Developments in the Use of Mathematica for Computable General Equilibrium Analysis
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
University of California, San Diego
UNIVERSITY OF CALIFORNIA, SAN DIEGO Computational Methods for Parameter Estimation in Nonlinear Models A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics with a Specialization in Computational Physics by Bryan Andrew Toth Committee in charge: Professor Henry D. I. Abarbanel, Chair Professor Philip Gill Professor Julius Kuti Professor Gabriel Silva Professor Frank Wuerthwein 2011 Copyright Bryan Andrew Toth, 2011 All rights reserved. The dissertation of Bryan Andrew Toth is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2011 iii DEDICATION To my grandparents, August and Virginia Toth and Willem and Jane Keur, who helped put me on a lifelong path of learning. iv EPIGRAPH An Expert: One who knows more and more about less and less, until eventually he knows everything about nothing. |Source Unknown v TABLE OF CONTENTS Signature Page . iii Dedication . iv Epigraph . v Table of Contents . vi List of Figures . ix List of Tables . x Acknowledgements . xi Vita and Publications . xii Abstract of the Dissertation . xiii Chapter 1 Introduction . 1 1.1 Dynamical Systems . 1 1.1.1 Linear and Nonlinear Dynamics . 2 1.1.2 Chaos . 4 1.1.3 Synchronization . 6 1.2 Parameter Estimation . 8 1.2.1 Kalman Filters . 8 1.2.2 Variational Methods . 9 1.2.3 Parameter Estimation in Nonlinear Systems . 9 1.3 Dissertation Preview . 10 Chapter 2 Dynamical State and Parameter Estimation . 11 2.1 Introduction . 11 2.2 DSPE Overview . 11 2.3 Formulation . 12 2.3.1 Least Squares Minimization . -
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. -
Julia, My New Friend for Computing and Optimization? Pierre Haessig, Lilian Besson
Julia, my new friend for computing and optimization? Pierre Haessig, Lilian Besson To cite this version: Pierre Haessig, Lilian Besson. Julia, my new friend for computing and optimization?. Master. France. 2018. cel-01830248 HAL Id: cel-01830248 https://hal.archives-ouvertes.fr/cel-01830248 Submitted on 4 Jul 2018 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. « Julia, my new computing friend? » | 14 June 2018, IETR@Vannes | By: L. Besson & P. Haessig 1 « Julia, my New frieNd for computiNg aNd optimizatioN? » Intro to the Julia programming language, for MATLAB users Date: 14th of June 2018 Who: Lilian Besson & Pierre Haessig (SCEE & AUT team @ IETR / CentraleSupélec campus Rennes) « Julia, my new computing friend? » | 14 June 2018, IETR@Vannes | By: L. Besson & P. Haessig 2 AgeNda for today [30 miN] 1. What is Julia? [5 miN] 2. ComparisoN with MATLAB [5 miN] 3. Two examples of problems solved Julia [5 miN] 4. LoNger ex. oN optimizatioN with JuMP [13miN] 5. LiNks for more iNformatioN ? [2 miN] « Julia, my new computing friend? » | 14 June 2018, IETR@Vannes | By: L. Besson & P. Haessig 3 1. What is Julia ? Open-source and free programming language (MIT license) Developed since 2012 (creators: MIT researchers) Growing popularity worldwide, in research, data science, finance etc… Multi-platform: Windows, Mac OS X, GNU/Linux.. -
[20Pt]Algorithms for Constrained Optimization: [ 5Pt]
SOL Optimization 1970s 1980s 1990s 2000s 2010s Summary 2020s Algorithms for Constrained Optimization: The Benefits of General-purpose Software Michael Saunders MS&E and ICME, Stanford University California, USA 3rd AI+IoT Business Conference Shenzhen, China, April 25, 2019 Optimization Software 3rd AI+IoT Business Conference, Shenzhen, April 25, 2019 1/39 SOL Optimization 1970s 1980s 1990s 2000s 2010s Summary 2020s SOL Systems Optimization Laboratory George Dantzig, Stanford University, 1974 Inventor of the Simplex Method Father of linear programming Large-scale optimization: Algorithms, software, applications Optimization Software 3rd AI+IoT Business Conference, Shenzhen, April 25, 2019 2/39 SOL Optimization 1970s 1980s 1990s 2000s 2010s Summary 2020s SOL history 1974 Dantzig and Cottle start SOL 1974{78 John Tomlin, LP/MIP expert 1974{2005 Alan Manne, nonlinear economic models 1975{76 MS, MINOS first version 1979{87 Philip Gill, Walter Murray, MS, Margaret Wright (Gang of 4!) 