COIN-OR: Revving up the Engine
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Nonlinear Optimization Using the Generalized Reduced Gradient Method Revue Française D’Automatique, Informatique, Recherche Opé- Rationnelle
REVUE FRANÇAISE D’AUTOMATIQUE, INFORMATIQUE, RECHERCHE OPÉRATIONNELLE.RECHERCHE OPÉRATIONNELLE LEON S. LASDON RICHARD L. FOX MARGERY W. RATNER Nonlinear optimization using the generalized reduced gradient method Revue française d’automatique, informatique, recherche opé- rationnelle. Recherche opérationnelle, tome 8, no V3 (1974), p. 73-103 <http://www.numdam.org/item?id=RO_1974__8_3_73_0> © AFCET, 1974, tous droits réservés. L’accès aux archives de la revue « Revue française d’automatique, in- formatique, recherche opérationnelle. Recherche opérationnelle » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ R.A.LR.O. (8* année, novembre 1974, V-3, p. 73 à 104) NONLINEAR OPTIMIZATION USING THE-GENERALIZED REDUCED GRADIENT METHOD (*) by Léon S. LÀSDON, Richard L. Fox and Margery W. RATNER Abstract. — This paper describes the principles and logic o f a System of computer programs for solving nonlinear optimization problems using a Generalized Reduced Gradient Algorithm, The work is based on earlier work of Âbadie (2). Since this paper was written, many changes have been made in the logic, and significant computational expérience has been obtained* We hope to report on this in a later paper. 1. INTRODUCTION Generalized Reduced Gradient methods are algorithms for solving non- linear programs of gênerai structure. This paper discusses the basic principles of GRG, and constructs a spécifie GRG algorithm. The logic of a computer program implementing this algorithm is presented by means of flow charts and discussion. -
Combinatorial Structures in Nonlinear Programming
Combinatorial Structures in Nonlinear Programming Stefan Scholtes¤ April 2002 Abstract Non-smoothness and non-convexity in optimization problems often arise because a combinatorial structure is imposed on smooth or convex data. The combinatorial aspect can be explicit, e.g. through the use of ”max”, ”min”, or ”if” statements in a model, or implicit as in the case of bilevel optimization where the combinatorial structure arises from the possible choices of active constraints in the lower level problem. In analyzing such problems, it is desirable to decouple the combinatorial from the nonlinear aspect and deal with them separately. This paper suggests a problem formulation which explicitly decouples the two aspects. We show that such combinatorial nonlinear programs, despite their inherent non-convexity, allow for a convex first order local optimality condition which is generic and tight. The stationarity condition can be phrased in terms of Lagrange multipliers which allows an extension of the popular sequential quadratic programming (SQP) approach to solve these problems. We show that the favorable local convergence properties of SQP are retained in this setting. The computational effectiveness of the method depends on our ability to solve the subproblems efficiently which, in turn, depends on the representation of the governing combinatorial structure. We illustrate the potential of the approach by applying it to optimization problems with max-min constraints which arise, for example, in robust optimization. 1 Introduction Nonlinear programming is nowadays regarded as a mature field. A combination of important algorithmic developments and increased computing power over the past decades have advanced the field to a stage where the majority of practical prob- lems can be solved efficiently by commercial software. -
Hyperheuristics in Logistics Kassem Danach
Hyperheuristics in Logistics Kassem Danach To cite this version: Kassem Danach. Hyperheuristics in Logistics. Combinatorics [math.CO]. Ecole Centrale de Lille, 2016. English. NNT : 2016ECLI0025. tel-01485160 HAL Id: tel-01485160 https://tel.archives-ouvertes.fr/tel-01485160 Submitted on 8 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. No d’ordre: 315 Centrale Lille THÈSE présentée en vue d’obtenir le grade de DOCTEUR en Automatique, Génie Informatique, Traitement du Signal et des Images par Kassem Danach DOCTORAT DELIVRE PAR CENTRALE LILLE Hyperheuristiques pour des problèmes d’optimisation en logistique Hyperheuristics in Logistics Soutenue le 21 decembre 2016 devant le jury d’examen: President: Pr. Laetitia Jourdan Université de Lille 1, France Rapporteurs: Pr. Adnan Yassine Université du Havre, France Dr. Reza Abdi University of Bradford, United Kingdom Examinateurs: Pr. Saïd Hanafi Université de Valenciennes, France Dr. Abbas Tarhini Lebanese American University, Lebanon Dr. Rahimeh Neamatian Monemin University Road, United Kingdom Directeur de thèse: Pr. Frédéric Semet Ecole Centrale de Lille, France Co-encadrant: Dr. Shahin Gelareh Université de l’ Artois, France Invited Professor: Dr. Wissam Khalil Université Libanais, Lebanon Thèse préparée dans le Laboratoire CRYStAL École Doctorale SPI 072 (EC Lille) 2 Acknowledgements Firstly, I would like to express my sincere gratitude to my advisor Prof. -
