Nonlinear Integer Programming ∗
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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. -
How to Use Your Favorite MIP Solver: Modeling, Solving, Cannibalizing
How to use your favorite MIP Solver: modeling, solving, cannibalizing Andrea Lodi University of Bologna, Italy [email protected] January-February, 2012 @ Universit¨atWien A. Lodi, How to use your favorite MIP Solver Setting • We consider a general Mixed Integer Program in the form: T maxfc x : Ax ≤ b; x ≥ 0; xj 2 Z; 8j 2 Ig (1) where matrix A does not have a special structure. A. Lodi, How to use your favorite MIP Solver 1 Setting • We consider a general Mixed Integer Program in the form: T maxfc x : Ax ≤ b; x ≥ 0; xj 2 Z; 8j 2 Ig (1) where matrix A does not have a special structure. • Thus, the problem is solved through branch-and-bound and the bounds are computed by iteratively solving the LP relaxations through a general-purpose LP solver. A. Lodi, How to use your favorite MIP Solver 1 Setting • We consider a general Mixed Integer Program in the form: T maxfc x : Ax ≤ b; x ≥ 0; xj 2 Z; 8j 2 Ig (1) where matrix A does not have a special structure. • Thus, the problem is solved through branch-and-bound and the bounds are computed by iteratively solving the LP relaxations through a general-purpose LP solver. • The course basically covers the MIP but we will try to discuss when possible how crucial is the LP component (the engine), and how much the whole framework is built on top the capability of effectively solving LPs. • Roughly speaking, using the LP computation as a tool, MIP solvers integrate the branch-and-bound and the cutting plane algorithms through variations of the general branch-and-cut scheme [Padberg & Rinaldi 1987] developed in the context of the Traveling Salesman Problem (TSP). -
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. -
Functional Description of MINTO, a Mixed Integer Optimizer
Functional description of MINTO, a mixed integer optimizer Citation for published version (APA): Savelsbergh, M. W. P., Sigismondi, G. C., & Nemhauser, G. L. (1991). Functional description of MINTO, a mixed integer optimizer. (1st version ed.) (Memorandum COSOR; Vol. 9117). Technische Universiteit Eindhoven. Document status and date: Published: 01/01/1991 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. -
A Tutorial on Linear and Bilinear Matrix Inequalities
Journal of Process Control 10 (2000) 363±385 www.elsevier.com/locate/jprocont Review A tutorial on linear and bilinear matrix inequalities Jeremy G. VanAntwerp, Richard D. Braatz* Large Scale Systems Research Laboratory, Department of Chemical Engineering, University of Illinois at Urbana-Champaign, 600 South Mathews Avenue, Box C-3, Urbana, Illinois 61801-3792, USA Abstract This is a tutorial on the mathematical theory and process control applications of linear matrix inequalities (LMIs) and bilinear matrix inequalities (BMIs). Many convex inequalities common in process control applications are shown to be LMIs. Proofs are included to familiarize the reader with the mathematics of LMIs and BMIs. LMIs and BMIs are applied to several important process control applications including control structure selection, robust controller analysis and design, and optimal design of experiments. Software for solving LMI and BMI problems is reviewed. # 2000 Published by Elsevier Science Ltd. All rights reserved. 1. Introduction One of the main reasons for this is that process control engineers are generally unfamiliar with the mathematics A linear matrix inequality (LMI) is a convex con- of LMI/BMIs, and there is no introductory text avail- straint. Consequently, optimization problems with con- able to aid the control engineer in learning these vex objective functions and LMI constraints are mathematics. As of the writing of this paper, the only solvable relatively eciently with o-the-shelf software. text that covers LMIs in any depth is the research The form of an LMI is very general. Linear inequalities, monograph of Boyd and co-workers [22]. Although this convex quadratic inequalities, matrix norm inequalities, monograph is a useful roadmap for locating LMI and various constraints from control theory such as results scattered throughout the electrical engineering Lyapunov and Riccati inequalities can all be written as literature, it is not a textbook for teaching the concepts LMIs. -
