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) Minimal Error (Parameter Calibration) Maximal Profit (Management) Maximal Utility (Economics)

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, …

Existing Nature-Inspired Algorithms

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

Algorithm from Music Phenomenon

Algorithm from Jazz Improvisation Click Below

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

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

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

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.

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

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}

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

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

Ensemble Considering

Rule Violation (Parallel 5th)

Example of Harmony Search

Initial Harmony Memory

Next Harmony Memory

With Three Operators {1, 2, 3, 4, 5} +1 f = 6 1 4 2

HS Applications for Benchmark Problems

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

Multi-Modal Function

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

HS Applications for Real-World Problems

Sudoku Puzzle 6 1 4 2 5 3 8 7 9

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

Project Scheduling (TCTP)

University Timetabling

Internet Routing

Web-Based Parameter Calibration RMSE: 1.305 (Powell), 0.969 (GA), 0.948 (HS)

Truss Structure Design GA = 546.01, HS = 484.85

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

Generalized Orienteering Problem Max. Multi-Objectives 1. Natural Beauty 2. Historical Significance 3. Cultural Attraction 4. Business Opportunity Case1 Case2 Case3 Case4 Case5 ANN 12.38 13.05 12.51 12.78 12.36 HS 13.08 12.56 12.40

Water Distribution Network Design 1 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

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

Multiple Dam Operation Max. Benefit (Power, Irrigation) GA = 400.5, HS = 401.3 (GO)

Hydrologic Parameter Calibration Wedge Storage = K x (I - O) Prism Storage = K O O I Mathematical = 143.60, GA = 38.23, HS = 36.78

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

Satellite Heat Pipe Design

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

Oceanic Oil Structure Mooring

RNA Structure Prediction

Medical Imaging

Radiation Oncology

Astronomical Data Analysis

All that Jazz Robotics Visual Tracking Internet Searching Management Science Et Cetera

Paradigm Shift a change in basic assumptions within the ruling theory of science

Stochastic Partial Derivative of HS

Stochastic Co-Derivative of HS

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

Wikipedia (Web Encyclopedia)

Books on Harmony Search

Visitor Clustering (As of Nov. 2010)

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

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