CasADi tutorial Introduction
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1 Lund, 6 December 2011 CasADi tutorial Introduction Joel Andersson Department of Electrical Engineering (ESAT-SCD) & Optimization in Engineering Center (OPTEC) Katholieke Universiteit Leuven OPTEC (ESAT SCD) Katholieke Universiteit Leuven
2 OPTEC Optimization in Engineering Interdiciplinary center at Katholieke Universiteit Leuven, Belgium (Flemish region) since professors, 10 postdocs and 60+ PhD students from Mech.Eng, Elec.Eng, Civ.Eng, Comp.Sc. Principle investigator: Moritz Diehl Aim: Bridge the gap between state-of-the optimization algorithms and applications Applications: Kite power, river networks, robot control, chemical reactors, etc.
3 Outline 1 Optimization an overview 2 What is CasADi? 3 Schedule of tutorials 4 Exercise: Getting started
4 Optimization an overview What is optimization? Optimization = search for the best solution in mathematical terms: minimization or maximization of an objective function f (x) depending on variables x subject to constraints Equivalence of maximization and minimization problems: (from now on only minimization) f(x) -f(x) x* Maximum x x x* Minimum
5 Optimization an overview Constrained optimization Often variable x shall satisfy certain constraints, e.g.: x 0, x x 2 2 = C. General formulation minimize f (x) subject to g(x) 0, h(x) = 0 (1) f objective (cost) function g inequality constraint function h equality constraint function
6 Optimization an overview Feasible set = collection of all points that satisfy all constraints: Example feasible set is intersection of grey and blue area x x x2 0 h ( x) : = 1 h ( x) : = 2 1 x x x 0 0
7 Optimization an overview Local and global minima Global minimizer: x Ω and x Ω : f (x) f (x ) Local minimizer: x Ω and there exists a neighborhood N of x such that x Ω N : f (x) f (x ) f(x) Local Minimum Local Minimum Global Minimum:
8 Optimization an overview Derivatives First and second derivatives of the objective function or the constraints play an important role in optimization The first order derivatives are called the gradient f (x) = [ f, f,..., f ] T (2) x 1 x 2 x n When f is vector valued we have the Jacobian: J(x) = ( f (x)) T The Hessian matrix contains second order derivatives: 2 f 2 f x 2 x 1 1 x f x 1 x n 2 2 f 2 f f (x) = x 2 x f x2 2 x n x n f x n x 1 2 f x n x f xn 2 (3)
9 Optimization an overview Methods to calculate derivatives Finite differences Low accuracy, relatively slow Symbolic differentiation (in e.g. Maple) Only for small expressions Automatic/algorithmic differentiation (AD) Fast & accurate Tools: ADOL-C, CppAD, (CasADi)
10 Optimization an overview Automatic/algorithmic differentiation (AD) Given a function f : R N R M, AD cheaply delivers: Forward directional derivatives: y fsens = f x x fseed (4) Adjoint directional derivatives: x asens = ( ) T f y aseed (5) x Complete Jacobians/Hessians (cheap if sparse): J = f x (6)
11 Optimization an overview Optimality conditions Unconstrained problem: minimize f (x), x R n (7) Assume that f is twice differentiable. We want to test a point x for local optimality Necessary condition f (x ) = 0 (stationarity) Sufficient condition x stationary and 2 f (x ) positive definite Necessary Necessary & sufficient
12 Optimization an overview Stationary points (a)-(c) x* is stationary: f(x*)=0 2 f(x*) positive definite: local minimum 2 f(x*) negative definite: local maximum 2 f(x*) indefinite: saddle point
13 Optimization an overview Local and global optimization Sometimes there are many local minima Global optimization is a very hard issue - most algorithms find only the next local minimum. Exception: Convex optimization problems
14 Optimization an overview Classes of optimal control problems Linear programming (LP) minimize subject to c T x x min x x max (8) g min A x g max Quadratic programming (QP) minimize subject to x T H x + c T x x min x x max (9) g min A x g max Convex QP H is positive definite Tools: quadprog, CPLEX, OOQP, qpoases,...
