ISCTE/FCUL - Mestrado Matemática Financeira. Aula de Janeiro de 2009 Ano lectivo: 2008/2009. Diana Aldea Mendes

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1 ISCTE/FCUL - Mestrado Matemática Financeira Aula 5 17 de Janeiro de 2009 Ano lectivo: 2008/2009 Diana Aldea Mendes Departamento de Métodos Quantitativos, IBS - ISCTE Business School Gab. 207 AA, diana.mendes@iscte.pt, deam 1

2 Optimization Toolbox Matlab >> [xsol,fopt,exitflag,output,grad,hessian] = fminunc (fun,x0,options) Input arguments: - fun: a Matlab function m-file that contains the function to be minimzed - x0: Startvector for the algorithm, if known, else [ ] options: options are set using the optimset funciton, they determine what algorism to use,etc. Output arguments: 2

3 - xsol: optimal solution - fopt: optimal value of the objective function; i.e. f(xopt) -exitflag: tells whether the algorithm converged or not, exitflag > 0 means convergence - output: a structure for number of iterations, algorithm used and PCG iterations(when LargeScale=on) - grad: gradient vector at the optimal point xsol. - hessian: hessian matrix at the optimal point xsol. 3

4 To display the type of options that are available and can be used with the fminunc.m use >>optimset( fminunc ) Hence, from the list of option parameters displayed, you can easily see that some of them have default values. However, you can adjust these values depending on the type of problem you want to solve. However, when you change the default values of some of the parameters, Matlab might adjust other parameters automatically. There are two types of algorithms that you can use with fminunc.m 4

5 (i) Medium-scale algorithms: The medium-scale algorithms under fminunc.m are based on the Quasi-Newton method. This options used to solve problems of smaller dimensions. As usual this is set using >>OldOptions=optimset( fminunc ); >>Options=optimset(OldOptions, LargeScale, off ); With medium Scale algorithm you can also decide how the search direction dk be determined by adjusting the parameter HessUpdate by using one of: >>Options=optimset(OldOptions, LargeScale, off, HessUpdate, bfgs ); >>Options=optimset(OldOptions, LargeScale, off, HessUpdate, dfp ); 5

6 >>Options=optimset(OldOptions, LargeScale, off, HessUpdate, steepdesc ); (ii) Large-scale algorithms: By default the LargeScale option parameter of Matlab is always on. However, you can set it using >>OldOptions=optimset( fminunc ); >>Options=optimset(OldOptions, LargeScale, on ); When the LargeScale is set on, then fminunc.m solves the given unconstrained problem using the trust-region method. Usually, the large-scale option of fminunc is used to solve problems with very large number of variables or with sparse hessian matrices. Such problem, for instance, might arise from discretized optimalcontrolproblems,someinverse-problemsinsignalprocessing,etc. 6

7 However, to use the large-scale algorithm under fminunc.m, the gradient of the objective function must be provided by the user and the parameter GradObj must be set on using: >>Options=optimset(OldOptions, LargeScale, on, GradObj, on ); Hence, for the large-scale option, you can define your objective and gradient functions in a single function m-file : function [fun,grad]=myfun(x) fun =...; if nargout > 1 7

8 grad =...; end However, if you fail to provide the gradient of the objective function, then fminunc uses the medium-scale algorithm to solve the problem. Experiment: Write programs to solve the following problem with fminunc.m using both the medium and large-scale options and compare the results: minimize f(x 1,x 2,x 3 )=x x2 2 +5x2 3 Solution Define the problem in an m-file, including the derivative in case if you want to use the LargeScale option. 8

9 function [f,g]=fun1(x) %Objective function for example (a) %Defines an unconstrained optimization problem to be solved with fminunc f=x(1)ˆ2+3*x(2)ˆ2+5*x(3)ˆ2; if nargout > 1 g(1)=2*x(1); g(2)=6*x(2); 9

10 g(3)=10*x(3); end Next you can write a Matlab m-file to call fminunc to solve the problem. function [xopt,fopt,exitflag]=unconstex1 options=optimset( fminunc ); options.largescale= off ; options.hessupdate= bfgs ; % assuming the function is defined in the 10

