Exact and Fast Algorithms for Mixed-Integer Nonlinear Programming
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An Algorithm Based on Semidefinite Programming for Finding Minimax
Takustr. 7 Zuse Institute Berlin 14195 Berlin Germany BELMIRO P.M. DUARTE,GUILLAUME SAGNOL,WENG KEE WONG An algorithm based on Semidefinite Programming for finding minimax optimal designs ZIB Report 18-01 (December 2017) Zuse Institute Berlin Takustr. 7 14195 Berlin Germany Telephone: +49 30-84185-0 Telefax: +49 30-84185-125 E-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 An algorithm based on Semidefinite Programming for finding minimax optimal designs Belmiro P.M. Duarte a,b, Guillaume Sagnolc, Weng Kee Wongd aPolytechnic Institute of Coimbra, ISEC, Department of Chemical and Biological Engineering, Portugal. bCIEPQPF, Department of Chemical Engineering, University of Coimbra, Portugal. cTechnische Universität Berlin, Institut für Mathematik, Germany. dDepartment of Biostatistics, Fielding School of Public Health, UCLA, U.S.A. Abstract An algorithm based on a delayed constraint generation method for solving semi- infinite programs for constructing minimax optimal designs for nonlinear models is proposed. The outer optimization level of the minimax optimization problem is solved using a semidefinite programming based approach that requires the de- sign space be discretized. A nonlinear programming solver is then used to solve the inner program to determine the combination of the parameters that yields the worst-case value of the design criterion. The proposed algorithm is applied to find minimax optimal designs for the logistic model, the flexible 4-parameter Hill homoscedastic model and the general nth order consecutive reaction model, and shows that it (i) produces designs that compare well with minimax D−optimal de- signs obtained from semi-infinite programming method in the literature; (ii) can be applied to semidefinite representable optimality criteria, that include the com- mon A−; E−; G−; I− and D-optimality criteria; (iii) can tackle design problems with arbitrary linear constraints on the weights; and (iv) is fast and relatively easy to use. -
Xpress Nonlinear Manual This Material Is the Confidential, Proprietary, and Unpublished Property of Fair Isaac Corporation
R FICO Xpress Optimization Last update 01 May, 2018 version 33.01 FICO R Xpress Nonlinear Manual This material is the confidential, proprietary, and unpublished property of Fair Isaac Corporation. Receipt or possession of this material does not convey rights to divulge, reproduce, use, or allow others to use it without the specific written authorization of Fair Isaac Corporation and use must conform strictly to the license agreement. The information in this document is subject to change without notice. If you find any problems in this documentation, please report them to us in writing. Neither Fair Isaac Corporation nor its affiliates warrant that this documentation is error-free, nor are there any other warranties with respect to the documentation except as may be provided in the license agreement. ©1983–2018 Fair Isaac Corporation. All rights reserved. Permission to use this software and its documentation is governed by the software license agreement between the licensee and Fair Isaac Corporation (or its affiliate). Portions of the program may contain copyright of various authors and may be licensed under certain third-party licenses identified in the software, documentation, or both. In no event shall Fair Isaac Corporation or its affiliates be liable to any person for direct, indirect, special, incidental, or consequential damages, including lost profits, arising out of the use of this software and its documentation, even if Fair Isaac Corporation or its affiliates have been advised of the possibility of such damage. The rights and allocation of risk between the licensee and Fair Isaac Corporation (or its affiliates) are governed by the respective identified licenses in the software, documentation, or both. -
Solving Mixed Integer Linear and Nonlinear Problems Using the SCIP Optimization Suite
