part 2 chapter 7 - caucau.ac.kr/~jjang14/nae/chap7.pdfchapter objectives • understanding why and...
TRANSCRIPT
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Part 2Chapter 7
Optimization
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PowerPoints organized by Dr. Michael R. Gustafson II, Duke UniversityRevised by Prof. Jang, CAU
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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.
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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− = + + − −
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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)
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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.
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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.
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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)
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Code for Golden-Section Search
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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
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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( )[ ]
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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.
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Multidimensional Visualization• Functions of two-dimensions may be
visualized using contour or surface/mesh plots.
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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.
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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