Jump: a Modeling Language for Mathematical Optimization∗
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Multi-Objective Optimization of Unidirectional Non-Isolated Dc/Dcconverters
MULTI-OBJECTIVE OPTIMIZATION OF UNIDIRECTIONAL NON-ISOLATED DC/DC CONVERTERS by Andrija Stupar A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Edward S. Rogers Department of Electrical and Computer Engineering University of Toronto © Copyright 2017 by Andrija Stupar Abstract Multi-Objective Optimization of Unidirectional Non-Isolated DC/DC Converters Andrija Stupar Doctor of Philosophy Graduate Department of Edward S. Rogers Department of Electrical and Computer Engineering University of Toronto 2017 Engineers have to fulfill multiple requirements and strive towards often competing goals while designing power electronic systems. This can be an analytically complex and computationally intensive task since the relationship between the parameters of a system’s design space is not always obvious. Furthermore, a number of possible solutions to a particular problem may exist. To find an optimal system, many different possible designs must be evaluated. Literature on power electronics optimization focuses on the modeling and design of partic- ular converters, with little thought given to the mathematical formulation of the optimization problem. Therefore, converter optimization has generally been a slow process, with exhaus- tive search, the execution time of which is exponential in the number of design variables, the prevalent approach. In this thesis, geometric programming (GP), a type of convex optimization, the execution time of which is polynomial in the number of design variables, is proposed and demonstrated as an efficient and comprehensive framework for the multi-objective optimization of non-isolated unidirectional DC/DC converters. A GP model of multilevel flying capacitor step-down convert- ers is developed and experimentally verified on a 15-to-3.3 V, 9.9 W discrete prototype, with sets of loss-volume Pareto optimal designs generated in under one minute. -
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). -
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.. -
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. -
[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. -
Pysp: Modeling and Solving Stochastic Programs in Python
Noname manuscript No. (will be inserted by the editor) PySP: Modeling and Solving Stochastic Programs in Python Jean-Paul Watson · David L. Woodruff · William E. Hart Received: September 6, 2010. Abstract Although stochastic programming is a powerful tool for modeling decision- making under uncertainty, various impediments have historically prevented its wide- spread use. One key factor involves the ability of non-specialists to easily express stochastic programming problems as extensions of deterministic models, which are often formulated first. A second key factor relates to the difficulty of solving stochastic programming models, particularly the general mixed-integer, multi-stage case. Intri- cate, configurable, and parallel decomposition strategies are frequently required to achieve tractable run-times. We simultaneously address both of these factors in our PySP software package, which is part of the COIN-OR Coopr open-source Python project for optimization. To formulate a stochastic program in PySP, the user speci- fies both the deterministic base model and the scenario tree with associated uncertain parameters in the Pyomo open-source algebraic modeling language. Given these two models, PySP provides two paths for solution of the corresponding stochastic program. The first alternative involves writing the extensive form and invoking a standard deter- ministic (mixed-integer) solver. For more complex stochastic programs, we provide an implementation of Rockafellar and Wets’ Progressive Hedging algorithm. Our particu- lar focus is on the use of Progressive Hedging as an effective heuristic for approximating Jean-Paul Watson Sandia National Laboratories Discrete Math and Complex Systems Department PO Box 5800, MS 1318 Albuquerque, NM 87185-1318 E-mail: [email protected] David L. -
Process Optimization
