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7/24/2019 AE 5301 HW 3 Solution
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Prob 1
1a) Himmelblau function: Himmelblau function is a mathematical
function with multiple minima and is used to check the performance of
optimization schemes. It is named after David Himmelblau and it is
referenced in a 1972 book named pplied !onlinear "ro#rammin#.
$athematicall%& the function is de'ned as
f(x , y)=(x2+ y11)2+(x+y27 )2
(he function is usuall% evaluated in ran#e
x , y(6,6)
1b) Plot Contours
)or all the contour plots& the *(+,ontour"lot function was used within the
mentioned ran#es
Code for contours :
Function
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Plot :
Fig 1 Contour plot for HimmelBlau function
1c) -olve usin# fminunc times :
"/0,D*/:
(he function is sou#ht to be minimized usin# the fminunc built in
matlab function. (he minimum function value and correspondin# variable values will
depend on the initial #uess provided since fminunc 'nds the closest
local minima. (he second step is repeated usin# points 3&34 & 53&34 & 3&534 and 5
3&534 (he values of minimum 6 & % and fmin are presented in a table below
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(he same procedure is followed for 2c and c for solvin# the respective
functions with the fminunc matlab function.
Contour
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
x
y
Fig. Contour plot with (66) as initial point and (!") as local minima
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-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
x
y
Fig. Contour plot with (#66) as starting point and (#".$%&1!.1!1!)as local minima
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-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
x
y
Fig. Contour plot with ( 6#6) as starting point and ( !.&$'' #1.$'$1)
as local minima
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
x
y
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Fig. Plot with (#6#) as starting point and (#!.!#!."$!") as local
minima
Code
1d) *+,+-C /0*-2H3
#enetic al#orithm 84 is a method for solvin# both constrained and
unconstrained optimization problems based on a natural selection process
that mimics biolo#ical evolution1. t each step& the #enetic al#orithm
randoml% selects individuals from the current population and uses them as
parents to produce the children for the ne6t #eneration with which the
population evolves toward an optimal solution1.
(he matlab #a optimation toolbo6 was used to perform the #a optimization
process for the d subsections of this homework
"/0,D*/
(he obective function is created in a separate matlab 'le and is savedin the workin# director%
(he #eneric al#orithm optimization method was chosen durin#
initialization of the toolbo6 (he function was included throu#h the 'le created earlier and the
number of variables was #iven to be 2 (he upper and lower bound ran#es were provided and the al#orithm
was e6ecuted (he #lobal minimum of the function alon# with the respective value of
6 and % were #iven as output.
(00;
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)i#. 8 toolbo6 output for "rob. 1
/-*;(-
6 > ? % > ? fmin >
1.1@137@97ABA757
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Prob "
"a) 4i5 Hump Camel Bac Function : (he si#ni'cance of new
optimization al#orithms depends on the eCectiveness of solvin# practical
problems. -i6 Hump camel back function is used to test the performance and
reliabilit% of optimization al#orithms. -i6 Hump
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Plot
Fig." Contour Plots for 4i5 Hump Camel Bac Function
"c) -olve usin# fminunc:
"d) *+,+-C /0*-2H3
6> A.A9 %> 5A.71 fmin> 51.A132@91BAAB
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)i# : 0utput from $atlab 8 for "rob. 2
Prob. !
!a) Branin rcos function : It is a #lobal optimization test function. (he
function has three #lobal optima.
)unction de'nition :
f(x , y )=1(y 5.14 2x2+
5
x6)
2
+10(1 18 )cosx+10
/an#e for the function is
5 x 10,0 y 15
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!b) Contour plot
Code
Plot
Fig !. Contour plot of Branin rcos function
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!c) -olve usin# fminunc:
!d) *+,+-C /0*-2H3
Fig: $atlab 8 output for "rob.
= > .1 %> 2.27B )min>A.97@@32@37133
-eference
718Ehttp:FFwww.mathworks.comFdiscover%F#enetic5al#orithm.htmlE