Mixed Integer Nonlinear Programming
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Mixed Integer Nonlinear Programming
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Author : Jon Lee
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-12-02
Mixed Integer Nonlinear Programming written by Jon Lee and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-12-02 with Mathematics categories.
Many engineering, operations, and scientific applications include a mixture of discrete and continuous decision variables and nonlinear relationships involving the decision variables that have a pronounced effect on the set of feasible and optimal solutions. Mixed-integer nonlinear programming (MINLP) problems combine the numerical difficulties of handling nonlinear functions with the challenge of optimizing in the context of nonconvex functions and discrete variables. MINLP is one of the most flexible modeling paradigms available for optimization; but because its scope is so broad, in the most general cases it is hopelessly intractable. Nonetheless, an expanding body of researchers and practitioners — including chemical engineers, operations researchers, industrial engineers, mechanical engineers, economists, statisticians, computer scientists, operations managers, and mathematical programmers — are interested in solving large-scale MINLP instances.
Relaxation And Decomposition Methods For Mixed Integer Nonlinear Programming
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Author : Ivo Nowak
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-08-15
Relaxation And Decomposition Methods For Mixed Integer Nonlinear Programming written by Ivo Nowak and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-08-15 with Computers categories.
Nonlinearoptimizationproblemscontainingbothcontinuousanddiscretevariables are called mixed integer nonlinear programs (MINLP). Such problems arise in many ?elds, such as process industry, engineering design, communications, and ?nance. There is currently a huge gap between MINLP and mixed integer linear programming(MIP) solvertechnology.With a modernstate-of-the-artMIP solver itispossibletosolvemodelswithmillionsofvariablesandconstraints,whereasthe dimensionofsolvableMINLPsisoftenlimitedbyanumberthatissmallerbythree or four orders of magnitude. It is theoretically possible to approximate a general MINLP by a MIP with arbitrary precision. However, good MIP approximations are usually much larger than the original problem. Moreover, the approximation of nonlinear functions by piecewise linear functions can be di?cult and ti- consuming. In this book relaxation and decomposition methods for solving nonconvex structured MINLPs are proposed. In particular, a generic branch-cut-and-price (BCP) framework for MINLP is presented. BCP is the underlying concept in almost all modern MIP solvers. Providing a powerful decomposition framework for both sequential and parallel solvers, it made the success of the current MIP technology possible. So far generic BCP frameworks have been developed only for MIP, for example,COIN/BCP (IBM, 2003) andABACUS (OREAS GmbH, 1999). In order to generalize MIP-BCP to MINLP-BCP, the following points have to be taken into account: • A given (sparse) MINLP is reformulated as a block-separable program with linear coupling constraints.The block structure makes it possible to generate Lagrangian cuts and to apply Lagrangian heuristics. • In order to facilitate the generation of polyhedral relaxations, nonlinear c- vex relaxations are constructed. • The MINLP separation and pricing subproblems for generating cuts and columns are solved with specialized MINLP solvers.
Convexification And Global Optimization In Continuous And Mixed Integer Nonlinear Programming
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Author : Mohit Tawarmalani
language : en
Publisher: Springer Science & Business Media
Release Date : 2002-10-31
Convexification And Global Optimization In Continuous And Mixed Integer Nonlinear Programming written by Mohit Tawarmalani and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002-10-31 with Business & Economics categories.
This book provides an insightful and comprehensive treatment of convexification and global optimization of continuous and mixed-integer nonlinear programs. Developed for students, researchers, and practitioners, the book covers theory, algorithms, software, and applications. This thought-provoking book: -develops a powerful and widely-applicable framework for constructing closed-form expressions of convex envelopes of nonlinear functions; -presents a systematic treatment of branch-and-bound, while providing acceleration mechanisms and enhancements; -unifies ideas at the interface between operations research and computer science, devising efficient algorithmic implementation for global optimization; offers students, modelers, and algorithm developers a rich collection of models, applications, and numerical examples; -elucidates through geometric interpretations the concepts discussed throughout the book; -shows how optimization theory can lead to breakthroughs in diverse application areas, including molecular design, process and product design, facility location, and supply chain design and operation; -demonstrates that the BARON software developed by the authors can solve global optimization problems heretofore considered intractable, in an entirely automated manner on a personal computer. Audience: This book will be of interest to researchers in operations research, management science, applied mathematics, computer science, computational chemistry, and all branches of engineering. In addition, the book can be used in graduate level courses in nonlinear optimization, integer programming, global optimization, convex analysis, applied mathematics, and engineering design.
