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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 -
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Click to edit Master title style APMonitor Modeling Language John Hedengren Brigham Young University Advanced Process Solutions, LLC http://apmonitor.com Overview of APM Software as a service accessible through: MATLAB, Python, Web-browser interface Linux / Windows / Mac OS / Android platforms Solvers 1 1 2 3 3 APOPT , BPOPT , IPOPT , SNOPT , MINOS Problem characteristics: min J (x, y,u, z) Large-scale x s.t. 0 f , x, y,u, z Nonlinear Programming (NLP) t Mixed Integer NLP (MINLP) 0 g(x, y,u, z) Multi-objective 0 h(x, y,u, z) n m m Real-time systems x, y u z Differential Algebraic Equations (DAEs) 1 – APS, LLC 2 – EPL 3 – SBS, Inc. Oct 14, 2012 APMonitor.com Advanced Process Solutions, LLC Overview of APM Vector / matrix algebra with set notation Automatic Differentiation st nd Exact 1 and 2 Derivatives Large-scale, sparse systems of equations Object-oriented access Thermo-physical properties Database of preprogrammed models Parallel processing Optimization with uncertain parameters Custom solver or model connections Oct 14, 2012 APMonitor.com Advanced Process Solutions, LLC Unique Features of APM Initialization with nonlinear presolve minJ(x, y,u) x s.t. 0 f ,x, y,u min J (x, y,u) t 0 g(x, y,u) 0h(x, y,u) x minJ(x, y,u) x s.t. 0 f ,x, y,u s.t. 0 f , x, y,u t 0 g(x, y,u) t 0 h(x, y,u) minJ(x, y,u) x s.t. 0 f ,x, y,u t 0g(x, y,u) 0h(x, y,u) 0 g(x, y,u) minJ(x, y,u) x s.t. -
Notes 1: Introduction to Optimization Models
Notes 1: Introduction to Optimization Models IND E 599 September 29, 2010 IND E 599 Notes 1 Slide 1 Course Objectives I Survey of optimization models and formulations, with focus on modeling, not on algorithms I Include a variety of applications, such as, industrial, mechanical, civil and electrical engineering, financial optimization models, health care systems, environmental ecology, and forestry I Include many types of optimization models, such as, linear programming, integer programming, quadratic assignment problem, nonlinear convex problems and black-box models I Include many common formulations, such as, facility location, vehicle routing, job shop scheduling, flow shop scheduling, production scheduling (min make span, min max lateness), knapsack/multi-knapsack, traveling salesman, capacitated assignment problem, set covering/packing, network flow, shortest path, and max flow. IND E 599 Notes 1 Slide 2 Tentative Topics Each topic is an introduction to what could be a complete course: 1. basic linear models (LP) with sensitivity analysis 2. integer models (IP), such as the assignment problem, knapsack problem and the traveling salesman problem 3. mixed integer formulations 4. quadratic assignment problems 5. include uncertainty with chance-constraints, stochastic programming scenario-based formulations, and robust optimization 6. multi-objective formulations 7. nonlinear formulations, as often found in engineering design 8. brief introduction to constraint logic programming 9. brief introduction to dynamic programming IND E 599 Notes 1 Slide 3 Computer Software I Catalyst Tools (https://catalyst.uw.edu/) I AIMMS - optimization software (http://www.aimms.com/) Ming Fang - AIMMS software consultant IND E 599 Notes 1 Slide 4 What is Mathematical Programming? Mathematical programming refers to \programming" as a \planning" activity: as in I linear programming (LP) I integer programming (IP) I mixed integer linear programming (MILP) I non-linear programming (NLP) \Optimization" is becoming more common, e.g. -
Treball (1.484Mb)
Treball Final de Màster MÀSTER EN ENGINYERIA INFORMÀTICA Escola Politècnica Superior Universitat de Lleida Mòdul d’Optimització per a Recursos del Transport Adrià Vall-llaura Salas Tutors: Antonio Llubes, Josep Lluís Lérida Data: Juny 2017 Pròleg Aquest projecte s’ha desenvolupat per donar solució a un problema de l’ordre del dia d’una empresa de transports. Es basa en el disseny i implementació d’un model matemàtic que ha de permetre optimitzar i automatitzar el sistema de planificació de viatges de l’empresa. Per tal de poder implementar l’algoritme s’han hagut de crear diversos mòduls que extreuen les dades del sistema ERP, les tracten, les envien a un servei web (REST) i aquest retorna un emparellament òptim entre els vehicles de l’empresa i les ordres dels clients. La primera fase del projecte, la teòrica, ha estat llarga en comparació amb les altres. En aquesta fase s’ha estudiat l’estat de l’art en la matèria i s’han repassat molts dels models més importants relacionats amb el transport per comprendre’n les seves particularitats. Amb els conceptes ben estudiats, s’ha procedit a desenvolupar un nou model matemàtic adaptat a les necessitats de la lògica de negoci de l’empresa de transports objecte d’aquest treball. Posteriorment s’ha passat a la fase d’implementació dels mòduls. En aquesta fase m’he trobat amb diferents limitacions tecnològiques degudes a l’antiguitat de l’ERP i a l’ús del sistema operatiu Windows. També han sorgit diferents problemes de rendiment que m’han fet redissenyar l’extracció de dades de l’ERP, el càlcul de distàncies i el mòdul d’optimització. -