Takustraße 7 Konrad-Zuse-Zentrum D-14195 Berlin-Dahlem fur¨ Informationstechnik Berlin Germany TIMO BERTHOLD GERALD GAMRATH AMBROS M. GLEIXNER STEFAN HEINZ THORSTEN KOCH YUJI SHINANO Solving mixed integer linear and nonlinear problems using the SCIP Optimization Suite Supported by the DFG Research Center MATHEON Mathematics for key technologies in Berlin. ZIB-Report 12-27 (July 2012) Herausgegeben vom Konrad-Zuse-Zentrum f¨urInformationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Telefon: 030-84185-0 Telefax: 030-84185-125 e-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 Solving mixed integer linear and nonlinear problems using the SCIP Optimization Suite∗ Timo Berthold Gerald Gamrath Ambros M. Gleixner Stefan Heinz Thorsten Koch Yuji Shinano Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany, fberthold,gamrath,gleixner,heinz,koch,[email protected] July 31, 2012 Abstract This paper introduces the SCIP Optimization Suite and discusses the ca- pabilities of its three components: the modeling language Zimpl, the linear programming solver SoPlex, and the constraint integer programming frame- work SCIP. We explain how these can be used in concert to model and solve challenging mixed integer linear and nonlinear optimization problems. SCIP is currently one of the fastest non-commercial MIP and MINLP solvers. We demonstrate the usage of Zimpl, SCIP, and SoPlex by selected examples, give an overview of available interfaces, and outline plans for future development. ∗A Japanese translation of this paper will be published in the Proceedings of the 24th RAMP Symposium held at Tohoku University, Miyagi, Japan, 27{28 September 2012, see http://orsj.or. -
Of the American Mathematical Society August 2017 Volume 64, Number 7
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society August 2017 Volume 64, Number 7 The Mathematics of Gravitational Waves: A Two-Part Feature page 684 The Travel Ban: Affected Mathematicians Tell Their Stories page 678 The Global Math Project: Uplifting Mathematics for All page 712 2015–2016 Doctoral Degrees Conferred page 727 Gravitational waves are produced by black holes spiraling inward (see page 674). American Mathematical Society LEARNING ® MEDIA MATHSCINET ONLINE RESOURCES MATHEMATICS WASHINGTON, DC CONFERENCES MATHEMATICAL INCLUSION REVIEWS STUDENTS MENTORING PROFESSION GRAD PUBLISHING STUDENTS OUTREACH TOOLS EMPLOYMENT MATH VISUALIZATIONS EXCLUSION TEACHING CAREERS MATH STEM ART REVIEWS MEETINGS FUNDING WORKSHOPS BOOKS EDUCATION MATH ADVOCACY NETWORKING DIVERSITY blogs.ams.org Notices of the American Mathematical Society August 2017 FEATURED 684684 718 26 678 Gravitational Waves The Graduate Student The Travel Ban: Affected Introduction Section Mathematicians Tell Their by Christina Sormani Karen E. Smith Interview Stories How the Green Light was Given for by Laure Flapan Gravitational Wave Research by Alexander Diaz-Lopez, Allyn by C. Denson Hill and Paweł Nurowski WHAT IS...a CR Submanifold? Jackson, and Stephen Kennedy by Phillip S. Harrington and Andrew Gravitational Waves and Their Raich Mathematics by Lydia Bieri, David Garfinkle, and Nicolás Yunes This season of the Perseid meteor shower August 12 and the third sighting in June make our cover feature on the discovery of gravitational waves -
Hydraulic Optimization Demonstration for Groundwater Pump
Figure 5-18: Shallow Particles, Contain Shallow 20-ppb Plume, & 500 gpm for Deep 20-ppb plume (1573 gpm, 3 new wells, 1 existing well) Injection Well Well Layer 1 Well Layer 2 30000 25000 20000 15000 10000 5000 0 0 5000 10000 15000 A "+" symbol indicates that a particle starting at that location is captured by one of the remediation wells, based on particle tracking with MODPATH. Shallow particles originate half-way down in layer 1. Figure 5-19: Deep Particles, Contain Shallow 20-ppb Plume, & 500 gpm for Deep 20-ppb plume (1573 gpm, 3 new wells, 1 existing well) Injection Well Well Layer 1 Well Layer 2 30000 25000 20000 15000 10000 5000 0 0 5000 10000 15000 A "+" symbol indicates that a particle starting at that location is captured by one of the remediation wells, based on particle tracking with MODPATH. Deep particles originate half-way down in layer 2. Figure 5-20: Shallow Particles, Contain Shallow 20-ppb & 50-ppb Plumes, & 500 gpm for Deep 20-ppb plume (2620 gpm, 6 new wells, 0 existing wells) Injection Well Well Layer 1 Well Layer 2 30000 25000 20000 15000 10000 5000 0 0 5000 10000 15000 A "+" symbol indicates that a particle starting at that location is captured by one of the remediation wells, based on particle tracking with MODPATH. Shallow particles originate half-way down in layer 1. Figure 5-21: Deep Particles, Contain Shallow 20-ppb & 50-ppb Plumes, & 500 gpm for Deep 20-ppb plume (2620 gpm, 6 new wells, 0 existing wells) Injection Well Well Layer 1 Well Layer 2 30000 25000 20000 15000 10000 5000 0 0 5000 10000 15000 A "+" symbol indicates that a particle starting at that location is captured by one of the remediation wells, based on particle tracking with MODPATH. -