11 %in the m file fun1.m we call fminunc % with a starting point x0 x0=[1,1,1]; [xopt,fopt,exitflag]=fminunc(@fun1,x0,options); If you decide to use the Large-Scale algorithm on the problem, then you need to simply change the option parameter LargeScale to on. function [xopt,fopt,exitflag]=unconstex1 options=optimset( fminunc ); 11

12 options.largescale= on ; options.gradobj= on ; %assuming the function is defined as in fun1.m %we call fminunc with a starting point x0 x0=[1,1,1]; [xopt,fopt,exitflag]=fminunc(@fun1,x0,options); To compare the medium- and large-scale algorithms on the problem given above you can use the m-function TestFminunc 12

13 The Matlab fminsearch.m function uses the Nelder-Mead direct search (also called simplex search) algorithm. This method requires only function evaluations, but not derivatives. As such the method is useful when the derivative of the objective function is expensive to compute; exact first derivatives of f are difficult to compute or f has discontinuities; thevaluesoffare noisy. x = fminsearch(fun,x0) starts at X0 and attempts to find a local minimizer X of the function FUN. FUN is a function handle. FUN accepts input X and returns a scalar function value F evaluated at X. X0 can be a scalar, vector or matrix. 13

14 function f = myfun(x,c) f = x(1)ˆ2 + c*x(2)ˆ2; >> c = 1.5; % define parameter first >> x = fminsearch(@(x) myfun(x,c),[0.3;1]) x = lsqnonlin (FUN,X0) starts at the matrix X0 and finds a minimum X to the sum of squares of the functions in FUN. FUN accepts input X and returns a vector (or matrix) of function values F evaluated at X. NOTE: FUN should return FUN(X) and not the 14

15 sum-of-squares sum(fun(x).ˆ2)). (FUN(X) is summed and squared implicitly in the algorithm.) The default algorithm when OPTIONS.LargeScale = off is the Levenberg- Marquardt method with a mixed quadratic and cubic line search procedure. A Gauss-Newton method is selected by setting OPTIONS.LargeScale= off and OPTIONS.LevenbergMarquardt= off. function F = myfun(x,c) F = [ 2*x(1) - exp(c*x(1)) -x(1) - exp(c*x(2)) 15

16 x(1)-x(2)]; >> c =-1;%define parameter first >> x = lsqnonlin(@(x) myfun(x,c),[1;1]) Try bandem.m and datdemo.m Outros m-files Algoritmo de Powell: powell.m [xo,ot,ns]=powell(s,x0,ip,method,lb,ub,problem,tol,mxit) 16

17 - S: objective function - x0: initial point - ip: (0): no plot (default), (>0) plot figure ip with pause, (<0) plot figure ip - method: (0) Coggins (default), (1): Golden Section - Lb, Ub: lower and upper bound vectors to plot (default = x0*(1+/-2)) - problem: (-1): minimum (default), (1): maximum - tol: tolerance (default = 1e-4) 17

18 - mxit: maximum number of stages (default = 50*(1+4* (ip>0))) - xo: optimal point - Ot: optimal value of S - ns: number of objective function evaluations Para ver um exemplo correr o m-file: powell run.m Algoritmo de Newton: newtons.m (para sistemas não-lineares) - [x,fx,xx] = newtons(f,x0,tolx,maxiter,varargin) 18

19 Para exemplificar, correr o m-file: newtonsex.m M-file newton.m para minimizar funções não-lineares - [xo,ot,ns]=newton(s,x0,ip,g,h,lb,ub,problem,tol,mxit) Exercícios: Determine, caso existem, os mínimos das seguintes funções: 1. f (x) =x 1 x 2 2 x3 3 x4 4 exp( x 1 x 2 x 3 x 4 ) 2. f (x) =100(x 2 sin (x 1 )) x

20 3. f (x) =(x 1 +10x 2 ) 2 +5(x 3 x 4 ) 4 +10(x 1 x 4 ) 4 4. f (x) =e x 1+x 2 +1 e x 1 x e x