Getting Started with Ipopt in 90 Minutes

RC24790 (W0904-097) April 22, 2009 Mathematics IBM Research Report Short Tutorial: Getting Started With Ipopt in 90 Minutes Andreas Wächter IBM Research Division Thomas J. Watson Research Center P.O. Box 218 Yorktown Heights, NY 10598 Research Division Almaden - Austin - Beijing - Cambridge - 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 . Short Tutorial: Getting Started With Ipopt in 90 Minutes Andreas Wächter1 IBM T.J. Watson Research Center Department of Business Analytics and Mathematical Sciences 1101 Kitchawan Road, Yorktown Heights, NY 10598, USA [email protected] Abstract. Ipopt is an open-source software package for large-scale nonlinear optimization. This tutorial gives a short introduction that should allow the reader to install and test the package on a UNIX-like system, and to run simple examples in a short period of time. Keywords. Nonlinear Optimization, Tutorial, Ipopt, COIN-OR 1 Introduction Ipopt is an open-source software package for large-scale nonlinear optimization. It can be used to address general nonlinear programming problems of the form min f(x) (1a) n x∈R s.t. gL ≤ g(x) ≤ gU (1b) xL ≤ x ≤ xU , (1c) where x ∈ Rn are the optimization variables with lower and upper bounds, xL ∈ (R ∪ {−∞})n and xU ∈ (R ∪ {+∞})n, f : Rn −→ R is the objective function, and g : Rn −→ Rm are the constraints. The functions f(x) and g(x) can be linear or nonlinear and convex or non-convex, but should be sufficiently smooth (at least once, ideally twice continuously differentiable). The constraints, g(x), have lower and upper bounds, gL ∈ (R∪{−∞})m and gU ∈ (R∪{+∞})m. Note that equality constraints of the form gi(x) =g ¯i can be specified by setting L U gi = gi =g ¯i. Such optimization problems arise in a number of important engineering, fi- nancial, scientific, and medical applications, ranging from the optimal control of industrial processes (e.g., [1]) and the design of digital circuits (e.g., [2]) to port- folio optimization (e.g., [3]), from parameter identification in systems biology (e.g., [4]) to hyperthermia cancer treatment planning (e.g., [5]). Ipopt implements an interior-point line-search filter method; the mathematical details of the algorithm can be found in several publications [6,7,8,9,10]. This approach makes Ipopt particularly suitable for large problems with up 2 A. Wächter to millions of variables and constraints, assuming that the Jacobian matrix of constraint function is sparse, but also small and dense problems can be solved efficiently. It is important to keep in mind that the algorithm is only trying to find a local minimizer of the problem; if the problem is nonconvex, many station- ary points with different objective function values might exist, and it depends on the starting point and algorithmic choices which particular one the method converges to. In general, the computational effort during the optimization with Ipopt is typically concentrated in the solution of linear systems, or in the computation of the problem functions and their derivatives, depending on the particular application. With respect to both these tasks, research in Combinatorial Scientific Computing is of central importance: The KKT system is a saddle point problem; see [11] for a survey of the recent developments in this area, as well as progress in weighted graph matchings [12,13,14] and parallel partitioning tools, such as ParMetis[15], Scotch[16], and Zoltan[17]. Furthermore, the computation of derivatives can be facilitated by automatic differentiation tools, such as ADIFOR[18], ADOL-C[19] and OpenAD[20]. Optimization problems can be given to Ipopt either by using a modeling lan- guage, such AMPL1 or GAMS2, which allow one to specify the mathematical problem formulation in an easily readable text format, or by writing programming code (C++, C, Fortran 77, Matlab) that computes the problem functions f(x) and g(x) and their derivatives. This document provides only a short introduction to the Ipopt package. De- tailed information can be found on the Ipopt home page www.coin-or.org/Ipopt and from the Ipopt documentation available with the source code3. The instruction here are for Linux (or UNIX-like systems, including Cygwin); if you want to use Ipopt with the Windows Developer Studio instructions see the Ipopt documentation. The remainder of this tutorial is structured as follows: After presenting a motivating challenging example application in Section 2, easy-to-follow instal- lation instruction are provided in Section 3, which allow the reader to start experimenting with the code in Section 4. Section 5 gives some mathematical background of the underlying algorithm, providing the basis to explain the out- put (Section 6) and algorithmic options (Section 7). Some advise regarding good modeling practices is given in Section 8. Finally, Section 9 discusses how Ipopt can be used from programming code, and a coding exercise is provided for the interested reader. 