Neos & Htcondor: Optimizing Your World
Optimizing Your neos & World neos: Network-Enabled Optimization System Mathematical Formulation Maximize ∑j∈Vbjyj−α∗∑(i,j)∈Adijxij Subject to exiting node on tour ∑(i,j)∈Axij−yi=0,∀i∈V Subject to entering node on tour ∑(i,j)∈Axij−yi=0,∀j∈V AMPL Model set V; set LINKS := {i in V, j in V: i <> j}; param alpha >= 0; param d{LINKS} >= 0; param b{V} >= 0; #default benefit of visiting a bar neos param c{V} >= 0; # default cost of one drink param B default 30; # default maximum budget for drinks http://www.neos-guide.org/content/bar-crawl-optimization neos & HTCondor neos Workflow Optimization Job Solver Categories bco lp sdp neos co milp sio Results cp minco slp go miocp socp Browser Web kestrel nco uco lno ndo Solver Names AlphaECP csdp LOQO PGAPack ASA ddsip LRAMBO proxy BARON DICOPT MILES PSwarm BDMLP Domino MINLP QSopt_EX Email Email BiqMac DSDP MINOS RELAX4 BLMVM feaspump MINTO SBB Solver Pool bnbs FilMINT MOSEK scip at UW-Madison Bonmin filter MUSCOD-II SD bpmpd filterMPEC NLPEC SDPA Cbc Gurobi NMTR SDPLR Clp icos nsips SDPT3 concorde Ipopt OOQP SeDuMi Custom CONDOR KNITRO PATH SNOPT CONOPT LANCELOT PATHNLP SYMPHONY Couenne L-BFGS-B PENBMI TRON XMLRPC CPLEX LINDOGlobal PENSDP XpressMP Off-Site Solvers Argonne National Lab Solver Inputs Arizona State University AMPL jpg OSIL SPARSE Kestrel C LP RELAX4 SPARSE_SDPA (AMPL or GAMS) University of Klagenfurt CPLEX MATLAB_BINARY SDPA TSP University of Minho Fortran MOSEL SDPLR ZIMPL GAMS MPS SMPS Jobs Per Month 140000 120000 100000 80000 Minho 60000 Klagenfurt Arizona State 40000 Argonne 20000 -
Domain Reduction Techniques for Global NLP and MINLP Optimization 2
Domain reduction techniques for global NLP and MINLP optimization Y. Puranik1 and Nikolaos V. Sahinidis1 1Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA Abstract Optimization solvers routinely utilize presolve techniques, including model simplification, reformulation and domain reduction techniques. Domain reduction techniques are especially important in speeding up convergence to the global optimum for challenging nonconvex nonlinear programming (NLP) and mixed- integer nonlinear programming (MINLP) optimization problems. In this work, we survey the various techniques used for domain reduction of NLP and MINLP optimization problems. We also present a computational analysis of the impact of these techniques on the performance of various widely available global solvers on a collection of 1740 test problems. Keywords: Constraint propagation; feasibility-based bounds tightening; optimality-based bounds tightening; domain reduction 1 Introduction combinatorial complexity introduced by the integer variables, nonconvexities in objective function or We consider the following mixed-integer nonlinear the feasible region lead to multiple local minima programming problem (MINLP): and provide a challenge to the optimization of such problems. min f(~x) Branch-and-bound [138] based methods can be s.t. g(~~x) ≤ 0 exploited to solve Problem 1 to global optimal- (1) ~xl ≤ ~x ≤ ~xu ity. Inspired by branch-and-bound for discrete Rn−m Zm ~x ∈ × programming problems [109], branch-and-bound was arXiv:1706.08601v1 [cs.DS] 26 Jun 2017 adapted for continuous problems by Falk and MINLP is a very general representation for opti- Soland [58]. The algorithm proceeds by bounding mization problems and includes linear programming the global optimum by a valid lower and upper (LP), mixed-integer linear programming (MIP) and bound throughout the search process. -
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.