Numerical Methods School of Mechanical Engineering Chung-Ang University

Transcription

1 Part 2 Chapter 7 Optimization Prof. Hae-Jin Choi hjchoi@cau.ac.kr 1

2 Chapter Objectives l Understanding why and where optimization occurs in engineering and scientific problem solving. l Recognizing the difference between one-dimensional and multidimensional optimization. l Distinguishing between global and local optima. l Locating the optimum of a single-variable function with the golden-section search. l Locating the optimum of a single-variable function with parabolic interpolation. l Knowing how to apply the fminbnd function to determine the minimum of a one-dimensional function. l Knowing how to apply the fminsearch function to determine the minimum of a multidimensional function 2

3 Optimization l Optimization is the process of creating something that is as effective as possible. - Produce the greatest output for a given amount of input l From a mathematical perspective, optimization deals with finding the maxima and minima of a function that depends on one or more variables. m æ mg ö -( c / m) t mg z( t) = z0 + ç v0 + ( 1- e ) - t c è c ø c 3 Elevation as a function of time for an object initially projected upward with an initial velocity

4 Multidimensional Optimization l One-dimensional problems involve functions that depend on a single dependent variable -for example, f(x). l Multidimensional problems involve functions that depend on two or more dependent variables - for example, f(x,y) 4

5 Global vs. Local l A global optimum represents the very best solution for the whole range while a local optimum is better than its immediate neighbors. Cases that include local optima are called multimodal. l Generally desire to find the global optimum. 5

6 Golden-Section Search l Search algorithm for finding a minimum on an interval [x l x u ] which includes a single minimum (unimodal interval) l Uses the golden ratio f= to determine location of two interior points x 1 and x 2 ; by using the golden ratio, one of the interior points can be re-used in the next iteration. l + l l = l l l l 1 2 let f = f -f - 1 = f = = L 2 l This is similar to the bisection method (one middle point) except that two intermediate points are chosen according to the golden ratio. 6

7 Golden-Section Search (cont) d = ( f -1)( x - x ) x = x + d and x = x - d 1 l 2 x - x = x + d - ( x - d) = x - x - 2d 1 2 u = x - x + 2( f -1)( x - x ) = x - x + 2 f( x - x ) - 2( x - x ) = (2f - 3)( x - x ) > 0 x > x l u l u l u l u u l l u u l u l u l 1 2 l If f(x 1 )<f(x 2 ), x 2 becomes the new lower limit and x 1 becomes the new x 2 (as in figure). l If f(x 1 )>f(x 2 ), x 1 becomes the new upper limit and x 2 becomes the new x 1. l In either case, only one new interior point is needed and the function is only evaluated one more time. 7

8 Example7.2 l Q. Use the golden-section search to find the minimum of f(x) within the interval from x l = 0 to x u =4. For the first iteration d = (4-0) = x x 1 2 = = = = x f ( x) = - 2sin x f ( x2) = - 2sin(1.5279) = f ( x1 ) = - 2sin(2.4721) = i x l f(x l ) x 2 f(x 2 ) x 1 f(x 1 ) x u f(x u ) d

9 Absolute Errors in Golden-Section Search l For each iteration, the optimum will fall in upper interval (x 2,x 1,x u ) or lower interval (x l,x 2,x 1 ). l Since the interior points (x 1, x 2 ) are symmetrical, either case can be used to define the error. l Looking at the upper interval (x 2,x 1,x u ) - If the true value were at the far left, the maximum distance D x = x - x a 1 2 = x + ( f -1)( x - x ) - x + ( f -1)( x - x ) l u l u u l = ( x - x ) + 2( f -1)( x - x ) u l u l = (2f - 3)( x - x ) = ( x - x ) u l u l 9

10 Absolute Errors in Golden-Section Search(cont) l Looking at the upper interval (x 2,x 1,x u ) - If the true value were at the far right, the maximum distance D x = x - x l Normalization b u 1 = x - x - ( f -1)( x - x ) u l u l = ( x - x ) - ( f -1)( x - x ) u l u l = (2 -f)( x - x ) = ( x - x ) u l u l e a xu - xl = (2 - f) 100% Error Criterion x opt 10

