presentation optimization 2-libre

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Institute of Transportation Engineers, Purdue Student Chapter engineering.purdue.edu/ITE

Introduction to Optimization Page 1 of 18

Introduction to Optimization in

M TL B

Updated October 2, 2010 Compiled by Amit Kumar and Sushant Sharma

Table of Contents

MATLAB BASICS ..................................................................................................................................................... 2

WHAT IS MATLAB? ...................................................................................................................................................... 2

USAGEOFMATLAB .................................................................................................................................................... 2

MATLAB GETTING STARTED ................................................................................................................................... 2

WORKINGWITHMATRICES: ...................................................................................................................................... 5

CREATING LOOPS ............................................................................................................................................................ 7For loop ................................................................................................................................................................. 7

While Loop ............................................................................................................................................................ 8

CREATING LOGICAL OPERATIONS ....................................................................................................................................... 8

RIDING THE TIGER .................................................................................................................................................. 9

HOW TO WRITE CODE THAT RUNS FASTER .......................................................................................................................... 9

Preallocation ......................................................................................................................................................... 9

Vectorization ....................................................................................................................................................... 10

Logical Indexing .................................................................................................................................................. 10

MATLAB PROGRAMMING .................................................................................................................................... 11

WRITEYOUROWNCODE ......................................................................................................................................... 11

WRITE SIMPLE M-FILES .................................................................................................................................................. 11

OPTIMIZATION TOOLBOX .................................................................................................................................... 13

LINEAR CONSTRAINED OPTIMIZATION PROBLEM ................................................................................................................ 13

PROBLEM FORMULATION ............................................................................................................................................... 14

NON-LINEAR OPTIMIZATION PROBLEM ............................................................................................................................ 15

PRACTICE EXERCISE .............................................................................................................................................. 17

PROBLEM 1: ................................................................................................................................................................ 17


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MATLAB BASICS

What is MATLAB?

MATLAB is an interactive, matrix-based system for scientific and engineering numericcomputation and visualization. The word MATLAB stands for MATrix LABoratory. Each entry

is taken as a matrix in it, in particular scalar is considered as a 1 by 1 matrix. MATLAB is

available for a number of operating systems like Windows, Linux etc. MATLAB was originallywritten in FORTRAN and is licensed by The Math Works, Inc,

(http://www.mathworks.com).

USAGE OF MATLAB

MATLAB is well known software for numerical linear algebra and matrix computation. Industryuse it for research and to solve practical engineering and mathematical problems. Also, in

automatic control theory, statistics and digital signal processing (Time-Series Analysis) one can

use MATLAB. The following tool boxes make it useful in soft computing at various industrial

and scientific areas:(i) Neural Networks (ii) Optimization

(iii) Genetic Algorithms (iv) Wavelets

(v) Fuzzy Logic (vi) Control systems

(vi) Signal Processing

MATLAB GETTING STARTED

By clicking the MATLAB shortcut icon on the desktop of your computer (or selecting from the

program menu) you can access MATLAB. This results in getting MATLAB command window

with its prompt:

>>with a blinking cursor appearing right of the prompt, telling you that MATLAB is waiting to

perform a mathematical operation you would like to give.
http://www.mathworks.com/http://www.mathworks.com/http://www.mathworks.com/http://www.mathworks.com/


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Simple Math Calculations

( i ) If you want to add two numbers say 7 & 12, type as follows

>> 5+10

and press the ENTER or return key, you see the following output:

ans =

15

Here, ans stands for the answer of computation. Similarly, the following gives product anddifference of these numbers,

>> 5*10

ans = 50


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( ii ) If you want to store the values 5 and 10 in MATLAB variables a & b and store the values

of their product and division in c and d, do as follows:

>> a =5

a =5

>>b =10

b =

10

>>c= a*b

c =

50

>>d = 10/5 d =2

You can exit MATLAB with the command exit or quit. A computation can be stopped with

[ctrl-c]


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The basic arithmetic operations are given by:

Operation Symbol

Addition

a+b

+

Subtractiona-b

-

Multiplicationa.b

*

Divisiona/b

/ or \

Exponentialab

^

WORKING WITH MATRICES:

MATLAB works with essentially only one kind of objects, i.e. a rectangular numerical matrix with possibly

complex entries. All variables represent matrices.

