Jeff Edmonds · Cambridge University Press eBooks · 2024
Understand algorithms and their design with this revised student-friendly textbook. Unlike other algorithms books, this one is approachable, the methods it explains are straightforward, and the insights it provides are numerous and valuable. Without grinding through lots of formal proof, students will benefit from step-by-step methods for developing algorithms, expert guidance on common pitfalls, and an appreciation of the bigger picture. Revised and updated, this second edition includes a new chapter on machine learning algorithms, and concise key concept summaries at the end of each part for quick reference. Also new to this edition are more than 150 new exercises: selected solutions are included to let students check their progress, while a full solutions manual is available online for instructors. No other text explains complex topics such as loop invariants as clearly, helping students to think abstractly and preparing them for creating their own innovative ways to solve problems.
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Alexander Shen · Springer undergraduate texts in mathematics and technology · 2010
Cited by 24
Bernhard Körte, Jens Vygen · Algorithms and combinatorics · 2000
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Ruslan Shavkyatievich Uzbekov, Vitalii Vyacheslavovich Chelnokov · Программные системы и вычислительные методы · 2025
The subject of the research is nonlinear programming methods used to solve optimization problems in which the objective function, constraints, or both are nonlinear in nature. Unlike linear programming, nonlinear problems have a more complex structure, which increases their versatility, but at the same time complicates the analysis and search for solutions. The paper considers two key algorithms: Sequential quadratic programming (SQP) is a method based on the iterative solution of quadratic subproblems approximating the original nonlinear problem and is especially useful for problems where the objective function and constraints have a nonlinear relationship. The Interior Point Method is an efficient algorithm for problems with constraints, characterized by high convergence rate and versatility. In the course of the research, theoretical methods were used, including: Search and analysis of literature on nonlinear programming methods. Comparative analysis of SQP algorithms and internal points. The output of mathematical equations describing the optimization of chemical and technological processes using nonlinear programming. Special attention is paid to software implementations such as Interior Point OPTimizer, which uses the interior point method. The main conclusions of the study are: Nonlinear programming is a powerful tool for solving complex optimization problems, especially in chemical technology. The choice of method depends on the problem structure, the type of constraints, and the required accuracy: SQP is effective for problems with nonlinear objective functions and constraints, but requires precise definition of active constraints. The interior point method avoids this problem by using barrier functions, which makes it more stable. The main disadvantage of SQP is the need for complete identification of active constraints, whereas the interior point method demonstrates better convergence in complex cases. The novelty of the work lies in the comprehensive analysis and comparison of SQP methods and internal points in relation to the optimization of chemical and technological processes. The key features and limitations of each method are systematized. Their effectiveness is evaluated using the example of problems with nonlinear constraints. The practical aspects of implementing algorithms in modern software packages are considered. The results of the work can be useful for choosing the optimal method of nonlinear optimization in engineering and technological applications.
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Julie Darling, Diane Cools · 2024
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E.M.T. Hendrix, Boglárka G.-Tóth · Springer optimization and its applications · 2010
Cited by 1
Michael H. Veatch · 2021
Solving convex programs is a broad and more advanced topic. This final chapter provides a brief introduction and considers the convex optimization problem. It begins by presenting some motivating examples. The chapter describes the easier special case of separable programs, making a connection with linear programming approximations. It discusses the unconstrained convex programs. Another important class of problems, quadratic programs, is also introduced. For unconstrained problems, Newton’s method is equivalent to optimizing a quadratic function at each iteration. This relationship can be extended to problems with equality constraints. The chapter also presents the main algorithm: a primal-dual interior point method. While there are many algorithms for constrained convex programs, the primal-dual interior point algorithm performs well on many types of problems and has a good theoretical rate of convergence.
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Klimeš, Lubomír · Brno University of Technology Digital Library (Brno University of Technology) · 2010
Stochastické programování a optimalizace jsou mocnými nástroji pro řešení široké škály inženýrských problémů zahrnujících neurčitost. Algoritmus progressive hedging je efektivní dekompoziční metoda určená pro řešení scénářových stochastických úloh. Z důvodu vertikální dekompozice je možno tento algoritmus implementovat paralelně, čímž lze významně ušetřit výpočetní čas a ostatní prostředky. Teoretická část této diplomové práce se zabývá matematickým a zejména pak stochastickým programováním a detailně popisuje algoritmus progressive hedging. V praktické části je navržena a diskutována původní paralelní implementace algoritmu progressive hedging, která je pak otestována na jednoduchých úlohách. Dále je uvedená paralelní implementace použita pro řešení inženýrského problému plynulého odlévání ocelové bramy a na závěr jsou získané výsledky zhodnoceny.