1989{ Gerd Infanger, stochastic optimization 1979{ Walter Murray, MS, many students 2002{ Yinyu Ye, optimization algorithms, especially interior methods This week! UC Berkeley opened George B. Dantzig Auditorium Optimization Software 3rd AI+IoT Business Conference, Shenzhen, April 25, 2019 3/39 SOL Optimization 1970s 1980s 1990s 2000s 2010s Summary 2020s Optimization problems Minimize an objective function subject to constraints: 0 x 1 min '(x) st ` ≤ @ Ax A ≤ u c(x) x variables 0 1 A matrix c1(x) B . C c(x) nonlinear functions @ . A c (x) `; u bounds m Optimization -
Numericaloptimization
Numerical Optimization Alberto Bemporad http://cse.lab.imtlucca.it/~bemporad/teaching/numopt Academic year 2020-2021 Course objectives Solve complex decision problems by using numerical optimization Application domains: • Finance, management science, economics (portfolio optimization, business analytics, investment plans, resource allocation, logistics, ...) • Engineering (engineering design, process optimization, embedded control, ...) • Artificial intelligence (machine learning, data science, autonomous driving, ...) • Myriads of other applications (transportation, smart grids, water networks, sports scheduling, health-care, oil & gas, space, ...) ©2021 A. Bemporad - Numerical Optimization 2/102 Course objectives What this course is about: • How to formulate a decision problem as a numerical optimization problem? (modeling) • Which numerical algorithm is most appropriate to solve the problem? (algorithms) • What’s the theory behind the algorithm? (theory) ©2021 A. Bemporad - Numerical Optimization 3/102 Course contents • Optimization modeling – Linear models – Convex models • Optimization theory – Optimality conditions, sensitivity analysis – Duality • Optimization algorithms – Basics of numerical linear algebra – Convex programming – Nonlinear programming ©2021 A. Bemporad - Numerical Optimization 4/102 References i ©2021 A. Bemporad - Numerical Optimization 5/102 Other references • Stephen Boyd’s “Convex Optimization” courses at Stanford: http://ee364a.stanford.edu http://ee364b.stanford.edu • Lieven Vandenberghe’s courses at UCLA: http://www.seas.ucla.edu/~vandenbe/ • For more tutorials/books see http://plato.asu.edu/sub/tutorials.html ©2021 A. Bemporad - Numerical Optimization 6/102 Optimization modeling What is optimization? • Optimization = assign values to a set of decision variables so to optimize a certain objective function • Example: Which is the best velocity to minimize fuel consumption ? fuel [ℓ/km] velocity [km/h] 0 30 60 90 120 160 ©2021 A. -
Axiom / Fricas
Axiom / FriCAS Christoph Koutschan Research Institute for Symbolic Computation Johannes Kepler Universit¨atLinz, Austria Computer Algebra Systems 15.11.2010 Master's Thesis: The ISAC project • initiative at Graz University of Technology • Institute for Software Technology • Institute for Information Systems and Computer Media • experimental software assembling open source components with as little glue code as possible • feasibility study for a novel kind of transparent single-stepping software for applied mathematics • experimenting with concepts and technologies from • computer mathematics (theorem proving, symbolic computation, model based reasoning, etc.) • e-learning (knowledge space theory, usability engineering, computer-supported collaboration, etc.) • The development employs academic expertise from several disciplines. The challenge for research is interdisciplinary cooperation. Rewriting, a basic CAS technique This technique is used in simplification, equation solving, and many other CAS functions, and it is intuitively comprehensible. This would make rewriting useful for educational