GEKKO Documentation Release 1.0.1
GEKKO Documentation Release 1.0.1 Logan Beal, John Hedengren Aug 31, 2021 Contents 1 Overview 1 2 Installation 3 3 Project Support 5 4 Citing GEKKO 7 5 Contents 9 6 Overview of GEKKO 89 Index 91 i ii CHAPTER 1 Overview GEKKO is a Python package for machine learning and optimization of mixed-integer and differential algebraic equa- tions. It is coupled with large-scale solvers for linear, quadratic, nonlinear, and mixed integer programming (LP, QP, NLP, MILP, MINLP). Modes of operation include parameter regression, data reconciliation, real-time optimization, dynamic simulation, and nonlinear predictive control. GEKKO is an object-oriented Python library to facilitate local execution of APMonitor. More of the backend details are available at What does GEKKO do? and in the GEKKO Journal Article. Example applications are available to get started with GEKKO. 1 GEKKO Documentation, Release 1.0.1 2 Chapter 1. Overview CHAPTER 2 Installation A pip package is available: pip install gekko Use the —-user option to install if there is a permission error because Python is installed for all users and the account lacks administrative priviledge. The most recent version is 0.2. You can upgrade from the command line with the upgrade flag: pip install--upgrade gekko Another method is to install in a Jupyter notebook with !pip install gekko or with Python code, although this is not the preferred method: try: from pip import main as pipmain except: from pip._internal import main as pipmain pipmain(['install','gekko']) 3 GEKKO Documentation, Release 1.0.1 4 Chapter 2. Installation CHAPTER 3 Project Support There are GEKKO tutorials and documentation in: • GitHub Repository (examples folder) • Dynamic Optimization Course • APMonitor Documentation • GEKKO Documentation • 18 Example Applications with Videos For project specific help, search in the GEKKO topic tags on StackOverflow. -
OPTIMAL CONTROL of NONHOLONOMIC MECHANICAL SYSTEMS By
OPTIMAL CONTROL OF NONHOLONOMIC MECHANICAL SYSTEMS by Stuart Marcus Rogers A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics Department of Mathematical and Statistical Sciences University of Alberta ⃝c Stuart Marcus Rogers, 2017 Abstract This thesis investigates the optimal control of two nonholonomic mechanical systems, Suslov's problem and the rolling ball. Suslov's problem is a nonholonomic variation of the classical rotating free rigid body problem, in which the body angular velocity Ω(t) must always be orthogonal to a prescribed, time-varying body frame vector ξ(t), i.e. hΩ(t); ξ(t)i = 0. The motion of the rigid body in Suslov's problem is actuated via ξ(t), while the motion of the rolling ball is actuated via internal point masses that move along rails fixed within the ball. First, by applying Lagrange-d'Alembert's principle with Euler-Poincar´e'smethod, the uncontrolled equations of motion are derived. Then, by applying Pontryagin's minimum principle, the controlled equations of motion are derived, a solution of which obeys the uncontrolled equations of motion, satisfies prescribed initial and final conditions, and minimizes a prescribed performance index. Finally, the controlled equations of motion are solved numerically by a continuation method, starting from an initial solution obtained analytically (in the case of Suslov's problem) or via a direct method (in the case of the rolling ball). ii Preface This thesis contains material that has appeared in a pair of papers, one on Suslov's problem and the other on rolling ball robots, co-authored with my supervisor, Vakhtang Putkaradze. -
Clean Sky – Awacs Final Report
Clean Sky Joint Undertaking AWACs – Adaptation of WORHP to Avionics Constraints Clean Sky – AWACs Final Report Table of Contents 1 Executive Summary ......................................................................................................................... 2 2 Summary Description of Project Context and Objectives ............................................................... 3 3 Description of the main S&T results/foregrounds .......................................................................... 5 3.1 Analysis of Optimisation Environment .................................................................................... 5 3.1.1 Problem sparsity .............................................................................................................. 6 3.1.2 Certification aspects ........................................................................................................ 6 3.2 New solver interfaces .............................................................................................................. 6 3.3 Conservative iteration mode ................................................................................................... 6 3.4 Robustness to erroneous inputs ............................................................................................. 7 3.5 Direct multiple shooting .......................................................................................................... 7 3.6 Grid refinement ...................................................................................................................... -
