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CasADi: a software framework for nonlinear optimization and optimal control

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Abstract

We present CasADi, an open-source software framework for numerical optimization. CasADi is a general-purpose tool that can be used to model and solve optimization problems with a large degree of flexibility, larger than what is associated with popular algebraic modeling languages such as AMPL, GAMS, JuMP or Pyomo. Of special interest are problems constrained by differential equations, i.e. optimal control problems. CasADi is written in self-contained C++, but is most conveniently used via full-featured interfaces to Python, MATLAB or Octave. Since its inception in late 2009, it has been used successfully for academic teaching as well as in applications from multiple fields, including process control, robotics and aerospace. This article gives an up-to-date and accessible introduction to the CasADi framework, which has undergone numerous design improvements over the last 7 years.

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Notes

  1. One may run these snippets on a Windows, Linux or Mac platform by following the install instructions for CasADi 3.1 at http://install31.casadi.org/.

  2. A paper draft is available at http://www.optimization-online.org/DB_HTML/2018/05/6642.html.

  3. Support for finite differences, with automatic step-size selection, was added in CasADi 3.3.

  4. Cf. http://live.casadi.org (Python) and http://live-octave.casadi.org (Octave).

  5. Two sparse direct linear solvers based on LDLT (for symmetric positive definite systems) and QR (for general sparse matrices), with support for C code generation, were added in CasADi 3.3.

  6. A new primal-dual active-set method for quadratic programming was added in CasADi 3.4.

  7. Support for automatic sensitivity analysis for NLPs was added in CasADi 3.4.

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Acknowledgements

The authors thank for the generous support that has made this work possible. In particular: the K.U. Leuven Research Council via CoE EF/05/006 Optimization in Engineering (OPTEC); the Flemish Government via FWO; the Belgian State via Science Policy programming (IAP VII, DYSCO); the European Union via HDMPC (223854), EMBOCON (248940), HIGHWIND (259166), TEMPO (607957), AWESCO (642682); the Helmholtz Association via vICERP; the German Federal Ministry for Economic Affairs and Energy (BMWi) via projects eco4wind and DyConPV; the German Research Foundation (DFG) via Research Unit FOR 2401; Flanders Make via MBSE4M, Drivetrain Co-design, Conceptdesign. We also thank our industrial partners, including GE Global Research and Johnson Controls International Inc. Finally, we thank the reviewers for valuable comments that helped to improve the final manuscript.

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Authors and Affiliations

  1. Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison, USA

    Joel A. E. Andersson & James B. Rawlings

  2. MECO Research Team, Department Mechanical Engineering, KU Leuven, Leuven, Belgium

    Joris Gillis

  3. DMMS Lab, Flanders Make, Leuven, Belgium

    Joris Gillis

  4. Kitty Hawk, Mountain View, USA

    Greg Horn

  5. Department of Microsystems Engineering IMTEK, University of Freiburg, Freiburg, Germany

    Moritz Diehl

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  1. Joel A. E. Andersson
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  2. Joris Gillis
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  3. Greg Horn
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  4. James B. Rawlings
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  5. Moritz Diehl
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Correspondence to Joel A. E. Andersson.

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The software reviewed as part of this submission was given the digital object identifier (DOI) https://doi.org/10.5281/zenodo.1257968.

Summary of main features of CasADi, version 3.4.4

Summary of main features of CasADi, version 3.4.4

In the following is a summary of the main features of CasADi 3.4.4, which was released in May 2018, after the acceptance of this paper. Newly added features, not present in CasADi 3.1 covered so far, are marked with footnotes. More details about these features can be found on the CasADi webpage, http://casadi.org, and in the paper Sensitivity Analysis for Nonlinear Programming in CasADi, which is currently in preparation.Footnote 2

1.1 Symbolic framework with algorithmic differentiation (AD)

A state-of-the-art implementation of algorithmic differentiation (AD), implemented within a symbolic framework, forms the backbone of CasADi. Users construct directed acyclic expression graphs using an everything-is-a-sparse-matrix syntax and expressions for derivatives are generated automatically using AD via source-code-transformation. CasADi implements the forward and reverse modes of AD and uses a graph coloring approach to construct large-and-sparse Jacobians and Hessians. Generated expressions are encapsulated in function objects that can be evaluated, numerically or symbolically, in virtual machines (VMs).