Process Optimization Mathematical Programming and Optimization of Multi-Plant Operations and Process Design Ralph W. Pike Director, Minerals Processing Research Institute Horton Professor of Chemical Engineering Louisiana State University Department of Chemical Engineering, Lamar University, April, 10, 2007 Process Optimization • Typical Industrial Problems • Mathematical Programming Software • Mathematical Basis for Optimization • Lagrange Multipliers and the Simplex Algorithm • Generalized Reduced Gradient Algorithm • On-Line Optimization • Mixed Integer Programming and the Branch and Bound Algorithm • Chemical Production Complex Optimization New Results • Using one computer language to write and run a program in another language • Cumulative probability distribution instead of an optimal point using Monte Carlo simulation for a multi-criteria, mixed integer nonlinear programming problem • Global optimization Design vs. Operations • Optimal Design −Uses flowsheet simulators and SQP – Heuristics for a design, a superstructure, an optimal design • Optimal Operations – On-line optimization – Plant optimal scheduling – Corporate supply chain optimization Plant Problem Size Contact Alkylation Ethylene 3,200 TPD 15,000 BPD 200 million lb/yr Units 14 76 ~200 Streams 35 110 ~4,000 Constraints Equality 761 1,579 ~400,000 Inequality 28 50 ~10,000 Variables Measured 43 125 ~300 Unmeasured 732 1,509 ~10,000 Parameters 11 64 ~100 Optimization Programming Languages • GAMS - General Algebraic Modeling System • LINDO - Widely used in business applications -
Specifying “Logical” Conditions in AMPL Optimization Models
Specifying “Logical” Conditions in AMPL Optimization Models Robert Fourer AMPL Optimization www.ampl.com — 773-336-AMPL INFORMS Annual Meeting Phoenix, Arizona — 14-17 October 2012 Session SA15, Software Demonstrations Robert Fourer, Logical Conditions in AMPL INFORMS Annual Meeting — 14-17 Oct 2012 — Session SA15, Software Demonstrations 1 New and Forthcoming Developments in the AMPL Modeling Language and System Optimization modelers are often stymied by the complications of converting problem logic into algebraic constraints suitable for solvers. The AMPL modeling language thus allows various logical conditions to be described directly. Additionally a new interface to the ILOG CP solver handles logic in a natural way not requiring conventional transformations. Robert Fourer, Logical Conditions in AMPL INFORMS Annual Meeting — 14-17 Oct 2012 — Session SA15, Software Demonstrations 2 AMPL News Free AMPL book chapters AMPL for Courses Extended function library Extended support for “logical” conditions AMPL driver for CPLEX Opt Studio “Concert” C++ interface Support for ILOG CP constraint programming solver Support for “logical” constraints in CPLEX INFORMS Impact Prize to . Originators of AIMMS, AMPL, GAMS, LINDO, MPL Awards presented Sunday 8:30-9:45, Conv Ctr West 101 Doors close 8:45! Robert Fourer, Logical Conditions in AMPL INFORMS Annual Meeting — 14-17 Oct 2012 — Session SA15, Software Demonstrations 3 AMPL Book Chapters now free for download www.ampl.com/BOOK/download.html Bound copies remain available purchase from usual -
COSMO: a Conic Operator Splitting Method for Convex Conic Problems
COSMO: A conic operator splitting method for convex conic problems Michael Garstka∗ Mark Cannon∗ Paul Goulart ∗ September 10, 2020 Abstract This paper describes the Conic Operator Splitting Method (COSMO) solver, an operator split- ting algorithm for convex optimisation problems with quadratic objective function and conic constraints. At each step the algorithm alternates between solving a quasi-definite linear sys- tem with a constant coefficient matrix and a projection onto convex sets. The low per-iteration computational cost makes the method particularly efficient for large problems, e.g. semidefi- nite programs that arise in portfolio optimisation, graph theory, and robust control. Moreover, the solver uses chordal decomposition techniques and a new clique merging algorithm to ef- fectively exploit sparsity in large, structured semidefinite programs. A number of benchmarks against other state-of-the-art solvers for a variety of problems show the effectiveness of our approach. Our Julia implementation is open-source, designed to be extended and customised by the user, and is integrated into the Julia optimisation ecosystem. 