Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha University.

1. Optimization Problems and
Algorithms
Dr. Mohammed M. Nasef
Mathematics Department, Faculty of Science, Menoufia University
Member at Scientific Research Group in Egypt(SERG)
Workshop on Intelligent System
and Applications (ISA’17)
Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics,
Benha University.
13 May 2017

2. Overview
 Definition of Optimization
 Definition of Optimization Problems
 Types of Optimization Techniques
 Meta-heuristic Algorithms
 An Example : Whale Optimization Algorithm
2
Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.

3. Definition of Optimization
3
Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
The process of finding the best values for the
variables of a particular problem to minimize or
maximize an objective function

4. Definition of Optimization Problem
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Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
Optimization
Problem
Variables
Continuous Discrete
Constraints
Constrained Unconstrained
Objective
Function
Single Multi

5. Definition of Optimization Problem (cont.)
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Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
𝐟 𝐱 𝟏, 𝐱𝟐 = 𝐱 𝟏
𝟐
+𝟐𝐱 𝟐
𝟐
-0.3cos(3 𝛑𝐱 𝟏)( 4 𝛑𝐱 𝟐)+0.3
𝐨𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝐦𝐢𝐧(𝐟)
𝐀𝐧 𝐞𝐱𝐚𝐦𝐩𝐥𝐞 ∶ 𝐬𝐢𝐧𝐠𝐥𝐞 𝐨𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧
𝐯𝐚𝐫𝐢𝐚𝐛𝐥𝐞𝐬 ∈ [𝟏𝟎, −𝟏𝟎]
𝐔𝐧𝐜𝐨𝐧𝐬𝐭𝐫𝐚𝐢𝐧𝐞𝐝 𝐏𝐫𝐨𝐛𝐥𝐞𝐦

6. Definition of Optimization Problem (cont.)
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Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
Min f(z1, z2, z3) = (-100-(z1-5)2 - (z2-5)2 +(z3-5)2)/100
Subject to;
h(z1, z2, z3) = (z1 - 3)2 + (z2 - 2)2 + (z3 - 5)2 – 0.0625 ≤ 0
where;
0 ≤ zi ≤ 10;
𝐀𝐧 𝐞𝐱𝐚𝐦𝐩𝐥𝐞 ∶ 𝐬𝐢𝐧𝐠𝐥𝐞 𝐨𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧
𝐂𝐨𝐧𝐬𝐭𝐫𝐚𝐢𝐧𝐞𝐝 𝐏𝐫𝐨𝐛𝐥𝐞𝐦

7. Definition of Optimization Problem (cont.)
7
Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
𝐀𝐧 𝐞𝐱𝐚𝐦𝐩𝐥𝐞 ∶ 𝐌𝐮𝐥𝐭𝐢 𝐨𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧
𝐔𝐧𝐜𝐨𝐧𝐬𝐭𝐫𝐚𝐢𝐧𝐞𝐝 𝐏𝐫𝐨𝐛𝐥𝐞𝐦
𝐨𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝐦𝐢𝐧(𝐟𝟏 ) & 𝐦𝐢𝐧(𝐟𝟐 ) & 𝐦𝐢𝐧(𝐟𝟑 )

8. Definition of Optimization Problem (cont.)
8
Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
𝐀𝐧 𝐞𝐱𝐚𝐦𝐩𝐥𝐞 ∶ 𝐌𝐮𝐥𝐭𝐢 𝐨𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧
𝐂𝐨𝐧𝐬𝐭𝐫𝐚𝐢𝐧𝐞𝐝 𝐏𝐫𝐨𝐛𝐥𝐞𝐦
𝒎𝒊𝒏 = {
𝐬𝐮𝐛𝐣𝐞𝐜𝐭 𝐭𝐨;

9. 9
Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
Types of Optimization Techniques
Optimization
Technique
Conventional
Mathematical
Programming
Calculus
Methods
Network
Methods
Nonconventional
Meta-heuristic
algorithms

10. 10
Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
Meta-heuristic Algorithms
Meta-heuristic is a general algorithmic framework
which can be applied to different optimization
problems with relatively few modifications to make
them adapted to a specific problem.

11. 11
Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
Meta-heuristic Algorithms (cont.)
Meta-heuristic
algorithms
Evolutionary
algorithms
GA GP
Physics-based
algorithms
CSS SA
Swarm-based
algorithms
Whale
Ant
Colony
Human-based
algorithms
TLBO EMA
Genetic Algorithm (GA) Genetic Programming (GP) Charged System Search (CSS)
Simulated Annealing (SA) Teaching Learning Based Optimization(TLBO) Exchange Market Algorithm (EMA)

12. 12
Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
An Example : Whale optimization algorithm
1- Encircling prey
2- Bubble-net attacking method (exploitation phase)
3- Search for prey (exploration phase)
Behavior of Whale

13. 13
Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
Whale optimization algorithm(cont.)
Mathematical Model
Where t is the current iteration, A and C are coefficient vectors, X* is the
position vector of the best solution, and X indicates the position vector of a
solution, | | is the absolute value.
1- Encircling prey

14. 14
Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
Whale optimization algorithm (cont.)
Where components of a are linearly decreased from 2 to 0 over the course of
iterations and r is random vector in [0; 1]
The vectors A and C are calculated as follows:
Mathematical Model (cont.)

15. 15
Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
2- Bubble-net mechanism (exploitation phase)
Whale optimization algorithm (cont.)
Mathematical Model (cont.)
Where the value of A is a random value in interval [-a, a] and the value of a is
decreased from 2 to 0 , D’ =| X*(t) - X(t) | is the distance between the prey (best
solution) and the ith whale, b is a constant, l is a random number in [-1; 1], and p is a
random number in [0; 1]

16. 16
Workshop on Intelligent System and Applications (ISA’17), Faculty of Computers and Informatics, Benha
University.
3- search for prey (exploration phase)
Whale optimization algorithm (cont.)
Mathematical Model (cont.)
Where Xrand is a random position vector chosen from the current population.
In order to force the search agent to move far a way from
reference whale, we use the A with values > 1 or < 1

17. Thanks and Acknowledgement
Workshop on Intelligent System and Applications (ISA’17),
Faculty of Computers and Informatics, Benha University.