all about introduction to optimization

1. Introduction to Optimization
Module 1

2. What is Optimization?
• The process of finding the best values for the
variables of a particular problem to minimize or
maximize an objective function
• The action of making the best or most effective use
of given situation - (Google dictionary)
• It is a technique of squeezing best performance out
of provided current state of model

3. Components of an Optimization Problem
• Objective function
An objective function express performance of a system, need
to be minimized or maximized
• Variable (Design or Decision Variable)
A set of unknowns, define the objective function and
constraints, can be continuous, discrete or boolean
• Constraints
They are conditions, allows the unknowns to take on certain
values but exclude others to render the design to be feasible

4. Components of an Optimization Problem
Optimization
Problem
Variables
Continuous Discrete
Constraints
Constrained Unconstrained
Objective
Function
Single Multi

5. Objective function
𝟏 𝟏 𝟐
𝐟 𝐱 , 𝐱𝟐 = 𝐱𝟐+𝟐𝐱𝟐-0.3cos(3 𝛑𝐱𝟏)( 4 𝛑𝐱𝟐)+0.3
𝐨𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝐦𝐢𝐧(𝐟)
𝐯𝐚𝐫𝐢𝐚𝐛𝐥𝐞𝐬 ∈ [𝟏𝟎, −𝟏𝟎]
𝐔𝐧𝐜𝐨𝐧𝐬𝐭𝐫𝐚𝐢𝐧𝐞𝐝 𝐏𝐫𝐨𝐛𝐥𝐞𝐦
𝐀𝐧 𝐞𝐱𝐚𝐦𝐩𝐥𝐞 ∶ 𝐬𝐢𝐧𝐠𝐥𝐞 𝐨𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧

6. Objective function
𝐀𝐧 𝐞𝐱𝐚𝐦𝐩𝐥𝐞 ∶ 𝐬𝐢𝐧𝐠𝐥𝐞 𝐨𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝐟𝐮𝐧𝐜𝐭
𝐢𝐨𝐧
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;
𝐂𝐨𝐧𝐬𝐭𝐫𝐚𝐢𝐧𝐞𝐝 𝐏𝐫𝐨𝐛𝐥𝐞𝐦

7. Objective function
𝐀𝐧 𝐞𝐱𝐚𝐦𝐩𝐥𝐞 ∶ 𝐌𝐮𝐥𝐭𝐢 𝐨𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧
𝐨𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝐦𝐢𝐧(𝐟𝟏 ) & 𝐦𝐢𝐧(𝐟𝟐 ) & 𝐦𝐢𝐧(𝐟𝟑 )
𝐔𝐧𝐜𝐨𝐧𝐬𝐭𝐫𝐚𝐢𝐧𝐞𝐝 𝐏𝐫𝐨𝐛𝐥𝐞𝐦

8. 𝐀𝐧 𝐞𝐱𝐚𝐦𝐩𝐥𝐞 ∶ 𝐌𝐮𝐥𝐭𝐢 𝐨𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧
𝒎𝒊𝒏 =
𝐬𝐮𝐛𝐣𝐞𝐜𝐭 𝐭𝐨;
𝐂𝐨𝐧𝐬𝐭𝐫𝐚𝐢𝐧𝐞𝐝 𝐏𝐫𝐨𝐛𝐥𝐞𝐦
Objective function
{

9. Type of Optimization Techniques
Optimization
Technique
Conventional
Mathematical
Programming
Calculus
Methods
Network
Methods
Nonconventional Meta-heuristic
algorithms

10. Meta-heuristicAlgorithms
• 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. Meta-heuristicAlgorithms
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. 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. An Example : Whale optimization algorithm
Mathematical Model
1- Encircling prey
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.

14. An Example : Whale optimization algorithm
Where components of a are linearly decreased from 2 to 0 over the course of
iterations and r is random vector in [0; 1]
Mathematical Model (cont.)
The vectors A and C are calculated asfollows:

15. An Example : Whale optimization algorithm
Mathematical Model (cont.)
2- Bubble-net mechanism (exploitationphase)
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. An Example : Whale optimization algorithm
Mathematical Model (cont.)
3- search for prey (exploration phase)
In order to force the search agent to move
far a way from reference whale, we use the A with
values > 1 or < 1
Where Xrand is a random position vector chosen from the current population.

17. Module 1: Introduction to Optimization
• END OF CONTENT MODULE