Compared to similar frameworks, CasADi scales well to higher dimensions, and offers a rich set of differentiable operations. Supported operations include common matrix-valued operations, serial or parallel function calls, (non)linear systems of equations, initial-value problems in ordinary differential equations (ODE) or differential-algebraic equations (DAE) and spline-based lookup tables. External code can be embedded with derivative information either user-provided or approximated by finite differences.Footnote 3

1.2 Core self-containment, auto-generated front-ends via SWIG

The symbolic core of CasADi is written in modern C++, with no external dependencies. While C++ offers great interoperability with other tools, high performance and multi-platform support, it lacks the interactivity and ease-of-use associated with scripting languages such as Python or MATLAB/Octave. CasADi was therefore designed to allow front-ends to be generated automatically using the open-source tool SWIG. At the time of this writing, Python, MATLAB and Octave were supported through full-featured and documented front-ends. The tool has also been successfully used from JAVA and Haskell.

1.3 License and availability

CasADi’s source code is hosted on Github and released under GNU Lesser General Public License (LGPL) on http://casadi.org. The relatively permissive LGPL allows CasADi to be used royalty-free in commercial and academic software. The code is built and tested on travis-ci, with full-featured binaries available for common Linux, Mac and Windows systems. In addition, the Python interface is available from pip. CasADi can also be run from a demo server.Footnote 4

1.4 C code generation, just-in-time compilation

A large subset of expressions can be exported as self-contained C code without memory allocation. This is useful for embedded applications or to speed up computations using just-in-time compilation (JIT). The C code can be compiled into shared libraries or be called directly from either MATLAB/Octave via a generated MEX interface or from the command line.

1.5 Plugin infrastructure

The core of CasADi supports a number of standard problems in numerical optimization, including initial-value problems in ODE or DAE, linear and nonlinear systems of equations, nonlinear programs (NLPs) and quadratic programs (QPs). The user specifies such problems in a uniform way and the solution is delegated to a solver plugin, loaded as a dynamically linked library (DLL) at runtime. Solver plugins include solvers that are distributed with CasADi and interfaces to third-party software packages.

1.6 Linear systems of equations

Linear systems of equations can be embedded into symbolic expressions via “backslash” nodes. Derivatives of such operations are calculated via chain rules for linear system solves. Supported plugins include LDLT and QRFootnote 5 as well as interfaces to CSPARSE and LAPACK.

1.7 Nonlinear systems of equations

Nonlinear systems of equations can be formulated and solved by defining rootfinder instances in CasADi. Derivatives of such objects are calculated analytically using the implicit function theorem and the chain rule for linear system solves. Supported plugins include standard Newton methods and KINSOL from the SUNDIALS suite.

1.8 Initial-value problems in ODE/DAE with automatic sensitivity analysis

Initial-value problems in ODE or DAE can be calculated using explicit or implicit Runge–Kutta methods or interfaces to IDAS/CVODES from the SUNDIALS suite. Derivatives of arbitrary order are calculated using automatically generated forward and adjoint sensitivity equations.

1.9 Quadratic programming

QPs can be formulated either using a traditional syntax, by explicitly providing the linear and quadratic terms, or using a syntax which mirrors that of NLP solvers. Solvers for quadratic programming include a primal-dual active-set methodFootnote 6 and interfaces to CPLEX, GUROBI, HPMPC, OOQP and qpOASES. A subset of the solvers support mixed-integer QPs.

1.10 Nonlinear programming with automatic sensitivity analysis, Optistack

NLPs can be solved using block structure or general sparsity exploiting sequential quadratic programming (SQP) or interfaces to IPOPT/BONMIN, BlockSQP, WORHP, KNITRO and SNOPT. Solution sensitivities can be calculated automatically by applying the implicit function theorem to the first order optimality conditions.Footnote 7 A subset of the solvers support mixed-integer NLPs.

Optistack, a simple but powerful abstraction layer, simplifies the formulation and solution of NLPs. It manages the creation and optimal-value retrieval of decision variables, allows a mathematical notation to specify constraints, and may identify problematic constraints when a solver reports infeasibility.

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Andersson, J.A.E., Gillis, J., Horn, G. et al. CasADi: a software framework for nonlinear optimization and optimal control. Math. Prog. Comp. 11, 1–36 (2019). https://doi.org/10.1007/s12532-018-0139-4

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