![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](https://image.slidesharecdn.com/module1introductiontooptimization-220816085026-741800bf/85/Introduction-to-Optimization-ppt-5-320.jpg)
![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:](https://image.slidesharecdn.com/module1introductiontooptimization-220816085026-741800bf/85/Introduction-to-Optimization-ppt-14-320.jpg)
![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]](https://image.slidesharecdn.com/module1introductiontooptimization-220816085026-741800bf/85/Introduction-to-Optimization-ppt-15-320.jpg)
Introduction to Optimization.ppt
- 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