Optimal Control for Constrained Hybrid System Computational Libraries and Applications
FINAL REPORT: FEUP2013.LTPAIVA.01 Optimal Control for Constrained Hybrid System Computational Libraries and Applications L.T. Paiva 1 1 Department of Electrical and Computer Engineering University of Porto, Faculty of Engineering - Rua Dr. Roberto Frias, s/n, 4200–465 Porto, Portugal ) [email protected] % 22 508 1450 February 28, 2013 Abstract This final report briefly describes the work carried out under the project PTDC/EEA- CRO/116014/2009 – “Optimal Control for Constrained Hybrid System”. The aim was to build and maintain a software platform to test an illustrate the use of the conceptual tools developed during the overall project: not only in academic examples but also in case studies in the areas of robotics, medicine and exploitation of renewable resources. The grand holder developed a critical hands–on knowledge of the available optimal control solvers as well as package based on non–linear programming solvers. 1 2 Contents 1 OC & NLP Interfaces 7 1.1 Introduction . .7 1.2 AMPL . .7 1.3 ACADO – Automatic Control And Dynamic Optimization . .8 1.4 BOCOP – The optimal control solver . .9 1.5 DIDO – Automatic Control And Dynamic Optimization . 10 1.6 ICLOCS – Imperial College London Optimal Control Software . 12 1.7 TACO – Toolkit for AMPL Control Optimization . 12 1.8 Pseudospectral Methods in Optimal Control . 13 2 NLP Solvers 17 2.1 Introduction . 17 2.2 IPOPT – Interior Point OPTimizer . 17 2.3 KNITRO . 25 2.4 WORHP – WORHP Optimises Really Huge Problems . 31 2.5 Other Commercial Packages . 33 3 Project webpage 35 4 Optimal Control Toolbox 37 5 Applications 39 5.1 Car–Like . -
Using SCIP to Solve Your Favorite Integer Optimization Problem
Using SCIP to Solve Your Favorite Integer Optimization Problem Gregor Hendel, [email protected] Hiroshima University October 5, 2018 Gregor Hendel, [email protected] – Using SCIP 1/76 What is SCIP? SCIP (Solving Constraint Integer Programs) … • provides a full-scale MIP and MINLP solver, • is constraint based, • incorporates • MIP features (cutting planes, LP relaxation), and • MINLP features (spatial branch-and-bound, OBBT) • CP features (domain propagation), • SAT-solving features (conflict analysis, restarts), • is a branch-cut-and-price framework, • has a modular structure via plugins, • is free for academic purposes, • and is available in source-code under http://scip.zib.de ! Gregor Hendel, [email protected] – Using SCIP 2/76 Meet the SCIP Team 31 active developers • 7 running Bachelor and Master projects • 16 running PhD projects • 8 postdocs and professors 4 development centers in Germany • ZIB: SCIP, SoPlex, UG, ZIMPL • TU Darmstadt: SCIP and SCIP-SDP • FAU Erlangen-Nürnberg: SCIP • RWTH Aachen: GCG Many international contributors and users • more than 10 000 downloads per year from 100+ countries Careers • 10 awards for Masters and PhD theses: MOS, EURO, GOR, DMV • 7 former developers are now building commercial optimization sotware at CPLEX, FICO Xpress, Gurobi, MOSEK, and GAMS Gregor Hendel, [email protected] – Using SCIP 3/76 Outline SCIP – Solving Constraint Integer Programs http://scip.zib.de Gregor Hendel, [email protected] – Using SCIP 4/76 Outline SCIP – Solving Constraint Integer Programs http://scip.zib.de Gregor Hendel, [email protected] – Using SCIP 5/76 ZIMPL • model and generate LPs, MIPs, and MINLPs SCIP • MIP, MINLP and CIP solver, branch-cut-and-price framework SoPlex • revised primal and dual simplex algorithm GCG • generic branch-cut-and-price solver UG • framework for parallelization of MIP and MINLP solvers SCIP Optimization Suite • Toolbox for generating and solving constraint integer programs, in particular Mixed Integer (Non-)Linear Programs. -
Component-Oriented Acausal Modeling of the Dynamical Systems in Python Language on the Example of the Model of the Sucker Rod String
