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Chapter 7 Optimization. Content. Introduction One dimensional unconstrained Multidimensional unconstrained Example. Introduction (1). Root finding & optimization are closely related !!. Optimization. Root finding. Introduction (2). An optimization problem - PowerPoint PPT Presentation
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Chapter 7Chapter 7
OptimizationOptimization
ContentContent
•Introduction•One dimensional unconstrained •Multidimensional unconstrained•Example
Introduction (1)Root finding & optimization are closely related !!
Optimization
Root finding
Introduction (2)An optimization problem
“Find x, which minimizes or maximizes f(x) subject to…
*,,2,1)(
*,,2,1)(
pibxe
miaxd
ii
ii
x is an n-dimensional design vector
f(x) is the objective function
di(x) are inequality constraints
ei(x) are equality constraints
ai and bi are constants ”
Introduction (3)Classification
Constrained optimization problem 1) Linear programming : f(x) & constraints are linear
2) Quadratic programming : f(x) is quadratic and constraints are linear
3) Nonlinear programming : f(x) is either quadratic or
nonlinear and/or
constraints are nonlinear
Unconstrained optimization problem
One-dimensional (1)
Multimodal function
interested in finding the absolute highest or lowest value of a function.
One-dimensional (2)
Graphing to gain insight into the behavior of the function.
Using randomly generated starting guesses and picking the largest of the optima as global.
Perturbing the starting point to see if the routine returns a better point or the same local minimum.
To distinguish global minimum (or maximum) fromlocal ones we can try
One-dimensional (3)Golden section search (for a unimodal function)A unimodal function has a single maximum or a
minimum in the a given interval. Step of GSS
Pick two points that will bracket your extremum [xl, xu].Pick an additional third point within this interval to
determine whether a maximum occurred.Then pick a fourth point to determine whether the
maximum has occurred within the first three or last three points
The key is making this approach efficient by choosing intermediate points wisely thus minimizing the function evaluations by replacing the old values with new values.
One-dimensional (4)
1
2
0
1
210
l
l
l
l
lll
GSS principle is always keep the ratio of section length equal @ each iteration
61803.02
15
2
)1(411
011
1 2
1
2
1
2
21
1
Rhence
R
RRR
R
l
lR
l
l
ll
l
One-dimensional (5)Step I:
Starts with two initial guesses (xl ,xu)
Step II:
Calculate points x1,x2 according to the golden ratio where
dxx
dxx
xxd
u
l
lu
2
1
)(2
15
Step III:
Evaluate function at x1 and x2
One-dimensional (6)Step IV:
if f(x1)> f(x2) xU = x1
if f(x1)< f(x2) xL= x2
Step V:
Calculate new x1
)(2
151 lul xxxx
x
Try with the following example !!
One-dimensional (7)Ex Use the GSS to find the minimum of
xx
xf sin210
)(2
assume that xl = 0 and xu = 4
Answer x = 1.4276
Multidimensional… (1)Techniques to find minimum and maximum of a
function of several variables.Classified as:
Requires derivative evaluationGradient or descent (or ascent) methods
Not require derivative evaluationNon-gradient or direct methods.
Multidimensional… (2)
Multidimensional… (3)DIRECT METHODS :Random SearchBased on evaluation of the function randomly at
selected values of the independent variables.If a sufficient number of samples are conducted,
the optimum will be eventually located.
Ex: Locate the maximum of a function
f (x, y)=y-x-2x2-2xy-y2
Multidimensional… (4)
Multidimensional… (5)Advantages/Works even for discontinuous and nondifferentiable
functions.Always finds the global optimum rather than the
global minimum.
Disadvantages/As the number of independent variables grows, the
task can become onerous.Not efficient, it does not account for the behavior of
underlying function.
Multidimensional… (6)Univariate and Pattern Searches
More efficient than random search and still doesn’t require derivative evaluation.
The basic strategy is:Change one variable at a time while the other
variables are held constant.Thus problem is reduced to a sequence of one-
dimensional searches that can be solved by variety of methods.
The search becomes less efficient as you approach the maximum.
Multidimensional… (7)Univariate and Pattern Searches
Multidimensional… (8)Gradient method
If f(x,y) is a two dimensional function, the gradient vector tells usWhat direction is the steepest ascend?How much we will gain by taking that step?
del fy
f
x
ff or ji
Directional derivative of f(x,y) at point x=a and y=b
Multidimensional… (9)Gradient method (cont’d)•For n dimensions
)(
)(
)(
)(2
1
x
x
x
x
nx
f
x
fx
f
f
Multidimensional… (10)Steepest ascent method
Use the gradient vector to change x
Says that the we should change with along h-axis
h
f
f
i
iii
)(
)(
)1(
)1()1()(
x
xxx
The gradient vector represents a direction of maximum rate of increase for the function f(x) at x .
Multidimensional… (11)Example
Minimize the function
2225),( yxyxf
where x = 0.6, y = 4
Multidimensional… (11)Example
Maximize the function
22 222),( yxxxyyxf
where x = -1, y = 1
Answer x = 2, y = 1
Multidimensional… (11)Example
Maximize the function
2225),( yxyxf
where x = 0.6, y = 4