Using the COIN-OR Server
Using the COIN-OR Server Your CoinEasy Team November 16, 2009 1 1 Overview This document is part of the CoinEasy project. See projects.coin-or.org/CoinEasy. In this document we describe the options available to users of COIN-OR who are interested in solving opti- mization problems but do not wish to compile source code in order to build the COIN-OR projects. In particular, we show how the user can send optimization problems to a COIN-OR server and get the solution result back. The COIN-OR server, webdss.ise.ufl.edu, is 2x Intel(R) Xeon(TM) CPU 3.06GHz 512MiB L2 1024MiB L3, 2GiB DRAM, 4x73GiB scsi disk 2xGigE machine. This server allows the user to directly access the following COIN-OR optimization solvers: • Bonmin { a solver for mixed-integer nonlinear optimization • Cbc { a solver for mixed-integer linear programs • Clp { a linear programming solver • Couenne { a solver for mixed-integer nonlinear optimization problems and is capable of global optiomization • DyLP { a linear programming solver • Ipopt { an interior point nonlinear optimization solver • SYMPHONY { mixed integer linear solver that can be executed in either parallel (dis- tributed or shared memory) or sequential modes • Vol { a linear programming solver All of these solvers on the COIN-OR server may be accessed through either the GAMS or AMPL modeling languages. In Section 2.1 we describe how to use the solvers using the GAMS modeling language. In Section 2.2 we describe how to call the solvers using the AMPL modeling language. In Section 3 we describe how to call the solvers using a command line executable pro- gram OSSolverService.exe (or OSSolverService for Linux/Mac OS X users { in the rest of the document we refer to this executable using a .exe extension). -
Linear Underestimators for Bivariate Functions with a Fixed Convexity Behavior
Takustraße 7 Konrad-Zuse-Zentrum D-14195 Berlin-Dahlem fur¨ Informationstechnik Berlin Germany MARTIN BALLERSTEINy AND DENNIS MICHAELSz AND STEFAN VIGERSKE Linear Underestimators for bivariate functions with a fixed convexity behavior This work is part of the Collaborative Research Centre “Integrated Chemical Processes in Liquid Multiphase Systems” (CRC/Transregio 63 “InPROMPT”) funded by the German Research Foundation (DFG). Main parts of this work has been finished while the second author was at the Institute for Operations Research at ETH Zurich and financially supported by DFG through CRC/Transregio 63. The first and second author thank the DFG for its financial support. Thethirdauthor was supported by the DFG Research Center MATHEON Mathematics for key technologies and the Research Campus MODAL Mathematical Optimization and Data Analysis Laboratories in Berlin. y Eidgenossische¨ Technische, Hochschule Zurich,¨ Institut fur¨ Operations Research, Ramistrasse¨ 101, 8092 Zurich (Switzerland) z Technische Universitat¨ Dortmund, Fakultat¨ fur¨ Mathematik, M/518, Vogelpothsweg 87, 44227 Dortmund (Germany) ZIB-Report 13-02 (revised) (February 2015) Herausgegeben vom Konrad-Zuse-Zentrum fur¨ Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Telefon: 030-84185-0 Telefax: 030-84185-125 e-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 Technical Report Linear Underestimators for bivariate functions with a fixed convexity behavior∗ A documentation for the SCIP constraint handler cons bivariate Martin Ballersteiny Dennis Michaelsz Stefan Vigerskex February 23, 2015 y Eidgenossische¨ Technische Hochschule Zurich¨ Institut fur¨ Operations Research Ramistrasse¨ 101, 8092 Zurich (Switzerland) z Technische Universitat¨ Dortmund Fakultat¨ fur¨ Mathematik, M/518 Vogelpothsweg 87, 44227 Dortmund (Germany) x Zuse Institute Berlin Takustr. -
SDDP.Jl: a Julia Package for Stochastic Dual Dynamic Programming
