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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 . -
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. -
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.. -
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. -
TOMLAB –Unique Features for Optimization in MATLAB
TOMLAB –Unique Features for Optimization in MATLAB Bad Honnef, Germany October 15, 2004 Kenneth Holmström Tomlab Optimization AB Västerås, Sweden [email protected] ´ Professor in Optimization Department of Mathematics and Physics Mälardalen University, Sweden http://tomlab.biz Outline of the talk • The TOMLAB Optimization Environment – Background and history – Technology available • Optimization in TOMLAB • Tests on customer supplied large-scale optimization examples • Customer cases, embedded solutions, consultant work • Business perspective • Box-bounded global non-convex optimization • Summary http://tomlab.biz Background • MATLAB – a high-level language for mathematical calculations, distributed by MathWorks Inc. • Matlab can be extended by Toolboxes that adds to the features in the software, e.g.: finance, statistics, control, and optimization • Why develop the TOMLAB Optimization Environment? – A uniform approach to optimization didn’t exist in MATLAB – Good optimization solvers were missing – Large-scale optimization was non-existent in MATLAB – Other toolboxes needed robust and fast optimization – Technical advantages from the MATLAB languages • Fast algorithm development and modeling of applied optimization problems • Many built in functions (ODE, linear algebra, …) • GUIdevelopmentfast • Interfaceable with C, Fortran, Java code http://tomlab.biz History of Tomlab • President and founder: Professor Kenneth Holmström • The company founded 1986, Development started 1989 • Two toolboxes NLPLIB och OPERA by 1995 • Integrated format for optimization 1996 • TOMLAB introduced, ISMP97 in Lausanne 1997 • TOMLAB v1.0 distributed for free until summer 1999 • TOMLAB v2.0 first commercial version fall 1999 • TOMLAB starting sales from web site March 2000 • TOMLAB v3.0 expanded with external solvers /SOL spring 2001 • Dash Optimization Ltd’s XpressMP added in TOMLAB fall 2001 • Tomlab Optimization Inc. -
Click to Edit Master Title Style
Click to edit Master title style MINLP with Combined Interior Point and Active Set Methods Jose L. Mojica Adam D. Lewis John D. Hedengren Brigham Young University INFORM 2013, Minneapolis, MN Presentation Overview NLP Benchmarking Hock-Schittkowski Dynamic optimization Biological models Combining Interior Point and Active Set MINLP Benchmarking MacMINLP MINLP Model Predictive Control Chiller Thermal Energy Storage Unmanned Aerial Systems Future Developments Oct 9, 2013 APMonitor.com APOPT.com Brigham Young University Overview of Benchmark Testing NLP Benchmark Testing 1 1 2 3 3 min J (x, y,u) APOPT , BPOPT , IPOPT , SNOPT , MINOS x Problem characteristics: s.t. 0 f , x, y,u t Hock Schittkowski, Dynamic Opt, SBML 0 g(x, y,u) Nonlinear Programming (NLP) Differential Algebraic Equations (DAEs) 0 h(x, y,u) n m APMonitor Modeling Language x, y u MINLP Benchmark Testing min J (x, y,u, z) 1 1 2 APOPT , BPOPT , BONMIN x s.t. 0 f , x, y,u, z Problem characteristics: t MacMINLP, Industrial Test Set 0 g(x, y,u, z) Mixed Integer Nonlinear Programming (MINLP) 0 h(x, y,u, z) Mixed Integer Differential Algebraic Equations (MIDAEs) x, y n u m z m APMonitor & AMPL Modeling Language 1–APS, LLC 2–EPL, 3–SBS, Inc. Oct 9, 2013 APMonitor.com APOPT.com Brigham Young University NLP Benchmark – Summary (494) 100 90 80 APOPT+BPOPT APOPT 70 1.0 BPOPT 1.0 60 IPOPT 3.10 IPOPT 50 2.3 SNOPT Percentage (%) 6.1 40 Benchmark Results MINOS 494 Problems 5.5 30 20 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Not worse than 2 times slower than -
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. -
IBM Research Report Branch and Bound, Integer and Non-Integer