Nonlinear Integer Programming ∗
Nonlinear Integer Programming ∗ Raymond Hemmecke, Matthias Koppe,¨ Jon Lee and Robert Weismantel Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Nu- merous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considera- tions and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research. Raymond Hemmecke Otto-von-Guericke-Universitat¨ Magdeburg, FMA/IMO, Universitatsplatz¨ 2, 39106 Magdeburg, Germany, e-mail: [email protected] Matthias Koppe¨ University of California, Davis, Dept. of Mathematics, One Shields Avenue, Davis, CA, 95616, USA, e-mail: [email protected] Jon Lee IBM T.J. -
An Improved Sequential Quadratic Programming Method Based on Positive Constraint Sets, Chemical Engineering Transactions, 81, 157-162 DOI:10.3303/CET2081027
157 A publication of CHEMICAL ENGINEERING TRANSACTIONS VOL. 81, 2020 The Italian Association of Chemical Engineering Online at www.cetjournal.it Guest Editors: Petar S. Varbanov, Qiuwang Wang, Min Zeng, Panos Seferlis, Ting Ma, Jiří J. Klemeš Copyright © 2020, AIDIC Servizi S.r.l. DOI: 10.3303/CET2081027 ISBN 978-88-95608-79-2; ISSN 2283-9216 An Improved Sequential Quadratic Programming Method Based on Positive Constraint Sets Li Xiaa,*, Jianyang Linga, Rongshan Bia, Wenying Zhaob, Xiaorong Caob, Shuguang Xianga a State Key Laboratory Base for Eco-Chemical Engineering, College of Chemical Engineering, Qingdao University of Science and Technology, Zhengzhou Road No. 53, Qingdao 266042, China b Chemistry and Chemistry Engineering Faculty,Qilu Normal University, jinan, 250013, Shandong, China [email protected] SQP (sequential quadratic programming) is an effective method to solve nonlinear constraint problems, widely used in chemical process simulation optimization. At present, most optimization methods in general chemical process simulation software have the problems of slow calculation and poor convergence. To solve these problems , an improved SQP optimization method based on positive constraint sets was developed. The new optimization method was used in the general chemical process simulation software named optimization engineers , adopting the traditional Armijo type step rule, L-BFGS (Limited-Memory BFGS) algorithm and rules of positive definite matrix. Compared to the traditional SQP method, the SQP optimization method based on positive constraint sets simplifies the constraint number of corresponding subproblems. L-BFGS algorithm and rules of positive definite matrix simplify the solution of the second derivative matrix, and reduce the amount of storage in the calculation process. -
Appendix a Solving Systems of Nonlinear Equations
Appendix A Solving Systems of Nonlinear Equations Chapter 4 of this book describes and analyzes the power flow problem. In its ac version, this problem is a system of nonlinear equations. This appendix describes the most common method for solving a system of nonlinear equations, namely, the Newton-Raphson method. This is an iterative method that uses initial values for the unknowns and, then, at each iteration, updates these values until no change occurs in two consecutive iterations. For the sake of clarity, we first describe the working of this method for the case of just one nonlinear equation with one unknown. Then, the general case of n nonlinear equations and n unknowns is considered. We also explain how to directly solve systems of nonlinear equations using appropriate software. A.1 Newton-Raphson Algorithm The Newton-Raphson algorithm is described in this section. A.1.1 One Unknown Consider a nonlinear function f .x/ W R ! R. We aim at finding a value of x so that: f .x/ D 0: (A.1) .0/ To do so, we first consider a given value of x, e.g., x . In general, we have that f x.0/ ¤ 0. Thus, it is necessary to find x.0/ so that f x.0/ C x.0/ D 0. © Springer International Publishing AG 2018 271 A.J. Conejo, L. Baringo, Power System Operations, Power Electronics and Power Systems, https://doi.org/10.1007/978-3-319-69407-8 272 A Solving Systems of Nonlinear Equations Using Taylor series, we can express f x.0/ C x.0/ as: Â Ã.0/ 2 Â Ã.0/ df .x/ x.0/ d2f .x/ f x.0/ C x.0/ D f x.0/ C x.0/ C C ::: dx 2 dx2 (A.2) .0/ Considering only the first two terms in Eq. -
A Sequential Quadratic Programming Algorithm with an Additional Equality Constrained Phase