systems|if one copes with the problem, that even elementary simplifications involve hundreds of rewrites. As an example see: http://www.ist.tugraz.at/projects/isac/www/content/ publications.html#DA-M02-main \Reverse rewriting" for comprehensible justification Many CAS functions can not be done by rewriting, for instance cancelling multivariate polynomials, factoring or integration. However, respective inverse problems can be done by rewriting and produce human readable derivations. As an example see: http://www.ist.tugraz.at/projects/isac/www/content/ publications.html#GGTs-von-Polynomen Equation solving made transparent Re-engineering equation solvers in \transparent single-stepping systems" leads to types of equations, arranged in a tree. ISAC's tree of equations are to be compared with what is produced by tracing facilities of Mathematica and/or Maple. -
Solution Software for CGE Modeling
Eleventh Floor, Menzies Building Monash University, Wellington Road CLAYTON Vic 3800 AUSTRALIA Telephone: from overseas: (03) 9905 2398, (03) 9905 5112 61 3 9905 2398 or 61 3 9905 5112 Fax: (03) 9905 2426 61 3 9905 2426 e-mail: [email protected] Internet home page: http//www.monash.edu.au/policy/ Solution Software for CGE Modeling by Mark HORRIDGE and Ken PEARSON Centre of Policy Studies, Monash University General Paper No. G-214 March 2011 ISSN 1 031 9034 ISBN 978 1 921654 21 3 The Centre of Policy Studies (COPS) is a research centre at Monash University devoted to economy-wide modelling of economic policy issues. Solution Software for CGE Modeling by Mark Horridge and Ken Pearson March 2011 Abstract We describe the progress of computable general equilibrium (CGE) modeling software since the 1980s and contrast the main systems used today: GAMS, MPSGE, and GEMPACK. The development of these general-purpose modeling systems has underpinned rapid growth in the use of CGE models and allowed models to be shared and their results replicated. We show how a very simple model may be implemented and solved in all 3 systems. We note that they produce the same numerical results but have different strengths. We conclude by considering some challenges for the future. JEL classification: C63, C68, D58. Keywords: CGE, software, GAMS, GEMPACK, MPSGE CONTENTS 1. Introduction 1 2. Early days 1 3. General-purpose software 3 3.1 Allow input intelligible to modelers 3 3.2 Handle a wide range of models 3 3.3 Insulate modelers from the solution method 4 3.4 Windows programs to control simulations and examine data and results 4 3.5 Run on different computers 5 4. -
Systematic Sensitivity Analysis with Respect to Correlated Variations in Parameters and Shocks J
Systematic Sensitivity Analysis with Respect to Correlated Variations in Parameters and Shocks J. Mark Horridge and Ken Pearson 27 January 2010, updated 20 December 2010 and 8 March 2011 Abstract We show how you can carry out systematic sensitivity analysis (SSA) with respect to parameters and/or shocks, which vary according to a specified covariance matrix. You can use the existing SSA tools in RunGTAP or RunGEM to do this if your model is implemented in GEMPACK. Those SSA tools assume that all parameters or shocks are varying independently (i.e., the distributions of all parameters or shocks are uncorrelated) or together (i.e., are completely correlated). The techniques in this paper remove those restrictions. However, users need to make small modifications to the TAB file for the model. Different modifications are needed for different SSA scenarios. Further, the standard SSA procedure built into RunGTAP and RunGEM allows you to compute the sensitivity of model results either with respect to variations in parameter values or with respect to variations in shock values, but you cannot vary both parameters and shocks at the same time. Our discussion concentrates on the parameter case. However, we later show how shock variation may be modelled as a type of parameter variation. This opens the door to simultaneous variation of shocks and parameters. We include worked examples of the techniques described, based on the standard GTAP model. CONTENTS 1. INTRODUCTION ...................................................................................................................... -
