part 2 chapter 7 - cauisdl.cau.ac.kr/education.data/numerical.analysis/lecture... · 2010-10-05 ·...

School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2

OptimizationOptimization

Prof. Prof. HaeHae--Jin ChoiJin Choi

[email protected]@cau.ac.kr

Part 2Part 2Chapter 7Chapter 7

1


Chapter ObjectivesChapter Objectivesl Understanding why and where optimization occurs in engineering

and scientific problem solving.l Recognizing the difference between one-dimensional and multi-

dimensional 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


OptimizationOptimizationl 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.

( )( / )0 0( ) 1 c m tm mg mgz t z v e tc c c

-æ ö= + + - -ç ÷è ø

Elevation as a function of time for an object initially projected upward with an initial velocity

3


Multidimensional OptimizationMultidimensional Optimizationl 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


Global vs. LocalGlobal vs. Locall A global optimum represents the very best solution for the whole

range while a local optimum is better than its immediateneighbors. Cases that include local optima are called multimodal.

l Generally desire to find the global optimum.

5


GoldenGolden--Section SearchSection Searchl Search algorithm for finding a minimum on an interval [xl xu]

which includes a single minimum (unimodal interval)l Uses the golden ratio f=1.6180… to determine location of two

interior points x1 and x2; by using the golden ratio, one of the interior points can be re-used in the next iteration.

l This is similar to the bisection method (one middle point) except that two intermediate points are chosen according to the golden ratio.

1 2 1

1 2

21

2

let 1 0

1 5 1.618033988749892

f f f

f

+=

= ® - - =

+= =

l l ll lll

L

6


GoldenGolden--Section Search (cont)Section Search (cont)

1 2

1 2

1 2

( 1)( ) and

( ) 22( 1)( ) 2 ( ) 2( )

(2 3)( ) 0

u l

l u

l u l u

l u u l l u u l u l

u l

d x xx x d x x dx x x d x d x x dx x x x x x x x x x

x x x x

f

f ff

= - -

= + = -

- = + - - = - -

= - + - - = - + - - -

= - - > ® >

l If f(x1)<f(x2), x2 becomes the new lower limit and x1 becomes the new x2 (as in figure).

l If f(x1)>f(x2), x1 becomes the new upper limit and x2 becomes the new x1.

l In either case, only one new interior point is needed and the function is only evaluated one more time.

7


Example7.2Example7.2l Q. Use the golden-section search to find the minimum of f(x)

within the interval from xl= 0 to xu=4.2

( ) 2sin10xf x x= -

For the first iteration

1

2

0.61803(4 0) 2.47210 2.4721 2.47214 2.4721 1.5279

dxx

= - == + == - =

2

2

2

1

1.5279( ) 2sin(1.5279) 1.764710

2.4721( ) 2sin(2.4721) 0.630010

f x

f x

= - = -

= - = -

i xl f(xl) x2 f(x2) x1 f(x1) xu f(xu) d 12345678

00

0.94430.94431.30501.30501.30501.3901

00

-1.5310-1.5310-1.7595-1.7595-1.7595-1.7742

1.52790.94431.52791.30501.52791.44271.39011.4427

-1.7647-1.5310-1.7647-1.7595-1.7647-1.7755-1.7742-1.7755

2.47211.52791.88851.52791.66561.52791.44271.4752

-0.6300-1.7647-1.5432-1.7647-1.7136-1.7647-1.7755-1.7732

4.00002.47212.47211.88851.88851.66561.52791.5279

3.1136-0.6300-0.6300-1.5432-1.5432-1.7136-1.7647-1.7647

2.47211.52790.94430.58360.36070.22290.13780.0851

8


Absolute Errors in GoldenAbsolute Errors in Golden--Section SearchSection Search

l For each iteration, the optimum will fall in upper interval (x2,x1,xu) or lower interval (xl,x2,x1).

l Since the interior points (x1, x2) are symmetrical, either case can be used to define the error.

l Looking at the upper interval (x2,x1,xu) - If the true value were at the far left, the maximum distance