A Sequential Quadratic Programming Algorithm with an Additional Equality Constrained Phase Jos´eLuis Morales∗ Jorge Nocedal † Yuchen Wu† December 28, 2008 Abstract A sequential quadratic programming (SQP) method is presented that aims to over- come some of the drawbacks of contemporary SQP methods. It avoids the difficulties associated with indefinite quadratic programming subproblems by defining this sub- problem to be always convex. The novel feature of the approach is the addition of an equality constrained phase that promotes fast convergence and improves performance in the presence of ill conditioning. This equality constrained phase uses exact second order information and can be implemented using either a direct solve or an iterative method. The paper studies the global and local convergence properties of the new algorithm and presents a set of numerical experiments to illustrate its practical performance. 1 Introduction Sequential quadratic programming (SQP) methods are very effective techniques for solv- ing small, medium-size and certain classes of large-scale nonlinear programming problems. They are often preferable to interior-point methods when a sequence of related problems must be solved (as in branch and bound methods) and more generally, when a good estimate of the solution is available. Some SQP methods employ convex quadratic programming sub- problems for the step computation (typically using quasi-Newton Hessian approximations) while other variants define the Hessian of the SQP model using second derivative informa- tion, which can lead to nonconvex quadratic subproblems; see [27, 1] for surveys on SQP methods. ∗Departamento de Matem´aticas, Instituto Tecnol´ogico Aut´onomode M´exico, M´exico. This author was supported by Asociaci´onMexicana de Cultura AC and CONACyT-NSF grant J110.388/2006. -
GRASP for NONLINEAR OPTIMIZATION in This Paper, We
GRASP FOR NONLINEAR OPTIMIZATION C.N. MENESES, P.M. PARDALOS, AND M.G.C. RESENDE ABSTRACT. We propose a Greedy Randomized Adaptive Search Procedure (GRASP) for solving continuous global optimization problems subject to box constraints. The method was tested on benchmark functions and the computational results show that our approach was able to find in a few seconds optimal solutions for all tested functions despite not using any gradient information about the function being tested. Most metaheuristcs found in the literature have not been capable of finding optimal solutions to the same collection of functions. 1. INTRODUCTION In this paper, we consider the global optimization problem: Find a point x∗ such that n f (x∗) f (x) for all x X, where X is a convex set defined by box constraints in R . For ≤ 2 instance, minimize f (x ;x ) = 100(x x2)2 +(1 x )2 such that 2 x 2 for i = 1;2. 1 2 2 − 1 − 1 − ≤ i ≤ Many optimization methods have been proposed to tackle continuous global optimiza- tion problems with box constraints. Some of these methods use information about the function being examined, e.g. the Hessian matrix or gradient vector. For some functions, however, this type of information can be difficult to obtain. In this paper, we pursue a heuristic approach to global optimization. Even though several heuristics have been designed and implemented for the problem discussed in this paper (see e.g. [1, 15, 16, 17]), we feel that there is room for yet an- other one because these heuristics were not capable of finding optimal solutions to several functions in standard benchmarks for continuous global optimization problems with box constraints. -
Parallel Technique for the Metaheuristic Algorithms Using Devoted Local Search and Manipulating the Solutions Space
applied sciences Article Parallel Technique for the Metaheuristic Algorithms Using Devoted Local Search and Manipulating the Solutions Space Dawid Połap 1,* ID , Karolina K˛esik 1, Marcin Wo´zniak 1 ID and Robertas Damaševiˇcius 2 ID 1 Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland; [email protected] (K.K.); [email protected] (M.W.) 2 Department of Software Engineering, Kaunas University of Technology, Studentu 50, LT-51368, Kaunas, Lithuania; [email protected] * Correspondence: [email protected] Received: 16 December 2017; Accepted: 13 February 2018 ; Published: 16 February 2018 Abstract: The increasing exploration of alternative methods for solving optimization problems causes that parallelization and modification of the existing algorithms are necessary. Obtaining the right solution using the meta-heuristic algorithm may require long operating time or a large number of iterations or individuals in a population. The higher the number, the longer the operation time. In order to minimize not only the time, but also the value of the parameters we suggest three proposition to increase the efficiency of classical methods. The first one is to use the method of searching through the neighborhood in order to minimize the solution space exploration. Moreover, task distribution between threads and CPU cores can affect the speed of the algorithm and therefore make it work more efficiently. The second proposition involves manipulating the solutions space to minimize the number of calculations. In addition, the third proposition is the combination of the previous two. All propositions has been described, tested and analyzed due to the use of various test functions. -
A New Successive Linearization Approach for Solving Nonlinear Programming Problems
