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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