Introduction to MATLAB

```MATLAB
Optimization
Greg Reese, Ph.D
Research Computing Support Group
Miami University
MATLAB
Optimization
2
Function Summary
Function
Parameter
Linearity
Constraints
Location
fgoalattain
fminbnd
fmincon
fminimax
fminsearch
fminunc
fzero
lsqlin
lsqnonlin
lsqnonneg
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Optimization
Optimization - finding value of a
parameter that maximizes or
minimizes a function with that
parameter
optimization, not optimization of computer
code!
– "function" is mathematical function, not
MATLAB language function
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Optimization
Optimization - finding value of a parameter
that maximizes or minimizes a function with
that parameter
– Can have multiple parameters
– Can have multiple functions
– Parameters can appear linearly or nonlinearly
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Linear programming
Linear programming
– "programming" means determining
feasible programs (plans, schedules,
allocations) that are optimal with respect
to a certain criterion
– Most often used kind of optimization
–Tremendous number of practical
applications\
Won't discuss further
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fminbnd
The basic optimizing 1D routine is
fminbnd, which attempts to find a
minimum of a function of one variable
within a fixed interval.
• function must be continuous
• only real values (not complex)
• only scalar values (not vector)
• might only find local minimum
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fminbnd
x = fminbnd( @fun, x1, x2 )
x1 < x2 is the minimization interval
x is the x-value in [x1,x2] where the minimum occurs
fun is the function whose minimum we’re looking
for. fun must accept exactly one scalar argument
and must return exactly one scalar value
Because there are no restrictions on the value of
fun, finding the minimum is called
unconstrained optimization.
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fminbnd
x = fminbnd( @fun, x1, x2 )
@fun is called a function handle. You
must include the “@” when you call
fminbnd.
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fminbnd
Try It
Find the minimum of f(x) = ( x – 3 )2 – 1
on the interval [0,5]
Step 1 – make an m-file with the function
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fminbnd
Try It
Step 2 – call fminbnd with your function
>> x = fminbnd( @parabola, 0, 5 )
x = 3
If you want to get the value of the function at
its minimum, call the function with the
minimum x value returned by fminbnd
>> parabola(x)
ans =
-1
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fminbnd
Try It
Step 3 – verify result is a global minimum by
plotting function over a range, say [-10 10]
>> ezplot( @parabola, [-10 10] )
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fminunc
The multidimensional equivalent of
fminbnd is fminunc. It attempts to
find a local minimum of a function of one
or more variables.
• function must be continuous
• only real values (not complex)
• only scalar values (not vector)
• might only find local minimum
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fminunc
x = fminunc( @fun, x0 )
x0 is your guess of where the minimum is
fun is the function whose minimum we’re
looking for. fun must accept exactly one
scalar or vector argument and must return
exactly one scalar value
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fminunc
x = fminunc( @fun, x0 )
fun only has one argument. If the
argument is a vector, each element
represents the corresponding
independent variable in the function
Example
f(x,y,z) = x2 + y2 + z2
function w = f( x )
w = x(1)^2 + x(2)^2 + x(3)^2
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fminunc
Optimizing routines such as fminunc are
often sensitive to the initial value. For bad
initial values they may converge to the wrong
answer, or not converge at all.
Tip – if fun is a function of one or two
variables, plot it first to get an idea of where
the minimum is
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fminunc
Try It
Find the minimum of f(x) = ( x – π )2 – 1
Step 1 – make an m-file with the function
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fminunc
Try It
Step 2 – plot the function to get an estimate
of the location of its minimum
Looks like min is
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fminunc
Try It
Step 3 – call fminunc with your function and
an initial guess of 3
Ignore
this
stuff
>> xmin = fminunc( @pi_parabola, 3 )
Warning: Gradient must be provided for
trust-region method;
> In fminunc at 247
Optimization terminated: relative
options.TolFun.
xmin = 3.1416
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fminunc
Try It
Find the minimum of
f(x,y) = ( x–22.22 )2 + ( y-44.44 )2 - 17
Step 1 – make an m-file with the function
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fminunc
Try It
Step 2 – plot the function to get an estimate
of the location of its minimum
ezsurf expects a function that takes two
arguments. Since parabola only takes one,
we need to make a different version that
takes two. Let's call it paraboloid_xy
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fminunc
Try It
Step 2 – plot the function to get an estimate
of the location of its minimum
>> ezsurf( @parabola_xy, [ 0 50 0 50] )
After zooming and
panning some, it
looks like the min
and y=50
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fminunc
Try It
Step 3 – call fminunc with your function and
an initial guess of [20 50]
>> x = fminunc( @paraboloid, [20 50] )
x =
22.2200
44.4400
>> paraboloid( x )
ans = -17.0000
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fminunc
Try It
Find the minimum of a 5D hypersphere of
centered at [ 1 2 3 4 5 ]
f(x,y,z,a,b) = (x–1)2 + (y–2)2 + (z–3)2 +
(a–4)2 + (b–5)2
Step 1 – make an m-file with the function
Use 3
dots to
continue a
line in a
file.
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fminunc
Try It
Step 2 – can't plot 5D, so guess [ 0 0 0 0 0]
Step 3 – call fminunc with your function and an
initial guess of [0 0 0 0 0]
>> xmin = fminunc( @hypersphere, [0 0 0 0 0] )
xmin =
1.0000
2.0000
3.0000
4.0000
5.0000
>> hypersphere( xmin )
ans = 3.9790e-012
≈0
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MATLAB
Optimization
Questions?
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The End
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