Scalars - 1 by 1 matrix

Row vector - matrix with one row

Column vector - matrix with one column.

If you want to store a matrix

1 2 3

4 5 6

7 8 9

you type the following in the MATLAB prompt.

>> a = [1 2 3; 4 5 6; 7 8 9]

a =

1 2 3

4 5 6

7 8 9


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The rows are separated by semicolons and elements are separated by space or by comma. To transpose

a matrix and store in b we can run following command

>> b = a

b =

1 4 7

2 5 8

3 6 9

Matrix Operations available in MATLAB

Function Task

+ Addition

- Subtraction

* Multiplication

^ Power

Transpose

\ Left Division

/ Right Division

Examples matrix addition:

>> a1 = a+b (matrices a and bare added and stores in a1)

a1 =

2 6 10

6 10 14

10 14 18

Examples matrix multiplication:


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>>a2= a * b

= 14 32 50

32 77 122

50 122 194

>>a3= a.*b

= 1 8 21

8 25 48

21 48 81

Creating loops

Similar to other programming language MATLAB also has two looping constructs: for loops and while

loops.

For loop

For loops are useful for repeating a process or a set of processes (or operations) a certain number of

times. Each for loop should have counter variable, starting value of the counter variable, increment

(default 1 if not specified) and upper limit of counter variable. In addition all for loops should be closed

by an end statement.

An example of for loop

fori=1:1:10disp(i);

end

Here, initial value of counter variable iis 1, increment is defined as 1 and upper limit is 10. Hence it willexecute the loop ten times. In each loop it will display the value of the counter variable. An another loop

can be nested inside a loop as below-

fori=1:1:10forj=1:1:3disp(i);

endend

Now this nested loop will execute the value of counter variable ithree times.


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While Loop

While loops are useful for repeating a process or a set of processes (or operations) until a specified

condition(s) is (are) met. Following is the example of a while loop

i=1;while~(i==10)

disp(i);i=i+1;

end

In this example of while loop, the loop runs 9 times as opposed to for loop. The while loop first

evaluates the conditional statement and then executes the processes inside the loop if conditional

statement is true else the loop is terminated. So in this example as soon as the value of iwill turn 10 the

loop will be terminated.

So, each while loop needs a logical test and it runs while the answer to the logical test is valid. Hence

next we discuss how to create the logical statements.

Creating Logical Operations

The logical statements can be combination of variables, arithmetic operators and relational operators

and results in true or false. Following table shows the list of relational operators.

For more details visit the mathworks web page

http://www.mathworks.com/help/techdoc/matlab_prog/f0-40063.html

The local operations can be used by inbuilt function logical or under the if(else) statement. Following

example shows use of inbuilt function logical.

clear; %clear all variables

clc; %clear screena=[1 2 3 2 4 2 5];disp('values of vector "a" before');disp(a);b=logical(a==2);disp('after modifying "a" using logical operations');a(b)=99;disp(a);

Operator Description

= Greater than or equal to

== Equal to

~=Not equal to
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Following is an example of using logical operation in an if-else statement-

clear; %clear all variablesclc; %clear screena=[1 2 3 2 4 2 5];disp('values of vector "a" before');

disp(a);m=length(a);fori=1:1:m

ifa(i)==2a(i)=999;

elsea(i)=0;

endenddisp('after modifying "a" using logical operations');disp(a);

RIDING THE TIGER

How to Write Code that Runs Faster

There are three important techniques to speed up the execution of MATLAB code namely,preallocation,

vectorizationand indexingby logical expression.

Preallocation

It means defining the size of a variable (matrix or vector) and giving some initial values. Run the

following two codes below see the difference in execution time-

Code with preallocation Code without preallocation%clc;clear;tic; %start of measuring timea=zeros(50000,1); %preallocationsum=0;fori=1:1:50000

a(i)=i;sum=sum+a(i);

enddisp('sum of 1 to 50,000 is =');

disp(sum);toc; %End of measuring time

clc;clear;tic; %start of measuring time%a=zeros(50000); % No preallocationsum=0;fori=1:1:50000

a(i)=i;sum=sum+a(i);

enddisp('sum of 1 to 50,000 is =');

disp(sum);toc;%End of measuring time


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Vectorization

Vectorization gives faster performance compared to loop. Run the following codes to see the benefit of

vectorization.