11 Code for Golden-Section Search [1] function [x,fx,ea,iter]=goldmin(f,xl,xu,es,maxit,varargin) % function goldmin: [x,fx,ea,iter]=goldmin(f,xl,xu,es,maxit,varargin) minimization golden section search % goldmin: [xopt,fopt,ea,iter]=goldmin(f,xl,xu,es,maxit,p1,p2,...): minimization golden section search % [xopt,fopt,ea,iter]=goldmin(f,xl,xu,es,maxit,p1,p2,...): uses golden section search to find the minimum of f % input: uses golden section search to find the minimum of f % input: f = name of function % xl, f = xu name = lower of function and upper guesses % es xl, = xu desired = lower relative and upper error guesses (default = %) % maxit es = desired = maximum relative allowable error (default iterations = %) (default = 50) % maxit p1,p2,... = maximum = additional allowable parameters iterations used by (default f = 50) % output: p1,p2,... = additional parameters used by f % output: x = location of minimum % x = location of minimum 11

12 Code for Golden-Section Search [2] % fx = minimum function value % ea = approximate relative error (%) % iter = number of iterations if nargin<3,error('at least 3 input arguments required'),end if nargin<4 isempty(es), es=0.0001;end if nargin<5 isempty(maxit), maxit=50;end phi=(1+sqrt(5))/2; iter=0; while(1) d = (phi-1)*(xu - xl); x1 = xl + d; x2 = xu - d; 12

13 Code for Golden-Section Search [3] if f(x1,varargin{:}) < f(x2,varargin{:}) xopt = x1; xl = x2; else xopt = x2; xu = x1; end iter=iter+1; if xopt~=0, ea = (2 - phi) * abs((xu - xl) / xopt) * 100;end if ea <= es iter >= maxit,break,end end x=xopt;fx=f(xopt,varargin{:}); 13

14 Solve Example 7.1 by Golden Section Search >> g=9.81; v0=55; m=80;c=15;z0=100; >> (z0+m/c*(v0+m*g/c)*(1-exp(-c/m*t))-m*g/c*t); >> [xmin,fmin,ea,iter]=goldmin(z,0,8) xmin = fmin = ea = e-005 Note : To transform the finding maximum value to finding minimum value, minus (-) sign is added to the equation. 14

15 Parabolic Interpolation l Another algorithm uses parabolic interpolation of three points to estimate optimum location. l The location of the maximum/minimum of a parabola defined as the interpolation of three points (x1, x2, and x3) is: x 4 = x ( x 2 - x 1 ) 2 [ f ( x 2 )- f ( x 3 )]- x 2 - x 3 ( x 2 - x 1 )[ f ( x 2 )- f ( x 3 )]- x 2 - x 3 l The new point x 4 and the two surrounding it (either x 1 and x 2 or x 2 and x 3 ) are used for the next iteration of the algorithm. ( ) 2 f x 2 ( ) f x 2 [ ( )- f ( x 1 )] ( )- f ( x 1 ) [ ] 15

16 fminbnd Function l MATLAB has a built-in function, fminbnd, which combines the golden-section search and the parabolic interpolation. >> [xmin, fval] = fminbnd(function, x1, x2) l Options may be passed through a fourth argument using optimset, similar to fzero. >> g=9.81; v0=55; m=80;c=15;z0=100; >> z=@(t) (z0+m/c*(v0+m*g/c)*(1-exp(-c/m*t))-m*g/c*t); >> [x,f]=fminbnd(z,0,8) x = f =

17 >> options = optimset ('display', 'iter') ; >> fminbnd(z, 0, 8, options) Func-count x f(x) Procedure initial golden golden parabolic parabolic parabolic parabolic parabolic parabolic Optimization terminated: the current x satisfies the termination criteria using OPTIONS.TolX of e-004 ans =

18 Multidimensional Visualization l Functions of two-dimensions may be visualized using contour or surface/mesh plots. 18

19 x=linspace(-2,0,40);y=linspace(0,3,40); [X,Y] = meshgrid(x,y); Z=2+X-Y+2*X.^2+2*X.*Y+Y.^2; subplot (1,2,1); cs=contour(x,y,z); clabel(cs); xlabel('x_1'); ylabel('x_2'); title('(a) Contour plot') ;grid; subplot(1,2,2); cs=surfc(x,y,z); zmin=floor(min(z)); zmax=ceil(max(z)); xlabel('x_1'); ylabel ('x_2'); zlabel('f(x_1,x_2)'); title ('(b) Mesh plot ) ; 19

20 fminsearch Function l MATLAB has a built-in function, fminsearch, that can be used to determine the minimum of a multidimensional function. l The function must be written in terms of a single variable, where different dimensions are represented by different indices of that variable. >> f=@(x) 2+x(1)-x(2)+2*x(1)^2+2*x(1)*x(2)+x(2)^2; >> [x, fval]=fminsearch(f, [-0.5, 0.5]) x = fval =