Arxiv:2102.02679V1 [Cs.LO] 4 Feb 2021 HOL-ODE [10], Which Supports Reasoning for Systems of Ordinary Differential Equations (Sodes)
Certifying Differential Equation Solutions from Computer Algebra Systems in Isabelle/HOL Thomas Hickman, Christian Pardillo Laursen, and Simon Foster University of York Abstract. The Isabelle/HOL proof assistant has a powerful library for continuous analysis, which provides the foundation for verification of hybrid systems. However, Isabelle lacks automated proof support for continuous artifacts, which means that verification is often manual. In contrast, Computer Algebra Systems (CAS), such as Mathematica and SageMath, contain a wealth of efficient algorithms for matrices, differen- tial equations, and other related artifacts. Nevertheless, these algorithms are not verified, and thus their outputs cannot, of themselves, be trusted for use in a safety critical system. In this paper we integrate two CAS systems into Isabelle, with the aim of certifying symbolic solutions to or- dinary differential equations. This supports a verification technique that is both automated and trustworthy. 1 Introduction Verification of Cyber-Physical and Autonomous Systems requires that we can verify both discrete control, and continuous evolution, as envisaged by the hy- brid systems domain [1]. Whilst powerful bespoke verification tools exist, such as the KeYmaera X [2] proof assistant, software engineering requires a gen- eral framework, which can support a variety of notations and paradigms [3]. Isabelle/HOL [4] is a such a framework. Its combination of an extensible fron- tend for syntax processing, and a plug-in oriented backend, based in ML, which supports a wealth of heterogeneous semantic models and proof tools, supports a flexible platform for software development, verification, and assurance [5,6,7]. Verification of hybrid systems in Isabelle is supported by several detailed li- braries of Analysis, including Multivariate Analysis [8], Affine Arithmetic [9], and arXiv:2102.02679v1 [cs.LO] 4 Feb 2021 HOL-ODE [10], which supports reasoning for Systems of Ordinary Differential Equations (SODEs). -
MATH2010-1 Logiciels Mathématiques
MATH2010-1 Logiciels mathématiques Émilie Charlier Département de Mathématique Université de Liège 10 février 2020 Calculatrices électroniques I La HP-35, commercialisée en janvier 1972 par Hewlett-Packard. I Première calculatrice scientifique. I Devient célèbre sous le nom de “règle à calcul électronique”. I 395 dollars (moitié du salaire mensuel d’un enseignant de l’époque). https://fr.wikipedia.org/wiki/HP-35 Calculatrices graphiques I La TI-89, commercialisée par Texas Instruments en 1998. I Calcul formel. I Possibilités de programmation. http://fr.wikipedia.org/wiki/Calculatrice (Une multitude de) logiciels mathématiques Depuis 1960, au moins 45 logiciels mathématiques : Axiom FORM Magnus MuPAD SyMAT Cadabra FriCAS Maple OpenAxiom SymbolicC++ Calcinator FxSolver Mathcad PARI/GP Symbolism CoCoA-4 GAP Mathematica Reduce Symengine CoCoA-5 GiNaC MathHandbook Scilab SymPy Derive KANT/KASH Mathics SageMath TI-Nspire DataMelt Macaulay2 Mathomatic SINGULAR Wolfram Alpha Erable Macsyma Maxima SMath Xcas/Giac Fermat Magma MuMATH Symbolic Yacas http://en.wikipedia.org/wiki/List_of_computer_algebra_systems Logiciels de mathématiques Quelques logiciels commerciaux : I Maple, Waterloo Maple Inc., Maplesoft, depuis 1985. I Mathematica, Wolfram Research, depuis 1988. I Matlab, MathWorks, depuis 1989 I Magma, University of Sydney, depuis 1990 Quelques logiciels libres : I Maxima, W. Schelter et coll., depuis 1967 : Calcul symbolique I Singular, U. Kaiserslautern, depuis 1984 : Polynômes I PARI/GP, U. Bordeaux 1, depuis 1985 : Théorie des nombres I GAP, GAP Group, depuis 1986 : Théorie des groupes I R, U. Auckland, New Zealand, depuis 1993 : Statistiques Deux distinctions importantes I Calcul numérique vs calcul symbolique I Logiciel libre vs logiciel commercial Dans ce cours, présentation de 4 logiciels : I Mathematica : commercial, calcul symbolique I SymPy : libre, calcul symbolique I Geogebra : gratuit, géométrie et algèbre I Calc : gratuit, tableurs Remarque : gratuit 6= libre. -