Available at http://pvamu.edu/aam Applications and Applied Appl. Appl. Math. Mathematics: ISSN: 1932-9466 An International Journal (AAM) Vol. 14, Issue 1 (June 2019), pp. 437 – 451 A New Successive Linearization Approach for Solving Nonlinear Programming Problems 1Inci Albayrak, 2Mustafa Sivri and 3Gizem Temelcan 1,2 Department of Mathematical Engineering Yildiz Technical University Davutpasa, Istanbul, Turkey [email protected]; [email protected] 3 Department of Computer Programming Istanbul Aydin University Kucukcekmece, Istanbul, Turkey [email protected] Abstract In this paper, we focused on general nonlinear programming (NLP) problems having m nonlinear (or linear) algebraic inequality (or equality or mixed) constraints with a nonlinear (or linear) algebraic objective function in n variables. We proposed a new two-phase-successive linearization approach for solving NLP problems. Aim of this proposed approach is to find a solution of the NLP problem, based on optimal solution of linear programming (LP) problems, satisfying the nonlinear constraints oversensitively. This approach leads to novel methods. Numerical examples are given to illustrate the approach. Keywords: Nonlinear programming problems; Taylor series; Linear programming problems; Hessian matrix; Maclaurin series; Linearization approach MSC 2010 No.: 90C05, 41A58, 93B18 1. Introduction Optimization occurs in many fields. Constructing a mathematical model for real life problems is important for optimizers to find optimal strategies effectively. Optimization problems can be classified according to the nature of the objective function and constraints. An optimization problem can be defined as min (or max) of a single (or multi) objective function, subject to (or not to) single (or multi) nonlinear (or linear) inequality (or equality or mixed) constraints. -
Cocircuits of Vector Matroids
RICE UNIVERSITY Cocircuits of Vector Matroids by John David Arellano A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE Master of Arts APPROVED, THESIS COMMITTEE: ?~ Ill::f;Cks, Chair Associate Professor of Computational and Applied Mathematics ~~~~ Richard A. Tapia Maxfield-Oshman Professor of Engineering University Professor of Computational and Applied Mathematics Wotao Yin Assistant Professor of Computational and Applied Mathematics Houston, Texas August, 2011 ABSTRACT Cocircuits of Vector Matroids by John David Arellano In this thesis, I present a set covering problem (SCP) formulation of the matroid cogirth problem, finding the cardinality of the smallest cocircuit of a matroid. Ad dressing the matroid cogirth problem can lead to significantly enhancing the design process of sensor networks. The solution to the matroid cogirth problem provides the degree of redundancy of the corresponding sensor network, and allows for the evalu ation of the quality of the network. I provide an introduction to matroids, and their relation to the degree of redundancy problem. I also discuss existing methods devel oped to solve the matroid cogirth problem and the SCP. Computational results are provided to validate a branch-and-cut algorithm that addresses the SCP formulation. Acknowledgments I would like to thank my parents and family for the love and support throughout my graduate career. I would like to thank my advisor, Dr. Illya Hicks, and the rest of my committee for their guidance and support. I would also like to thank Dr. Maria Cristina Villalobos for her guidance and support during my undergraduate career. A thanks also goes to Nabor Reyna, Dr. -
Generic Branch-Cut-And-Price
Generic Branch-Cut-and-Price Diplomarbeit bei PD Dr. M. L¨ubbecke vorgelegt von Gerald Gamrath 1 Fachbereich Mathematik der Technischen Universit¨atBerlin Berlin, 16. M¨arz2010 1Konrad-Zuse-Zentrum f¨urInformationstechnik Berlin, [email protected] 2 Contents Acknowledgments iii 1 Introduction 1 1.1 Definitions . .3 1.2 A Brief History of Branch-and-Price . .6 2 Dantzig-Wolfe Decomposition for MIPs 9 2.1 The Convexification Approach . 11 2.2 The Discretization Approach . 13 2.3 Quality of the Relaxation . 21 3 Extending SCIP to a Generic Branch-Cut-and-Price Solver 25 3.1 SCIP|a MIP Solver . 25 3.2 GCG|a Generic Branch-Cut-and-Price Solver . 27 3.3 Computational Environment . 35 4 Solving the Master Problem 39 4.1 Basics in Column Generation . 39 4.1.1 Reduced Cost Pricing . 42 4.1.2 Farkas Pricing . 43 4.1.3 Finiteness and Correctness . 44 4.2 Solving the Dantzig-Wolfe Master Problem . 45 4.3 Implementation Details . 48 4.3.1 Farkas Pricing . 49 4.3.2 Reduced Cost Pricing . 52 4.3.3 Making Use of Bounds . 54 4.4 Computational Results . 58 4.4.1 Farkas Pricing . 59 4.4.2 Reduced Cost Pricing . 65 5 Branching 71 5.1 Branching on Original Variables . 73 5.2 Branching on Variables of the Extended Problem . 77 5.3 Branching on Aggregated Variables . 78 5.4 Ryan and Foster Branching . 79 i ii Contents 5.5 Other Branching Rules . 82 5.6 Implementation Details . 85 5.6.1 Branching on Original Variables . 87 5.6.2 Ryan and Foster Branching .