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Papers Admin, UJ Exam · Figshare · 2019
First semester
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Bernhard Körte, Jens Vygen · Algorithms and combinatorics · 2002
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Donald Goldfarb · 1983
Abstract : Several algorithms for solving problems in linear, quadratic, and nonlinear programming, network flows, and facilities location were developed and analyzed. Results include: (1) The analysis of the computational complexity of the problem of determining an optimally sparse representation of the null space of a matrix, and the development of worst-case bounds for the shadow-vertex simplex algorithm and several heuristics for distance constrained discrete facility location problems and conditional covering problems; (2) The development of efficient algorithms for dense and sparse assignment problems, strictly convex quadratic programming problems, and a nonlinear programming problem that arises when maximizing a correlation coefficient subject to linear constraints; and (3) The application of iterative methods to large sparse equality-constraind quadratic programs and the development of multiple constraint deletion strategies for active-set algorithms for linearly consrained nonlinear programming problems. (Author)
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J. Mutuku · UNAM Scholarly Repository (University of Namibia) · 2010
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Kružík, Jakub · DSpace VŠB-TUO (VŠB-TUO) · 2024
The main objective of this thesis is to present improvements in quadratic programming algorithms. These improvements speed up the solution of quadratic programming problems, with or without constraints, which are key in various fields, including, but not limited to, economics, engineering, and machine learning. The main improvements are for solving box-constrained quadratic programming problems. The MPRGP (Modified Proportioning with Reduced Gradient Projections) algorithm is analyzed and, based on this analysis, improved. The analysis reveals that the expansion of the active set through the reduced gradient projections is the most expensive part of the algorithm. Our modification of the expansion step, using the projected conjugate gradient, proves significantly superior to the original algorithm in most cases. The presented fallback steps and criteria can be used to guarantee convergence or even ensure that the convergence rate is at least as good as that of the standard MPRGP algorithm. Another presented modification is to use the Spectral Projected Gradient (SPG) method as the expansion step. This proves to be extremely effective in certain cases, but a little less so in others. Numerical experiments showcasing the effectiveness of the proposed methods, as well as a comparison with the SPG method, are presented on a number of benchmarks. Another improvement of MPRGP is in preconditioning, which is not straightforward to implement when the problem is constrained. Our improvement is to cheaply approximate the preconditioning in face, which must be recomputed every time the active set changes, with a preconditioner that is set up only once. The numerical experiments show speedups between $5.1$ and $13.4$ compared to the unpreconditioned algorithm. The previous expansion step modification is a key ingredient for an effective preconditioned algorithm. An error analysis comparing the standard and the approximate variant of the preconditioning in face is provided to complement the numerical experiments. Further improvements include the scalability of FETI (Finite Element Tearing and Interconnecting) domain decomposition methods, which allow us to solve problems with more than one billion unknowns using tens of thousands of computational cores on large supercomputers. Most of the presented algorithms are implemented in the PERMON library, of which the author of this thesis was the maintainer and the main contributor throughout their doctoral studies. The main aim of PERMON is to provide a scalable framework for the solution of large-scale quadratic problems. Another software contribution was the multilevel deflation preconditioner, PCDEFLATION, in the PETSc library for scientific computing.
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János Mayer · 2022
A computationally oriented comparison of solution algorithms for two stage and jointly chance constrained stochastic linear programming problems, this is the first book to present comparative computational results with several major stochastic programming solution approaches. The following methods are considered: regularized decomposition, stochastic decomposition and successive discrete approximation methods for two stage problems; cutting plane methods, and a reduced gradient method for jointly chance constrained problems. The first part of the book introduces the algorithms, including a unified approach to decomposition methods and their regularized counterparts. The second part addresses computer implementation of the methods, describes a testing environment based on a model management system, and presents comparative computational results with the various algorithms. Emphasis is on the computational behavior of the algorithms.
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Jeff Kennington, Richard V. Helgason · CERN Document Server (European Organization for Nuclear Research) · 1980
Cited by 493
John H. Seinfeld, Leon Lapidus · Industrial & Engineering Chemistry Process Design and Development · 1968
ADVERTISEMENT RETURN TO ISSUEPREVArticleDiscrete Dynamic Programming AlgorithmsJ. H. Seinfeld and Leon LapidusCite this: Ind. Eng. Chem. Process Des. Dev. 1968, 7, 3, 479–480Publication Date (Print):July 1, 1968Publication History Published online1 May 2002Published inissue 1 July 1968https://pubs.acs.org/doi/10.1021/i260027a028https://doi.org/10.1021/i260027a028research-articleACS PublicationsRequest reuse permissionsArticle Views46Altmetric-Citations-LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access options Get e-Alerts
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Martin Schönert, Hans Ulrich Besche, Thomas Breuer et al. · 1994
GAP can be copied and distributed freely for any non-commercial purpose. If you copy GAP for somebody else, you may ask this person for refund of your expenses. This should cover cost of media, copying and shipping. You are not allowed to ask for more than this. In any case you must give a copy of this copyright notice along with the program. If you obtain GAP please send us a short notice to that effect, e.g., an e-mail message to the address gap@samson.math.rwth-aachen.de, containing your full name and address. This allows us to keep track of the number of GAP users. If you publish a mathematical result that was partly obtained using GAP, please cite GAP, just as you would cite another paper that you used. Also we would appreciate it if you could inform us about such a paper. You are permitted to modify and redistribute GAP, but you are not allowed to restrict further redistribution. That is to say proprietary modifications will not be allowed. We want all versions of GAP to remain free. If you modify any part of GAP and redistribute it, you must supply a README document. This should specify what modifications you made in which files. We do not want to take
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Kedar N. Swadi, Walid Taha, Oleg Kiselyov · 2005
Applications of dynamic programming (DP) algorithms are numerous, and include genetic engineering and operations research problems. At a high level, DP algorithms are specified as a system of recursive equations implemented using memoization. The recursive nature of these equations suggests that they can be written naturally in a functional language. However, the requirement for memoization poses a subtle challenge: memoization can be implemented using monads, but a systematic treatment introduces several layers of abstraction that can have a prohibitive runtime overhead. Inspired by other researchers' experience with automatic specialization (partial evaluation), this paper investigates the feasibility of explicitly staging DP algorithms in the functional setting. We find that the key challenge is code duplication (which is automatically handled by partial evaluators), and show that a key source of code duplication can be isolated and addressed once and for all. The result is a simple combinator library. We use this library to implement several standard DP algorithms including ones in standard algorithm textbooks (e.g.
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Papers Admin, UJ Exam · University of Johannesburg · 2019
First semester
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Papers Admin, UJ Exam · University of Johannesburg · 2019
First semester
Cited by 0Open access