Linking Global CGE Models with Sectoral Models to Generate Baseline Scenarios: Approaches, Challenges, and Opportunities
Journal of Global Economic Analysis, Volume 5 (2020), No. 1, pp. 162-195. Linking Global CGE models with Sectoral Models to Generate Baseline Scenarios: Approaches, Challenges, and Opportunities BY RUTH DELZEITa, ROBERT BEACHb, RUBEN BIBASc, WOLFGANG BRITZd, JEAN CHATEAUe, FLORIAN FREUNDf, JULIEN LEFEVREg, FRANZISKA SCHUENEMANNh,TIMOTHY SULSERi, HUGO VALINj, BAS VAN RUIJVENk, MATTHIAS WEITZELl, DIRK WILLENBOCKELm, AND KRZYSZTOF WOJTOWICZn a Kiel Institute for the World Economy, Kiellinie 66, 24105 Kiel, Germany (e-mail: [email protected]) b RTI International, 3040 Cornwallis Road, Research Triangle Park, NC, USA (e-mail: [email protected]) c Organisation for Economic Cooperation and Development, 2 Rue André Pascal, 75016 Paris, France (e-mail: [email protected]) d Institute for Food and Resource Economics, University of Bonn, Nussallee 21, 53115 Bonn, Germany (e-mail: [email protected]) e Organisation for Economic Cooperation and Development, 2 Rue André Pascal, 75016 Paris, France (e-mail: [email protected]) f Thünen Institut, Bundesallee 50, 38116 Braunschweig, Germany (e-mail: [email protected]) g Centre international de recherche sur l'environnement et le développement, 45bis Avenue de la Belle Gabrielle, 94130 Nogent-sur-Marne, France (e-mail: jlefevre@centre- cired.fr) h Kiel Institute for the World Economy, Kiellinie 66, 24105 Kiel, Germany (e-mail: [email protected]) i International Food Policy Research Institute, Eye Street, 1201 I St NW, Washington, DC 20005, USA (e-mail: [email protected]) j International Institute for Applied Systems Analysis, Schloßpl. 1, 2361 Laxenburg, Austria (e-mail: [email protected]) k International Institute for Applied Systems Analysis, Schloßpl. -
Collaborative Use of Ketcindy and Free Cass for Making Materials 1
Collaborative Use of KeTCindy and Free CASs for Making Materials Setsuo Takato1, Alasdair McAndrew2, Masataka Kaneko3 1 Toho University, Japan, [email protected] 2 Victoria University, Australia, [email protected] 3 Toho University, Japan, [email protected] 1 Introduction Many mathematics teachers at collegiate level use LATEX to write materials for dis- tribution to their classes. As is well known, LATEX can typeset complex mathe- matical formulas. On the other hand, it has poor ability to create figures. One possibility might be to use a third-party package, such as TiKZ. However, TiKZ coding is complicated and is not easy to read even for the following simple figure. y y = x y = sinx x O Figure 1 A simple example TiKZ can in fact produce figures of great complexity, but its power comes at the cost of a steep learning curve. In order to provide a system for easy creation of publication quality figures, the first author has developed KETpic, the first version of which was released in 2006. KETpic is a macro package of mathematical soft- ware such as Maple, Mathematica, Scilab and R. The recent version uses Scilab mainly and R secondarily. The flow of generating and inserting graphs with KETpic is as follows. 1. KETpic and Scilab commands are listed in a Scilab editor and executed by Scilab. 2. Scilab generates a LATEX file composed of codes for drawing figures. 3. That file can be inserted into a LATEX document with \input command. 4. The document can be compiled to produce a pdf file.