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Music-Inspired Optimization Algorithm Harmony Search

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

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Music-Inspired Optimization Algorithm Harmony Search

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  1. Music-Inspired Optimization AlgorithmHarmony 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 Mi, Fa, Sol Do, Re, Mi Sol, La, Si = Do = Mi = Sol f (100, 300, 500) 100mm 200mm 300mm 300mm 400mm 500mm 500mm 600mm 700mm = 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 forBenchmark Problems

  21. Six-Hump Camel Back Function f*(-0.08983, 0.7126) = -1.0316285 (Exact) f (-0.08975, 0.7127) = -1.0316285 (HS)

  22. Multi-Modal Function

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

  24. HS Applications forReal-World Problems

  25. Truss Structure Design GA = 546.01, HS = 484.85

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

  27. 1 1 15 2 15 2 14 3 14 3 4 13 4 13 19 5 12 18 5 6 12 17 18 6 7 11 7 11 8 19 8 10 20 9 21 20 9 10 16 16 17 Water Distribution Network Design • 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 = 401.3 (GO)

  30. I O Wedge Storage = K x (I - O) Prism Storage = K O O Hydrologic Parameter Calibration Mathematical = 143.60, 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 = 0.3808 W/K HS: Mass = 25.8 kg, Conductance = 0.3945 W/K

  35. Oceanic Oil Structure Mooring

  36. Internet Routing

  37. Music Composition – Medieval Organum

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

  39. Web-Based Parameter Calibration RMSE: 1.305 (Powell), 0.969 (GA), 0.948 (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 Literaturein 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 zwgeem@gmail.com

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