Code with vectorization Code without vectorization

% Vectorization example for%elementwise multiplication%clc;clear;tic; %start of measuring timea=ones(50000,1);b=ones(50000,1);c=zeros(50000,1);b=b*2;%for i=1:1:50000

c=a.*b;s=sum(c);disp(s);

%endtoc;%End of measuring time

% No_Vectorization example for%elementwise multiplication%clc;clear;tic; %start of measuring timea=ones(50000,1);b=ones(50000,1);c=zeros(50000,1);b=b*2;s=0;fori=1:1:50000

c(i)=a(i)*b(i);s=s+c(i);

enddisp(s);toc;%End of measuring time

Logical Indexing

Indexing also gives faster performance compared to loop. Run the following codes to see the benefit of

indexing-

Code with indexing Code without indexing%clc;clear;tic; %start of measuring timea=1:50000;

b=logical(rem(a,2)==0);a(b)=0;

s=sum(a);disp('sum of odd integers between 1and 50,000 is =');disp(s);toc;%End of measuring time

clc;clear;

tic; %start of measuring timea=1:50000;fori=1:1:50000

ifrem(a(i),2)==0a(i)=0;end

ends=sum(a);disp('sum of odd integers between 1and 50,000 is =');disp(s);toc;%End of measuring time


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MATLAB PROGRAMMINGWe can also do programming in MATLAB as we are doing in FORTRAN, C & C++. To make a file in

MATLAB e hae to lik o Ne i the file eu i euar. It ill ope a e file as e are doig

i word; ad if e sae this file alled -file, ill e saed i i folder of MATLAB.

Such files are alled M-files eause the hae a etesio of . i its fileae. Muh of our ork

with MATLABwill be creating and refining M-files.

There are two types of M-files: Script Files and Function Files.

WRITE YOUR OWN CODE

Write Simple m-files

A m-file consists of a sequence of normal MATLABstatements. If the file has the filename, say, test.m,

then the MATLAB command >> testwill cause the statements in the file to be executed. Variables in a

script file are global and will change the value of variables of the same name in the environment of the

current MATLAB session.

M files are often used to enter data into a large matrix; in such a file, entry errors can be easily edited

out.

Example:In a file test.m enter the following:

% This is a sample m-file

a = [4,2,3;0,1,2;1,2,3]

b =a;

c = a+b

d = inv(c)

save this file. Then to execute this file, type

>>test


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The % symbol indicates that the rest of the line is a comment. MATLAB will ignore the rest of the line.


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OPTIMIZATION TOOLBOX

Linear Constrained Optimization Problem

Example 1:

Apple manufactures multiple products including iphone and ipad. It has three manufacturing

bases. The products are assembled at assembling unit by the component parts arriving from

three manufacturing bases.

Following table describes the percentage of capacity available at the manufacturing bases due

to the other products being developed at same location. The table also shows percentage of

capacity required per million units production of iphone and ipad, in addition to the profit.

Table 1: Capacity of manufacturing base for iphone and ipad

Capacity Used (per million unit)

Manufacturing base iphone ipad Capacity Available

(per million unit)California 1 0 4

Illinois 0 2 12

Atlanta 3 2 18

Profit (in hundred million USD per

million unit)

3 5


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Problem Formulation

Objective function:

x1= number of iphones (in million) and x2= number of ipads (in million)

Goal :To maximize profit (Z)

Or,

Decision variable:

Constraints: See Table 1 to understand the constraints


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Non-Linear Optimization Problem

Create following M files

1.

Fun.m

functionz = fun(x)z = x(1)^2+x(2)^2+x(3)^2;end


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2. Constr.m

%function[g, h] = constr(x)g(1) = x(3)/x(2)-1;%h = x(1)-x(2)^2+x(2)*x(3)-4;%%EOF


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PRACTICE EXERCISE

Problem 1:

Write a MATLAB code for golden section method to solve the following optimization problem. You can

use the given flow chart of the method provided in this handout as a reference.

Given the initial interval of deisio ariale as , ad fial ofidee iteral to e ..


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Figure: Flow chart of golden section method

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