2 Example Application: PDE-Constrained Optimization One class of optimization problems that give rise to very large and sparse nonlinear optimization problems is the optimization of models described by partial- 1 www.ampl.com 2 www.gams.com 3 Also available online: www.coin-or.org/Ipopt/documentation Ipopt in 90 Minutes 3 Fig. 1. Heated Tumor (heated in red) In Hyperthermia Treatment Planning. differential equations (PDEs). Here, the “unknowns” are functions defined on a subset of R2 or R3 that are required to satisfy one or more PDEs, such as a temperature profile obeying a heat transfer equation. The degrees of freedom stem from a finite set of control parameters (e.g., intensities of a small number of microwave antennas) or from another function defined over in the full domain of the PDE or its boundary (e.g., material properties within the body conducting the heat, or a controllable temperature profile at the boundary). There are a number of ways to tackle such problems. Ignoring many subtle details, we consider here the reformulation of the originally infinite-dimensional problem into a discretized finite-dimensional version. In this process, the domain of the PDE is discretized into a finite number of well-distributed points, and the new optimization variables are the values of the original functions at those points. Furthermore, the original PDE is then expressed as a set of constraints for each such point, where the differential operators are approximated based on the function values at neighboring points, e.g., by means of finite differences. In this way, each of those constraints involves only a small number of variables, leading to sparse derivative matrices. At the same time, the approximation error made by discretizing the original problem is reduced by increasing the number of discretization points, and it is therefore very desirable to solve large instances of the reformulation. This makes this application very suitable for an interior-point algorithm such as Ipopt. A specific medical example application is hyperthermia treatment planning: Here, a patient is exposed to microwave radiation, emitted by several antennas, in order to heat a tumor; see Figure 1. The purpose of this procedure is to increase the tumor’s susceptibility to regular radiation or chemotherapy. 4 A. Wächter Mathematically, the problem can be formulated as this PDE-constrained optimization problem: Z Z 2 2 min (T − T ther) dΩ + (T − T health) dΩ (2a) x∈Ωt x6∈Ωt 2 σ X s.t.−∇ · (κ∇T ) + c ω(T − T ) = u E in Ω (2b) b b 2 i i i κ∂nT = qconst on ∂Ω (2c) T |Ω/Ωt ≤ T lim. (2d) Here, Ω is the considered region of the patient’s body, Ωt ⊂ Ω the domain of tumor tissue, and T the temperature profile. The constant κ is the heat diffusion coefficient, cb the specific heat of blood, w(T ) the temperature-dependent per- fusion, T b the arterial blood temperature, qconst the human vessel blood flux, σ the electrical conductivity, ui the complex-valued control of antenna i, and Ei the corresponding electrical field. The PDE is given in (2b) with a Neumann boundary condition (2c). The goal of the optimization is to find optimal controls ui in order to minimize the deviation from the desired temperature inside and outside the tumor (T ther and T health, respectively), as expressed in the objective function (2a). To avoid excessively high temperature in the healthy tissue, the bound constraint (2d) is explicitly included. Christen and Schenk[21] recently solved an instance of this problem with real patient data and 12 antennas, resulting in a discretized NLP with n = 854, 499 variables. The KKT matrix (see (6) below), which is factorized to compute the optimization steps, had more than 27 million nonzero elements. This is a consid- erably large nonlinear optimization problem, which required 182 Ipopt iterations and took 48 hours on a 8-core Xeon Linux workstation, using the parallel linear solver Pardiso4.

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