1 Introduction We consider convex optimisation problems in the form minimize f(x) subject to gi(x) 0; i = 1; : : : ; l (1) h (x)≤ = 0; i = 1; : : : ; k; arXiv:1901.10887v2 [math.OC] 9 Sep 2020 i where we assume that both the objective function f : Rn R and the inequality constraint n ! functions gi : R R are convex, and that the equality constraints hi(x) := ai>x bi are ! − affine. We will denote an optimal solution to this problem (if it exists) as x∗. Convex optimisa- tion problems feature heavily in a wide range of research areas and industries, including problems ∗The authors are with the Department of Engineering Science, University of Oxford, Oxford, OX1 3PJ, UK. -
Introduction to Mosek
Introduction to Mosek Modern Optimization in Energy, 28 June 2018 Micha l Adamaszek www.mosek.com MOSEK package overview • Started in 1999 by Erling Andersen • Convex conic optimization package + MIP • LP, QP, SOCP, SDP, other nonlinear cones • Low-level optimization API • C, Python, Java, .NET, Matlab, R, Julia • Object-oriented API Fusion • C++, Python, Java, .NET • 3rd party • GAMS, AMPL, CVXOPT, CVXPY, YALMIP, PICOS, GPkit • Conda package, .NET Core package • Upcoming v9 1 / 10 Example: 2D Total Variation Someone sends you the left signal but you receive noisy f (right): How to denoise/smoothen out/approximate u? P 2 P 2 minimize ij(ui;j − ui+1;j) + ij(ui;j − ui;j+1) P 2 subject to ij(ui;j − fi;j) ≤ σ: 2 / 10 Conic problems A conic problem in canonical form: min cT x s:t: Ax + b 2 K where K is a product of cones: • linear: K = R≥0 • quadratic: q n 2 2 K = fx 2 R : x1 ≥ x2 + ··· + xng • semidefinite: n×n T K = fX 2 R : X = FF g 3 / 10 Conic problems, cont. • exponential cone: 3 K = fx 2 R : x1 ≥ x2 exp(x3=x2); x2 > 0g • power cone: 3 p−1 p K = fx 2 R : x1 x2 ≥ jx3j ; x1; x2 ≥ 0g; p > 1 q 2 2 2 x1 ≥ x2 + x3; 2x1x2 ≥ x3 x1 ≥ x2 exp(x3=x2) 4 / 10 Conic representability Lots of functions and constraints are representable using these cones. T jxj; kxk1; kxk2; kxk1; kAx + bk2 ≤ c x + d !1=p 1 X xy ≥ z2; x ≥ ; x ≥ yp; t ≥ jx jp = kxk y i p i p 1=n t ≤ xy; t ≤ (x1 ··· xn) ; geometric programming (GP) X 1 t ≤ log x; t ≥ ex; t ≤ −x log x; t ≥ log exi ; t ≥ log 1 + x i 1=n det(X) ; t ≤ λmin(X); t ≥ λmax(X) T T convex (1=2)x Qx + c x + q 5 / 10 Challenge Find a • natural, • practical, • important, • convex optimization problem, which cannot be expressed in conic form. -
AIMMS 4 Release Notes
AIMMS 4 Release Notes Visit our web site www.aimms.com for regular updates Contents Contents 2 1 System Requirements 3 1.1 Hardware and operating system requirements ......... 3 1.2 ODBC and OLE DB database connectivity issues ........ 4 1.3 Viewing help files and documentation .............. 5 2 Installation Instructions 7 2.1 Installation instructions ....................... 7 2.2 Solver availability per platform ................... 8 2.3 Aimms licensing ............................ 9 2.3.1 Personal and machine nodelocks ............. 9 2.3.2 Installing an Aimms license ................ 12 2.3.3 Managing Aimms licenses ................. 14 2.3.4 Location of license files ................... 15 2.4 OpenSSL license ............................ 17 3 Project Conversion Instructions 20 3.1 Conversion of projects developed in Aimms 3.13 and before 20 3.2 Conversionof Aimms 3 projects to Aimms 4 ........... 22 3.2.1 Data management changes in Aimms 4.0 ........ 24 4 Getting Support 27 4.1 Reporting a problem ......................... 27 4.2 Known and reported issues ..................... 28 5 Release Notes 29 What’s new in Aimms 4 ........................ 29 Chapter 1 System Requirements This chapter discusses the system requirements necessary to run the various System components of your Win32 Aimms 4 system successfully. When a particular requirements requirement involves the installation of additional system software compo- nents, or an update thereof, the (optional) installation of such components will be part of the Aimms installation procedure. 1.1 Hardware and operating system requirements The following list provides the minimum hardware requirements to run your Hardware Aimms 4 system. requirements 1.6 Ghz or higher x86 or x64 processor XGA display adapter and monitor 1 Gb RAM 1 Gb free disk space Note, however, that performance depends on model size and type and can vary.