Component-oriented acausal modeling of the dynamical systems in Python language on the example of the model of the sucker rod string Volodymyr B. Kopei, Oleh R. Onysko and Vitalii G. Panchuk Department of Computerized Mechanical Engineering, Ivano-Frankivsk National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine ABSTRACT Typically, component-oriented acausal hybrid modeling of complex dynamic systems is implemented by specialized modeling languages. A well-known example is the Modelica language. The specialized nature, complexity of implementation and learning of such languages somewhat limits their development and wide use by developers who know only general-purpose languages. The paper suggests the principle of developing simple to understand and modify Modelica-like system based on the general-purpose programming language Python. The principle consists in: (1) Python classes are used to describe components and their systems, (2) declarative symbolic tools SymPy are used to describe components behavior by difference or differential equations, (3) the solution procedure uses a function initially created using the SymPy lambdify function and computes unknown values in the current step using known values from the previous step, (4) Python imperative constructs are used for simple events handling, (5) external solvers of differential-algebraic equations can optionally be applied via the Assimulo interface, (6) SymPy package allows to arbitrarily manipulate model equations, generate code and solve some equations symbolically. The basic set of mechanical components (1D translational “mass”, “spring-damper” and “force”) is developed. The models of a sucker rods string are developed and simulated using these components. The comparison of results of the sucker rod string simulations with practical dynamometer cards and Submitted 22 March 2019 Accepted 24 September 2019 Modelica results verify the adequacy of the models. -
On Scaled Stopping Criteria for a Safeguarded Augmented Lagrangian Method with Theoretical Guarantees
Noname manuscript No. (will be inserted by the editor) On scaled stopping criteria for a safeguarded augmented Lagrangian method with theoretical guarantees R. Andreani · G. Haeser · M. L. Schuverdt · L. D. Secchin · P. J. S. Silva Received: date / Accepted: date Abstract This paper discusses the use of a stopping criterion based on the scaling of the Karush-Kuhn-Tucker (KKT) conditions by the norm of the ap- proximate Lagrange multiplier in the ALGENCAN implementation of a safe- guarded augmented Lagrangian method. Such stopping criterion is already used in several nonlinear programming solvers, but it has not yet been con- sidered in ALGENCAN due to its firm commitment with finding a true KKT point even when the multiplier set is not bounded. In contrast with this view, we present a strong global convergence theory under the quasi-normality con- straint qualification, that allows for unbounded multiplier sets, accompanied by an extensive numerical test which shows that the scaled stopping criterion is more efficient in detecting convergence sooner. In particular, by scaling, ALGENCAN is able to recover a solution in some difficult problems where the original implementation fails, while the behavior of the algorithm in the easier instances is maintained. Furthermore, we show that, in some cases, a R. Andreani · P. J. S. Silva Department of Applied Mathematics, University of Campinas, Rua S´ergio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil. E-mail: [email protected], [email protected] G. Haeser Department of Applied Mathematics, University of S~aoPaulo, Rua do Mat~ao, 1010, Cidade Universit´aria,05508-090, S~aoPaulo, SP, Brazil. -
A Case Study to Help You Think About Your Project
Wind Turbine Optimization a case study to help you think about your project Andrew Ning ME 575 Three broad areas of optimization (with some overlap) Convex Gradient-Based Gradient-Free Today we will focus on gradient-based, but for some projects the other areas are more important. In higher dimensional space, analytic gradients become increasingly important #iterations # design vars There is a difference between developing for analysis vs optimization Vy(1 + a0) plane of rotation φ V (1 a) W x − There is a difference between developing for analysis vs optimization Vy(1 + a0) plane of rotation φ V (1 a) W x − 1 a 2 C = − c σ0 CT =4a(1 a) T sin φ n − ✓ ◆ 2 1+a0 2 2 C = c σ0λ CQ =4a0(1 + a0) tan φλ Q cos φ t r r ✓ ◆ There is a difference between developing for analysis vs optimization Vy(1 + a0) plane of rotation φ V (1 a) W x − 1 1 a = 2 a0 = 4sin φ +1 4sinφ cos φ 1 cnσ0 ctσ0 − There is a difference between developing for analysis vs optimization use Parameters y use ProgGen Converg Skew correction use WTP_Data n CALL GetCoefs Declarations reset induction !Converg & Return Initialize induction vars < MaxIter CALL FindZC CALL NewtRaph reset induction Initial guess (AxInd / TanInd) n & y CALL NewtRaph MaxIter ZFound Converg CALL BinSearch < Converg ! y n AxIndLo = AxInd -1 n AxIndHi = AxInd +1 AxIndLo = -0.5 Converg AxIndHi = 0.6 CALL BinSearch y n Skew correction Converg AxIndLo = -1.0 CALL BinSearch AxIndHi = -0.4 y CALL GetCoefs n Converg AxIndLo = 0.59 CALL BinSearch AxIndHi = 2.5 Return y y Converg Skew correction CALL GetCoefs Return n AxIndLo = -1.0 CALL BinSearch AxIndHi = 2.5 y Converg Skew correction CALL GetCoefs Return n Return Sometimes a new solution approach is needed… Vy(1 + a0) plane of rotation φ Vx(1 a) W − sin φ V cos φ (φ)= x =0 R 1 a − V (1 + a ) − y 0 Sometimes a new solution approach is needed… Algorithm Avg.