1 2

( 1)( ) ( 1)( )( ) 2( 1)( )(2 3)( ) 0.2361( )

a

l u l u u l

u l u l

u l u l

x x xx x x x x xx x x x

x x x x

f ff

f

D = -

= + - - - + - -

= - + - -

= - - = -

9


l Looking at the upper interval (x2,x1,xu) - If the true value were at the far right, the maximum distance

Absolute Errors in GoldenAbsolute Errors in Golden--Section Search(cont)Section Search(cont)

1

( 1)( )( ) ( 1)( )(2 )( ) 0.3820( )

b u

u l u l

u l u l

u l u l

x x xx x x xx x x x

x x x x

ff

f

D = -

= - - - -

= - - - -

= - - = -

l Normalization

(2 ) 100%u la

opt

x xx

e f-

= - ´ Error Criterion

10


Code for GoldenCode for Golden--Section Search [1]Section Search [1]

function [x,fx,ea,iter]=goldmin(f,xl,xu,es,maxit,varargin)% goldmin: 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:% f = name of function % xl, xu = lower and upper guesses% es = desired relative error (default = 0.0001%)% maxit = maximum allowable iterations (default = 50)% p1,p2,... = additional parameters used by f% output:% x = location of minimum

function [x,fx,ea,iter]=goldmin(f,xl,xu,es,maxit,varargin)% goldmin: 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:% f = name of function % xl, xu = lower and upper guesses% es = desired relative error (default = 0.0001%)% maxit = maximum allowable iterations (default = 50)% p1,p2,... = additional parameters used by f% output:% x = location of minimum

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% fx = minimum function value% ea = approximate relative error (%)% iter = number of iterationsif nargin<3,error('at least 3 input arguments required'),endif nargin<4|isempty(es), es=0.0001;endif nargin<5|isempty(maxit), maxit=50;endphi=(1+sqrt(5))/2;iter=0;while(1)d = (phi-1)*(xu - xl);x1 = xl + d;x2 = xu - d;


12


if f(x1,varargin{:}) < f(x2,varargin{:})xopt = x1;xl = x2;

elsexopt = x2;xu = x1;

enditer=iter+1;if xopt~=0, ea = (2 - phi) * abs((xu - xl) / xopt) * 100;endif ea <= es | iter >= maxit,break,end

endx=xopt;fx=f(xopt,varargin{:});


13


Solve Example 7.1 by Golden Section SearchSolve Example 7.1 by Golden Section Search

Note : To transform the finding maximum value to finding minimum value, minus (-) sign is added to the equation.

>> 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);>> [xmin,fmin,ea,iter]=goldmin(z,0,8)xmin =

3.8317fmin =-192.8609

ea = 6.9356e-005

14


Parabolic InterpolationParabolic Interpolationl 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:

l The new point x4 and the two surrounding it (either x1 and x2or x2 and x3) are used for the next iteration of the algorithm.

�

x4 = x2 -12x2 - x1( )2 f x2( )- f x3( )[ ]- x2 - x3( )2 f x2( )- f x1( )[ ]x2 - x1( ) f x2( )- f x3( )[ ]- x2 - x3( ) f x2( )- f x1( )[ ]

15


fminbndfminbnd FunctionFunction

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 =

3.8317f =-192.8609

16


>> options = optimset ('display', 'iter') ;>> fminbnd(z, 0, 8, options)

Func-count x f(x) Procedure1 3.05573 -189.759 initial2 4.94427 -187.19 golden3 1.88854 -171.871 golden4 3.87544 -192.851 parabolic5 3.85836 -192.857 parabolic6 3.83332 -192.861 parabolic7 3.83162 -192.861 parabolic8 3.83166 -192.861 parabolic9 3.83169 -192.861 parabolic

Optimization terminated:the current x satisfies the termination criteria usingOPTIONS.TolX of 1.000000e-004

ans =3.8317

17


Multidimensional VisualizationMultidimensional Visualizationl Functions of two-dimensions may be visualized using contour or

surface/mesh plots.

18


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


fminsearchfminsearch FunctionFunctionl 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 =

-1.0000 1.5000fval =

0.7500

20

part 2 chapter 7 - cauisdl.cau.ac.kr/education.data/numerical.analysis/lecture... · 2010-10-05 ·...

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