrilling areas, and will appeal to doctoral students, young graduates, and experienced researchers alike. The Handbook’s thirty-one chapters are organized into four parts: Theory, covering significant theoretical developments as well as the interactions between conic optimization and polynomial optimization; Algorithms, documenting the directions of current algorithmic development; Software, providing an overview of the state-of-the-art; Applications, dealing with the application areas where semidefinite and conic optimization has made a significant impact in recent years.
Book Synopsis Local Quadratic Convergence of Polynomial-time Interior-point Methods for Conic Optimization Problems by : Yu Nesterov
Download or read book Local Quadratic Convergence of Polynomial-time Interior-point Methods for Conic Optimization Problems written by Yu Nesterov and published by . This book was released on 2009 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Generalization of Primal-dual Interior-point Methods to Convex Optimization Problems in Conic Form by : Tunçel, Levent
Download or read book Generalization of Primal-dual Interior-point Methods to Convex Optimization Problems in Conic Form written by Tunçel, Levent and published by . This book was released on 1999 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Interior Point Methods and Simulated Annealing for Nonsymmetric Conic Optimization by : Riley Badenbroek
Download or read book Interior Point Methods and Simulated Annealing for Nonsymmetric Conic Optimization written by Riley Badenbroek and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Mixed-integer Convex Optimization by : Miles C. Lubin
Download or read book Mixed-integer Convex Optimization written by Miles C. Lubin and published by . This book was released on 2017 with total page 143 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we study mixed-integer convex optimization, or mixed-integer convex programming (MICP), the class of optimization problems where one seeks to minimize a convex objective function subject to convex constraints and integrality restrictions on a subset of the variables. We focus on two broad and complementary questions on MICP. The first question we address is, "what are efficient methods for solving MICP problems?" The methodology we develop is based on outer approximation, which allows us, for example, to reduce MICP to a sequence of mixed-integer linear programming (MILP) problems. By viewing MICP from the conic perspective of modern convex optimization as defined by Ben-Tal and Nemirovski, we obtain significant computational advances over the state of the art, e.g., by automating extended formulations by using disciplined convex programming. We develop the first finite-time outer approximation methods for problems in general mixed-integer conic form (which includes mixed-integer second-order-cone programming and mixed-integer semidefinite programming) and implement them in an open-source solver, Pajarito, obtaining competitive performance with the state of the art. The second question we address is, "which nonconvex constraints can be modeled with MICP?" This question is important for understanding both the modeling power gained in generalizing from MILP to MICP and the potential applicability of MICP to nonconvex optimization problems that may not be naturally represented with integer variables. Among our contributions, we completely characterize the case where the number of integer assignments is bounded (e.g., mixed-binary), and to address the more general case we develop the concept of "rationally unbounded" convex sets. We show that under this natural restriction, the projections of MICP feasible sets are well behaved and can be completely characterized in some settings.
