Part 2Chapter 7

Optimization

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PowerPoints organized by Dr. Michael R. Gustafson II, Duke UniversityRevised by Prof. Jang, CAU

Chapter Objectives• Understanding why and where optimization occurs in

engineering and scientific problem solving.• Recognizing the difference between one-dimensional

and multi-dimensional optimization.• Distinguishing between global and local optima.• Locating the optimum of a single-variable function

with the golden-section search.• Locating the optimum of a single-variable function

with parabolic interpolation.• Knowing how to apply the fminbnd function to

determine the minimum of a one-dimensional function.• Knowing how to apply the fminsearch function to

determine the minimum of a multidimensional function.

Optimization• Optimization is the process of creating something

that is as effective as possible.– Produce the greatest output for a given amount of input

• From a mathematical perspective, optimization deals with finding the maxima and minima of a functionthat depends on one or more variables.

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

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

Multidimensional Optimization• One-dimensional problems involve functions that depend

on a single dependent variable -for example, f(x).• Multidimensional problems involve functions that depend

on two or more dependent variables - for example, f(x,y)

In (a), x* both minimize f(x) and maximize –f(x)

Global vs. Local• 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.

• Generally desire to find the global optimum.

Golden-Section Search• Search algorithm for finding a minimum on an

interval [xl xu] which includes a single minimum(unimodal interval)

• Uses the golden ratio φ=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.

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

Golden-Section Search (cont)

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

• If f(x2)

Code for Golden-Section Search

Example 7.2Q. Use the golden-section search to find the minimum

of f(x) within the interval from xl= 0 to xu =4.

For the first iteration

2

( ) 2sin10xf x x= −

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

Parabolic Interpolation• Another algorithm uses

parabolic interpolation of three points to estimate optimum location.

• The location of the maximum/minimum of a parabola defined as the interpolation of three points (x1, x2, and x3) is:

• 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 −12

x2 − 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( )[ ]

fminbnd Function

• MATLAB has a built-in function, fminbnd, which combines the golden-section search and the parabolic interpolation.– [xmin, fval] = fminbnd(function, x1, x2)

• Options may be passed through a fourth argument using optimset, similar to fzero.

Multidimensional Visualization• Functions of two-dimensions may be

visualized using contour or surface/mesh plots.

fminsearch Function• MATLAB has a built-in function, fminsearch, that

can be used to determine the minimum of a multidimensional function.– [xmin, fval] = fminsearch(function, x0)

– xmin in this case will be a row vector containing the location of the minimum, while x0 is an initial guess. Note that x0 must contain as many entries as the function expects of it.

• The function must be written in terms of a single variable, where different dimensions are represented by different indices of that variable.

fminsearch Function

• To minimize f(x,y)=2+x-y+2x2+2xy+y2rewrite asf(x1, x2)=2+x1-x2+2(x1)2+2x1x2+(x2)2

• 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])

• Note that x0 has two entries - f is expecting it to contain two values.

• MATLAB reports the minimum value is 0.7500 at a location of [-1.000 1.5000]

Part 2�Chapter 7Chapter ObjectivesOptimizationMultidimensional OptimizationGlobal vs. LocalGolden-Section SearchGolden-Section Search (cont)Code for Golden-Section SearchExample 7.2Parabolic Interpolationfminbnd FunctionMultidimensional Visualizationfminsearch Functionfminsearch Function