Mixed Integer Nonlinear Programming
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Author :
language : en
Publisher: Springer
Release Date : 2011-12-02
Mixed Integer Nonlinear Programming written by and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-12-02 with categories.
Mixed Integer Nonlinear Programming Minlp
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Author : Pierre Bonami
language : en
Publisher:
Release Date : 2012
Mixed Integer Nonlinear Programming Minlp written by Pierre Bonami and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with categories.
Exact And Fast Algorithms For Mixed Integer Nonlinear Programming
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Author : Ambros Gleixner
language : en
Publisher:
Release Date : 2015
Exact And Fast Algorithms For Mixed Integer Nonlinear Programming written by Ambros Gleixner and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with Dissertations categories.
The discipline of mixed-integer nonlinear programming (MINLP) deals with finite-dimensional optimization problems featuring both discrete choices and nonlinear functions. By this combination, it facilitates more accurate models of real-world systems than possible with purely continuous or purely linear models alone. This book presents new methods that improve the numerical reliability and the computational performance of global MINLP solvers. The author addresses numerical accuracy directly at the linear programming level by means of LP iterative refinement: a new algorithm to solve linear programs to arbitrarily high levels of precision. The computational performance of LP-based MINLP solvers is enhanced by efficient methods to execute and approximate optimization-based bound tightening and by new branching rules that exploit the presence of nonlinear integer variables, i.e., variables both contained in nonlinear terms and required to be integral. The new algorithms help to solve problems which could not be solved before, either due to their numerical complexity or because of limited computing resources.
Relaxation And Decomposition Methods For Mixed Integer Nonlinear Programming
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Author : Ivo Nowak
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-03-28
Relaxation And Decomposition Methods For Mixed Integer Nonlinear Programming written by Ivo Nowak and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-03-28 with Computers categories.
Nonlinearoptimizationproblemscontainingbothcontinuousanddiscretevariables are called mixed integer nonlinear programs (MINLP). Such problems arise in many ?elds, such as process industry, engineering design, communications, and ?nance. There is currently a huge gap between MINLP and mixed integer linear programming(MIP) solvertechnology.With a modernstate-of-the-artMIP solver itispossibletosolvemodelswithmillionsofvariablesandconstraints,whereasthe dimensionofsolvableMINLPsisoftenlimitedbyanumberthatissmallerbythree or four orders of magnitude. It is theoretically possible to approximate a general MINLP by a MIP with arbitrary precision. However, good MIP approximations are usually much larger than the original problem. Moreover, the approximation of nonlinear functions by piecewise linear functions can be di?cult and ti- consuming. In this book relaxation and decomposition methods for solving nonconvex structured MINLPs are proposed. In particular, a generic branch-cut-and-price (BCP) framework for MINLP is presented. BCP is the underlying concept in almost all modern MIP solvers. Providing a powerful decomposition framework for both sequential and parallel solvers, it made the success of the current MIP technology possible. So far generic BCP frameworks have been developed only for MIP, for example,COIN/BCP (IBM, 2003) andABACUS (OREAS GmbH, 1999). In order to generalize MIP-BCP to MINLP-BCP, the following points have to be taken into account: • A given (sparse) MINLP is reformulated as a block-separable program with linear coupling constraints.The block structure makes it possible to generate Lagrangian cuts and to apply Lagrangian heuristics. • In order to facilitate the generation of polyhedral relaxations, nonlinear c- vex relaxations are constructed. • The MINLP separation and pricing subproblems for generating cuts and columns are solved with specialized MINLP solvers.