Optimization Online manuscript No. (will be inserted by the editor) SDDP.jl: a Julia package for Stochastic Dual Dynamic Programming Oscar Dowson · Lea Kapelevich Received: date / Accepted: date Abstract In this paper we present SDDP.jl, an open-source library for solving multistage stochastic optimization problems using the Stochastic Dual Dynamic Programming algorithm. SDDP.jl is built upon JuMP, an algebraic modelling lan- guage in Julia. This enables a high-level interface for the user, while simultaneously providing performance that is similar to implementations in low-level languages. We benchmark the performance of SDDP.jl against a C++ implementation of SDDP for the New Zealand Hydro-Thermal Scheduling Problem. On the bench- mark problem, SDDP.jl is approximately 30% slower than the C++ implementa- tion. However, this performance penalty is small when viewed in context of the generic nature of the SDDP.jl library compared to the single purpose C++ imple- mentation. Keywords julia · stochastic dual dynamic programming 1 Introduction Solving any mathematical optimization problem requires four steps: the formula- tion of the problem by the user; the communication of the problem to the com- puter; the efficient computational solution of the problem; and the communication of the computational solution back to the user. Over time, considerable effort has been made to improve each of these four steps for a variety of problem classes such linear, quadratic, mixed-integer, conic, and non-linear. For example, con- sider the evolution from early file-formats such as MPS [25] to modern algebraic modelling languages embedded in high-level languages such as JuMP [11], or the 73-fold speed-up in solving difficult mixed-integer linear programs in seven years by Gurobi [16]. -
Nonlinear Programming Techniques for Operative Planning in Large Drinking Water Networks Jens Burgschweiger1, Bernd Gnädig2 and Marc C
14 The Open Applied Mathematics Journal, 2009, 3, 14-28 Open Access Nonlinear Programming Techniques for Operative Planning in Large Drinking Water Networks Jens Burgschweiger1, Bernd Gnädig2 and Marc C. Steinbach3,* 1Berliner Wasserbetriebe, Abt. NA-G/W, 10864 Berlin, Germany 2Düsseldorf, Germany 3Leibniz Universität Hannover, IfAM, Welfengarten 1, 30167 Hannover, Germany Abstract: Mathematical decision support for operative planning in water supply systems is highly desirable; it leads, however, to very difficult optimization problems. We propose a nonlinear programming approach that yields practically satisfactory operating schedules in acceptable computing time even for large networks. Based on a carefully designed model supporting gradient-based optimization algorithms, this approach employs a special initialization strategy for convergence acceleration, special minimum up and down time constraints together with pump aggregation to handle switching decisions, and several network reduction techniques for further speed-up. Results for selected application scenarios at Berliner Wasserbetriebe demonstrate the success of the approach. Keywords: Water supply, large-scale nonlinear programming, convergence acceleration, discrete decisions, network reduction. INTRODUCTION • experiments with various optimization models and methods [36, 37], Stringent requirements on cost effectiveness and environmental compatibility generate an increased demand • a first nonlinear programming (NLP) model for model-based decision support tools for designing and developed under GAMS [38], operating municipal water supply systems. This paper deals • numerical results for a substantially reduced with the minimum cost operation of drinking water network graph using (under GAMS) the SQP codes networks. Operative planning in water networks is difficult: CONOPT, SNOPT, and the augmented Lagrangian a sound mathematical model leads to nonlinear mixed- code MINOS. -
ILOG AMPL CPLEX System Version 10.0 User's Guide
