1. Music-Inspired Optimization Algorithm Harmony Search
Zong Woo Geem

2. What is Optimization?
Procedure to make a system or design as effective, especially the mathematical techniques involved. ( Meta-Heuristics)
Finding Best Solution
Minimal Cost (Design)
Minimal Error (Parameter Calibration)
Maximal Profit (Management)
Maximal Utility (Economics)

3. Types of Optimization Algorithms
Mathematical Algorithms
Simplex (LP), BFGS (NLP), B&B (DP)
Drawbacks of Mathematical Algorithms
LP: Too Ideal (All Linear Functions)
NLP: Not for Discrete Var. or Complex Fn., Feasible Initial Vector, Local Optima
DP: Exhaustive Enumeration, Wrong Direction
Meta-Heuristic Algorithms
GA, SA, TS, ACO, PSO, …

4. Existing Nature-Inspired Algorithms

5. Existing Meta-Heuristic Algorithms
Definition & Synonym
Evolutionary, Soft computing, Stochastic
Evolutionary Algorithm (Evolution)
Simulated Annealing (Metal Annealing)
Tabu Search (Animal’s Brain)
Ant Algorithm (Ant’s Behavior)
Particle Swarm (Flock Migration)
Mimicking Natural or Behavioral Phenomena → Music Performance

6. Algorithm from Music Phenomenon

7. Algorithm from Jazz Improvisation

8. Analogy = Do = Mi = Sol = 100mm = 300mm = 500mm f (100, 300, 500)
Mi, Fa, Sol
Do, Re, Mi
Sol, La, Si
= Do
= Mi
= Sol
100mm
200mm
300mm
300mm 400mm
500mm
500mm
600mm
700mm
f (100, 300, 500)
= 100mm
= 300mm
= 500mm

9. Comparison Factors Musical Inst. → Decision Var.
Pitch Range → Value Range
Harmony → Solution Vector
Aesthetics → Objective Function
Practice → Iteration
Experience → Memory Matrix

10. Good Harmony & Bad Harmony
An Algorithm which Keeps Better Harmonies!

11. Procedures of Harmony Search
Step 0. Prepare a Harmony Memory.
Step 1. Improvise a new Harmony with Experience (HM) or Randomness (rather than Gradient).
Step 2. If the new Harmony is better, include it in Harmony Memory.
Step 3. Repeat Step 1 and Step 2.

12. HS Operators Random Playing Memory Considering Pitch Adjusting
Ensemble Considering
Dissonance Considering

13. Random Playing
x ∈ Playable Range = {E3, F3, G3, A3, B3, C4, D4, E4, F4, G4, A4, B4, C5, D6, E6, F6, G6, A6, B6, C7}

14. Memory Considering
x ∈ Preferred Note = {C4, E4, C4, G4, C4}

15. Pitch Adjusting
x+ or x-, x ∈ Preferred Note

16. Ensemble Considering

17. Rule Violation (Parallel 5th)

18. Stochastic Partial Derivative of HS

19. Parameter-Setting-Free HS
Overcoming Existing Drawbacks
Suitable for Discrete Variables; No need for Gradient Information or Feasible Initial Vector; Better Chance to Find Global Optimum
Drawbacks of Meta-Heuristic Algorithms
Requirement of Algorithm Parameters

20. HS Applications for Benchmark Problems

21. Six-Hump Camel Back Function
f*( , ) = (Exact)
f ( , ) = (HS)

22. Multi-Modal Function

23. Artificial Neural Network - XOR
   
          T F
Bias
Sum of Errors in BP = 0.010
Sum of Errors in HS = 0.003

24. HS Applications for Real-World Problems

25. Truss Structure Design
GA = , HS =

26. School Bus Routing Problem
Depot
School 1 2 3 4 5 6 7 8 9 10 15 20
Min C1 (# of Buses) + C2 (Travel Time)
s.t. Time Window & Bus Capacity
GA = $409,597, HS = $399,870

27. Water Distribution Network Design1 2 3 4 5 6 7 8 9 15 14 11 18 12 13 17 10 19 16 20 21
MP: $78.09M
GA: $38.64M (800,000)
SA: $38.80M (Unknown)
TS: $37.13M (Unknown)
Ant: $38.64M (7,014)
SFLA: $38.80M (21,569)
CE: $38.64M (70,000)
HS: $38.64M (3,373)
5 times out of 20 runs

28. Large-Scale Water Network Design
Huge Variables
(454 Pipes)
GA = 2.3M Euro
HS = 1.9M Euro

29. Multiple Dam Operation
Max. Benefit (Power, Irrigation)
GA = 400.5, HS = (GO)

30. Hydrologic Parameter Calibration
Wedge Storage
= K x (I - O)
Prism Storage
= K O O I
Mathematical = , GA = 38.23, HS = 36.78

31. Ecological Conservation
With 24 Sites, SA = 425, HS = 426

32. Heat Exchanger Design

33. Satellite Heat Pipe Design

34. Satellite Heat Pipe Design
BFGS HS
Minimize Mass
Maximize Conductance
BFGS: Mass =25.9 kg, Conductance = W/K
HS: Mass = 25.8 kg, Conductance = W/K

35. Oceanic Oil Structure Mooring

36. Internet Routing

37. Music Composition – Medieval Organum
Interval
Rank
Fourth 1
Fifth 2
Unison 3
Octave
Third 4
Sixth
Second 5
Seventh

38. Sudoku Puzzle 6 1 4 2 5 3 8 7 9

39. Web-Based Parameter Calibration
RMSE: (Powell), (GA), (HS)

40. All that Jazz Robotics Internet Searching Visual Tracking
Management Science
Project Scheduling
Medical Physics
Bioinformatics
Et Cetera

41. Wikipedia (Web Encyclopedia)

42. Books on Harmony Search

43. Visitor Clustering (As of Mar. 2009)

44. Growth in Major Literature in tantum ut si priora tua fuerint parva, et novissima tua multiplicentur nimis. Iob 8:7

45. Question for Harmony Search?
Contact to Zong Woo Geem