Topics In Mixed Integer Nonlinear Programming
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Author : Kumar Abhishek
language : en
Publisher:
Release Date : 2008
Topics In Mixed Integer Nonlinear Programming written by Kumar Abhishek and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with categories.
A Mixed Integer Nonlinear Program (MINLP) is the problem of minimizing a nonlinear function subject to nonlinear constraints and integrality restrictions on some or all of the decision variables. MINLP is one of the most fundamentally challenging problems in optimization, its difficulty arising from the combination of functional nonlinearities and nonconvexity induced by integrality restrictions. Nevertheless, MINLP is a tremendously important problem, with many important engineering and operations applications being formulated as MINLP models. The main focus of this thesis is on convex MINLPs, where the objective function is convex, and the points that satisfy the nonlinear constraints form a convex set. We aim to develop an efficient framework and set of methods for solving convex MINLPs, as well as to demonstrate the advantages of modeling applications as MINLPs.
Nonlinear And Mixed Integer Optimization
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Author : Christodoulos A. Floudas
language : en
Publisher: Oxford University Press
Release Date : 1995-10-05
Nonlinear And Mixed Integer Optimization written by Christodoulos A. Floudas and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995-10-05 with Science categories.
Filling a void in chemical engineering and optimization literature, this book presents the theory and methods for nonlinear and mixed-integer optimization, and their applications in the important area of process synthesis. Other topics include modeling issues in process synthesis, and optimization-based approaches in the synthesis of heat recovery systems, distillation-based systems, and reactor-based systems. The basics of convex analysis and nonlinear optimization are also covered and the elementary concepts of mixed-integer linear optimization are introduced. All chapters have several illustrations and geometrical interpretations of the material as well as suggested problems. Nonlinear and Mixed-Integer Optimization will prove to be an invaluable source--either as a textbook or a reference--for researchers and graduate students interested in continuous and discrete nonlinear optimization issues in engineering design, process synthesis, process operations, applied mathematics, operations research, industrial management, and systems engineering.
Integer Linear Programming Methods To Solve Some Mixed Integer Nonlinear Programs
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Author :
language : en
Publisher:
Release Date : 2016
Integer Linear Programming Methods To Solve Some Mixed Integer Nonlinear Programs written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016 with categories.
Mixed-Integer Nonlinear Programs (MINLPs) are used to model many applied problems. However, finding a global optimal solution to such problems is difficult due to the combined presence of integer variables and nonlinear functions. If a MINLP has special structure, one can exploit the structure and develop a specialized approach that has potential to solve the problem more efficiently than a general-purpose solver. In this thesis, we study MINLPs with three different structures. We first consider a class of MINLPs in which the set defined by the nonlinear constraint functions is convex, but the nonlinear functions may not be convex. Existing solution approaches for convex MINLPs are not guaranteed to find the global optimal solution of such problems. We propose two linearization-based methods to solve this class of problems. We prove the method finds an optimal solution using a finite number of nodes and cuts. Computational results show our methods are as efficient as the existing methods while solving this more general problem class. Second, we study a network design problem in which an operator wants to open facilities and determine their size to satisfy demand of customers. Customers face congestion at the facilities they send their demand to, which depends on total usage of facility. We model this problem as a Mixed-Integer Bilevel Program (MIBP) which can be reformulated as a nonconvex MINLP, but is difficult to solve by general purpose solvers. Hence, we propose a Lagrangian relaxation approach which finds a candidate feasible solution along with a lower bound that can be used to validate the solution quality. We find that the method can efficiently find provably high quality solutions even for large instances. Finally, we study optimization problems having a nonconvex monotone objective function that is computationally expensive to evaluate. Such problems can be solved using cuts known as "integer L-shaped cuts.'' We exploit the monotonicity structure to derive a stronger version of these cuts, and explore the use of mixing inequalities to further tighten the relaxation. Computational studies show that the strengthened cuts yield improved computational performance, but the impact of mixing inequalities is marginal.