ILOG AMPL CPLEX System Version 10.0 User’s Guide Standard (Command-line) Version Including CPLEX Directives January 2006 COPYRIGHT NOTICE Copyright © 1987-2006, by ILOG S.A., 9 Rue de Verdun, 94253 Gentilly Cedex, France, and ILOG, Inc., 1080 Linda Vista Ave., Mountain View, California 94043, USA. All rights reserved. General Use Restrictions This document and the software described in this document are the property of ILOG and are protected as ILOG trade secrets. They are furnished under a license or nondisclosure agreement, and may be used or copied only within the terms of such license or nondisclosure agreement. No part of this work may be reproduced or disseminated in any form or by any means, without the prior written permission of ILOG S.A, or ILOG, Inc. Trademarks ILOG, the ILOG design, CPLEX, and all other logos and product and service names of ILOG are registered trademarks or trademarks of ILOG in France, the U.S. and/or other countries. All other company and product names are trademarks or registered trademarks of their respective holders. Java and all Java-based marks are either trademarks or registered trademarks of Sun Microsystems, Inc. in the United States and other countries. Microsoft, Windows, and Windows NT are either trademarks or registered trademarks of Microsoft Corporation in the United States and other countries. document version 10.0 CO N T E N T S Table of Contents Chapter 1 Welcome to AMPL . 9 Using this Guide. .9 Installing AMPL . .10 Requirements. .10 Unix Installation . .10 Windows Installation . .11 AMPL and Parallel CPLEX. -
Non-Orientable Lagrangian Cobordisms Between Legendrian Knots
NON-ORIENTABLE LAGRANGIAN COBORDISMS BETWEEN LEGENDRIAN KNOTS ORSOLA CAPOVILLA-SEARLE AND LISA TRAYNOR 3 Abstract. In the symplectization of standard contact 3-space, R × R , it is known that an orientable Lagrangian cobordism between a Leg- endrian knot and itself, also known as an orientable Lagrangian endo- cobordism for the Legendrian knot, must have genus 0. We show that any Legendrian knot has a non-orientable Lagrangian endocobordism, and that the crosscap genus of such a non-orientable Lagrangian en- docobordism must be a positive multiple of 4. The more restrictive exact, non-orientable Lagrangian endocobordisms do not exist for any exactly fillable Legendrian knot but do exist for any stabilized Legen- drian knot. Moreover, the relation defined by exact, non-orientable La- grangian cobordism on the set of stabilized Legendrian knots is symmet- ric and defines an equivalence relation, a contrast to the non-symmetric relation defined by orientable Lagrangian cobordisms. 1. Introduction Smooth cobordisms are a common object of study in topology. Motivated by ideas in symplectic field theory, [19], Lagrangian cobordisms that are cylindrical over Legendrian submanifolds outside a compact set have been an active area of research interest. Throughout this paper, we will study 3 Lagrangian cobordisms in the symplectization of the standard contact R , 3 t namely the symplectic manifold (R×R ; d(e α)) where α = dz−ydx, that co- incide with the cylinders R×Λ+ (respectively, R×Λ−) when the R-coordinate is sufficiently positive (respectively, negative). Our focus will be on non- orientable Lagrangian cobordisms between Legendrian knots Λ+ and Λ− and non-orientable Lagrangian endocobordisms, which are non-orientable Lagrangian cobordisms with Λ+ = Λ−. -
54 OP08 Abstracts
54 OP08 Abstracts CP1 Dept. of Mathematics Improving Ultimate Convergence of An Aug- [email protected] mented Lagrangian Method Optimization methods that employ the classical Powell- CP1 Hestenes-Rockafellar Augmented Lagrangian are useful A Second-Derivative SQP Method for Noncon- tools for solving Nonlinear Programming problems. Their vex Optimization Problems with Inequality Con- reputation decreased in the last ten years due to the com- straints parative success of Interior-Point Newtonian algorithms, which are asymptotically faster. In the present research We consider a second-derivative 1 sequential quadratic a combination of both approaches is evaluated. The idea programming trust-region method for large-scale nonlin- is to produce a competitive method, being more robust ear non-convex optimization problems with inequality con- and efficient than its ”pure” counterparts for critical prob- straints. Trial steps are composed of two components; a lems. Moreover, an additional hybrid algorithm is defined, Cauchy globalization step and an SQP correction step. A in which the Interior Point method is replaced by the New- single linear artificial constraint is incorporated that en- tonian resolution of a KKT system identified by the Aug- sures non-accent in the SQP correction step, thus ”guiding” mented Lagrangian algorithm. the algorithm through areas of indefiniteness. A salient feature of our approach is feasibility of all subproblems. Ernesto G. Birgin IME-USP Daniel Robinson Department of Computer Science Oxford University [email protected]