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  T ECHNICAL  R EPORT DOTcvp: Dynamic Optimization Toolbox with Control Vector Parameterization approach for handling continuous and mixed-integerdynamic optimization problems  Authors: Tomáš Hirmajer a , Eva Balsa-Canto, and Julio R. Banga  E-mails: thirmajer@gmail.comebalsa@iim.csic.es julio@iim.csic.es Web page of the project: http://www.iim.csic.es/~dotcvp/ I NSTITUTO DE  I NVESTIGACIONES  M ARINAS , IIM-CSIC Process Engineering GroupSpanish Council for Scientific ResearchC/Eduardo Cabello 6, 36208 Vigo, Spain a Corresponding author January 9, 2010  DOTcvp: Dynamic Optimization Toolbox with CVP approach for handling continuous and mixed-integer DO problems Abstract: This report deals with the description of the dynamic optimization toolbox (DOTcvp) which is able to solveconstrained optimal control problems (OCP) with the control vector parameterization (CVP) approach. Thesrcinal continuous problem is transformed into the finite dimensional OCP. Then it is possible to solve theresulting nonlinear programming (NLP) problem with the help of any gradient method with the combination of initial value problem (IVP) solver. For all of this is used SUNDIALS tool which was modified for automaticgeneration of the necessary gradients in two ways. One of them is a finite difference and the second one, a moreaccurate method, is the sensitivity equations approach.The environment for several mixed-integer (MI) or only NLP solvers, deterministic, stochastic, and hybrid ablehandling continuous and mixed-integer variables was implemented in order to ensure higher accuracy not onlyfor small problems. It is assumed, that the optimized constrained or unconstrained system is described by a setof ordinary differential equations (ODEs). All of this is covered by MATLAB environment, which offers to usa comfortable control and a nice graphical output. The toolbox allows the combination of MATLAB with FOR-TRAN for the sake of efficiency FORTRAN, for enabling this option it is needed to have installed a FORTRANcompiler in a MATLAB environment.Key features:ã handling of a wide class of dynamic optimization problems, including constrained, unconstrained, fixed,and free terminal time problems described by ordinary differential equations (ODEs), as well as continuousand mixed integer decision variables;ã the inner initial value problem (IVP) is solved using the state-of-the-art methods available in SUNDIALS[24];ã theouter(MI)NLPproblemcanbesolvedusinganumberofadvancedsolvers, including localdeterministicmethods ,  stochastic global optimization methods , and  hybrid metaheuristics ;ã in addition to the traditional single optimization approach, the toolbox also offers more sophisticated strate-gies, like multistart, sucessive re-optimization [2], and hybrid strategies [4]; ã a graphical user interface (GUI) which makes the definition and edition of a problem more easy and clear;ã many output options for the results, including detailed figures. Keywords:  Dynamic optimization toolbox, control vector parameterization, sensitivity equations, ordinary dif-ferential equations, MATLAB Page – 1  Main Web Page It is possible to get the basic information about the toolbox from the following web page: http://www.iim.csic.es/~dotcvp/ The Toolbox Requirements MATLAB 7.0 or later is required, and the MATLAB Optimisation Toolbox and Symbolic Math Toolbox are highlyrecommended. The toolbox distribution includes most of the needed external solvers: IVP solver CVODE (partof SUNDIALS suite), and (MI)NLP solvers ACOmi, DE, IPOPT, MISQP, MITS and SRES. The OptimizationToolbox is needed if the user wants to use FMINCON as a NLP solver. FORTRAN compilation to speed-upcomputations is secured by a combination of gnumex and MinGW, packages which are distributed with thetoolbox as well. On the other hand, the Symbolic Math Toolbox is needed if automatic generation of sensitivitiesand Jacobian are desired (recommended). For more information about the MATLAB MEX files, please have alook on the following web link from the Mathworks web page: http://www.mathworks.com/support/tech-notes/1600/1605.html License The DOTcvp toolbox is completely free of charge under the creative commons license. It is possible to find theconditions of the license on the following web page: http://creativecommons.org/licenses/by-nc-nd/3.0/ Reference If you are using our toolbox to obtain results for a publication, we politely ask you for your support by citation of the following report: @TechReport{dotcvp,author = {Hirmajer, T. and Balsa-Canto, E. and Banga, J. R.},title = {{DOT}cvp: Dynamic Optimization Toolbox with CVP approach for handling continuous and mixed-integer DO problems},institution = {Instituto de Investigaciones Marinas, IIM CSIC},address = {C/Eduardo Cabello 6, 36208 Vigo, Spain},year = {2008},month = {4},note = {Available at: www.iim.csic.es/$\sim$dotcvp/}} T. HIRMAJER, E. BALSA-CANTO, AND J. R. BANGA,  DOTcvp: Dynamic optimization toolbox with controlvector parameterization approach for handling continuous and mixed-integer dynamic optimization problems ,Technical Report, Instituto de Investigaciones Marinas - CSIC, C/Eduardo Cabello 6, 36208 Vigo, Spain, 2008. Acknowledgments The authors would like to thank the whole CSIC group for their help with the implementation of the presenttoolbox, especially to José Alberto Egea Larrosa. Author Tomáš Hirmajer also thanks to Miroslav Fikar fromInstitute of Information Engineering, Automation, and Mathematics, Faculty of Chemical and Food Technology,Slovak University of Technology for several suggestions that have led to the improvement of the presentedtoolbox. The authors also would like to thank other people who contribute with their comments or ideas: MariánPodmajerský; Feedback If you find any problems, or you have only comments, questions, please do not hesitate to contact us!THANK YOU FOR USING THE DOTCVP TOOLBOX!  DOTcvp: Dynamic Optimization Toolbox with CVP approach for handling continuous and mixed-integer DO problems Contents 1 Introduction 8 1.1 Toolbox description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Description of main modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.1 Utility modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.2 Optimization modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Numerical optimization methods (NLP and MINLP solvers) . . . . . . . . . . . . . . . . . . . . 91.4 Numerical simulation method (IVP solvers) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Recommended operating procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Toolbox download . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.7 Toolbox installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Optimal control problem 13 2.1 System and cost description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Control vector parametrization 14 3.1 NLP formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Implemented gradient methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.1 Sensitivity equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.2 Finite differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Gradients with respect to time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Algorithm outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5 Brief information about the implemented modules . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5.1 Single optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5.2 Hybrid optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5.3 Sucessive re-optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5.4 Simulation module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Single optimization 18 4.1 From the ’input’ to the ’output’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1.1 Problem formulation for DOTcvp, a simple input file . . . . . . . . . . . . . . . . . . . . 184.1.2 Problem formulation for DOTcvp, a regular input file . . . . . . . . . . . . . . . . . . . . 194.1.3 Initialization, final results, and optimal trajectories . . . . . . . . . . . . . . . . . . . . . 22 5 Sucessive re-optimization 24 5.1 Application of the mesh refinement algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6 GUI for DOTcvp 26 6.1 Step by step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Page – 3

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