15 Optimization an overview Nonlinear programming (NLP) minimize subject to f (x) x min x x max (10) g min g(x) g max Nonconvex in general initial guess important Tools exist handling millions of variables & constraints
16 Optimization an overview Solution methods for NLP Sequential quadratic programming (SQP) At solution guess, quadratic approximation of f and g QP Solve QP to get a new a new solution guess Continue until necessary conditions for optimality satisfied Tools: fmincon, SNOPT, KNITRO,... Interior point or barrier methods (IP) Relaxation: penalize inequality constraint violation Solve equality constrained problem with Newton s method with increasing barrier Continue until necessary conditions for optimality satisfied, barrier infinite Tools: IPOPT, KNITRO,... Require first and (preferably) second order derivatives Exercise tomorrow
17 Optimization an overview Optimal control problem (OCP) (dynamic optimization) minimize subject to T t=0 L(x, u, p), dt + E(x(T ), p) ẋ(t) = f (x(t), u(t), p), t [0, T ] x(0) = x 0 (p) x min x(t) x max, t [0, T ] u min u(t) u max, t [0, T ] p min p p max (11) x: state, u: control, p: free parameters ẋ(t) = f (x(t), u(t), p), x(0) = x 0 (p): system dynamics L(x, u, p): Lagrange term, E(x, p) Meyer term
18 Optimization an overview More general OCPs Multiple dynamic stages Path constraints Differential algebraic equations (DAE) instead of ODE Explicit time dependence Multipoint constraints: r(x(t 0 ), x(t 0 ),..., x(t end )) = 0
19 Optimization an overview Solution methods for OCP Hamilton-Jacobi-Bellmann equation / Dynamic Programming Finds global optimum through smart exhaustive search Indirect Methods Solve necessary conditions for optimality (Pontryagin s Maximum Principle) Direct Methods (control discretization) Tools for OCP Reformulate OCP as an NLP JModelica.org, ACADO Toolkit, MUSCOD-II, DyOS, DIRCOL, GPOPS,...
20 What is CasADi? What is CasADi? An open-source (LGPL) symbolic framework for quick, yet efficient, implementation of derivative based algorithms for dynamic optimization Framework for writing OCP solvers Reformulate OCP NLP Solve NLP efficiently Used in JModelica.org
21 What is CasADi? Why write your own OCP solver? Learn about the methods Implement new OCP-algorithms Treat arbitrary complex OCP Full control if something goes wrong Alternative: Algebraic modelling languages Tools: AMPL, GAMS, Pyomo, OpenOpt High level formulation Less flexibility Only for collocation (Mostly) commercial
22 What is CasADi? Main components of CasADi A minimalistic Computer Algebra System (CAS) (cf. Symbolic Toolbox for Matlab) 8 flavors of automatic differentiation (AD) Forward and adjoint mode Symbolic and numeric Two different ways to represent expressions Interfaces Ipopt, Sundials, (KNITRO, OOQP, qpoases, CPLEX, LAPACK, CSparse, ACADO Toolkit... ) Front-ends C++, Python, (Octave) Model import from JModelica.org
23 What is CasADi? Main developers Myself Joris Gillis Contributing Carlo Savorgnan, Attila Kozma, Greg Horn, & Co.
24 Outline 1 Optimization an overview 2 What is CasADi? 3 Schedule of tutorials 4 Exercise: Getting started
25 Schedule of tutorials Tuesday 6 December Introduction Exercise: Getting started with CasADi Extra exercise: ODE/DAE sensitivity analysis Wednesday 7 December Exercise: NLP Thursday 8 December Optimal control methods Advanced concepts in CasADi Exercise: Matrix symbolics & OCP with CasADi JModelica.org introduction Exercise: OCP with JModelica.org
26 Outline 1 Optimization an overview 2 What is CasADi? 3 Schedule of tutorials 4 Exercise: Getting started
27 Exercise: Getting started Fundamental types SX scalar symbolic type SXMatrix and DMatrix sparse matrices FX and derived classes CasADi functions MX matrix symbolic type Thursday morning
28 Exercise: Getting started SX Represents symbolic expressions made up by a sequence of unary and binary operations Scalar type Used like symbolic variables in SymPy/Maple/Symbolic Toolbox/... Representation like types in AD-tools (cf. adouble in ADOL-C) Example: x y + sin(x y) + 4 x
29 Exercise: Getting started SXMatrix Sparse matrix containing SX scalars Everything is a matrix syntax Use this class instead of SX directly! DMatrix Sparse matrix containing double (= float in Python) Used internally
30 Exercise: Getting started FX and derived classes Base class for different kind of functions Defined by symbolic expressions ODE/DAE integrators Much more Defines a standard way of communicating with these functions Setting options Passing inputs Evaluation Directional derivatives Obtaining the Jacobian
31 Exercise: Getting started Exercise take-away Working with symbolic expressions AD in CasADi
32 Exercise: Getting started Extra exercise: ODE/DAE integration & sensitivity analysis Forward/adjoint sensitivity analysis in CasADi Open-source integrator suite Sundials (ODE: CVodes / DAE: IDAS) Usage: integrator = CVodesIntegrator(ode function) CasADi will formulate the necessary equations Sensitivity analysis works just like AD How to write your own integrator
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