RC22890 (W0309-010) September 3, 2003 Mathematics IBM Research Report Branch and Bound, Integer and Non-Integer Programming J. J. H. Forrest IBM Research Division Thomas J. Watson Research Center P.O. Box 218 Yorktown Heights, NY 10598 J. A. Tomlin IBM Research Division Almaden Research Center 650 Harry Road San Jose, CA 95120-6099 Research Division Almaden - Austin - Beijing - Delhi - Haifa - India - T. J. Watson - Tokyo - Zurich LIMITED DISTRIBUTION NOTICE: This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g. , payment of royalties). Copies may be requested from IBM T. J. Watson Research Center , P. O. Box 218, Yorktown Heights, NY 10598 USA (email: [email protected]). Some reports are available on the internet at http://domino.watson.ibm.com/library/CyberDig.nsf/home . Branch and Bound, Integer, and Non-Integer Programming J.J.H. Forrest IBM Watson Research Center, Yorktown Heights, NY 10598 J. A. Tomlin IBM Almaden Research Center, San Jose, CA 95120 {[email protected],[email protected]} Abstract In this note we review the development of the first commercial branch and bound codes at CEIR/Scicon, with particular emphasis on those features not commonly emphasized today—that is the use of branch and bound to handle discrete constructs, which, while often formally representable in terms of zero-one integer variables, can be handled more elegantly and effectively by applying branch and bound to the actual dichotomies in the model. -
Robert Fourer, David M. Gay INFORMS Annual Meeting
AMPL New Solver Support in the AMPL Modeling Language Robert Fourer, David M. Gay AMPL Optimization LLC, www.ampl.com Department of Industrial Engineering & Management Sciences, Northwestern University Sandia National Laboratories INFORMS Annual Meeting Pittsburgh, Pennsylvania, November 5-8, 2006 Robert Fourer, New Solver Support in the AMPL Modeling Language INFORMS Annual Meeting, November 5-8, 2006 1 AMPL What it is . A modeling language for optimization A system to support optimization modeling What I’ll cover . Brief examples, briefer history Recent developments . ¾ 64-bit versions ¾ floating license manager ¾ AMPL Studio graphical interface & Windows COM objects Solver support ¾ new KNITRO 5.1 features ¾ new CPLEX 10.1 features Robert Fourer, New Solver Support in the AMPL Modeling Language INFORMS Annual Meeting, November 5-8, 2006 2 Ex 1: Airline Fleet Assignment set FLEETS; set CITIES; set TIMES circular; set FLEET_LEGS within {f in FLEETS, c1 in CITIES, t1 in TIMES, c2 in CITIES, t2 in TIMES: c1 <> c2 and t1 <> t2}; # (f,c1,t1,c2,t2) represents the availability of fleet f # to cover the leg that leaves c1 at t1 and # whose arrival time plus turnaround time at c2 is t2 param leg_cost {FLEET_LEGS} >= 0; param fleet_size {FLEETS} >= 0; Robert Fourer, New Solver Support in the AMPL Modeling Language INFORMS Annual Meeting, November 5-8, 2006 3 Ex 1: Derived Sets set LEGS := setof {(f,c1,t1,c2,t2) in FLEET_LEGS} (c1,t1,c2,t2); # the set of all legs that can be covered by some fleet set SERV_CITIES {f in FLEETS} := union {(f,c1,c2,t1,t2) -
Julia: a Modern Language for Modern ML
Julia: A modern language for modern ML Dr. Viral Shah and Dr. Simon Byrne www.juliacomputing.com What we do: Modernize Technical Computing Today’s technical computing landscape: • Develop new learning algorithms • Run them in parallel on large datasets • Leverage accelerators like GPUs, Xeon Phis • Embed into intelligent products “Business as usual” will simply not do! General Micro-benchmarks: Julia performs almost as fast as C • 10X faster than Python • 100X faster than R & MATLAB Performance benchmark relative to C. A value of 1 means as fast as C. Lower values are better. A real application: Gillespie simulations in systems biology 745x faster than R • Gillespie simulations are used in the field of drug discovery. • Also used for simulations of epidemiological models to study disease propagation • Julia package (Gillespie.jl) is the state of the art in Gillespie simulations • https://github.com/openjournals/joss- papers/blob/master/joss.00042/10.21105.joss.00042.pdf Implementation Time per simulation (ms) R (GillespieSSA) 894.25 R (handcoded) 1087.94 Rcpp (handcoded) 1.31 Julia (Gillespie.jl) 3.99 Julia (Gillespie.jl, passing object) 1.78 Julia (handcoded) 1.2 Those who convert ideas to products fastest will win Computer Quants develop Scientists prepare algorithms The last 25 years for production (Python, R, SAS, DEPLOY (C++, C#, Java) Matlab) Quants and Computer Compress the Scientists DEPLOY innovation cycle collaborate on one platform - JULIA with Julia Julia offers competitive advantages to its users Julia is poised to become one of the Thank you for Julia. Yo u ' v e k i n d l ed leading tools deployed by developers serious excitement.