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Music-Inspired Optimization Algorithm Harmony Search Zong Woo Geem

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

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Types of Optimization Algorithms Mathematical Algorithms Mathematical Algorithms Simplex (LP), BFGS (NLP), B&B (DP) Drawbacks of Mathematical Algorithms 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 Meta-Heuristic Algorithms GA, SA, TS, ACO, PSO, …

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Existing Nature-Inspired Algorithms

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

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Algorithm from Music Phenomenon

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Algorithm from Jazz Improvisation Click Below

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

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

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

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

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HS Operators 1.Random Playing 2.Memory Considering 3.Pitch Adjusting 4.Ensemble Considering 5.Dissonance Considering

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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}

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

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

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Ensemble Considering

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Rule Violation (Parallel 5 th )

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Example of Harmony Search

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Initial Harmony Memory

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Next Harmony Memory

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With Three Operators 14 +1 {1, 2, 3, 4, 5} 2 f= 6

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HS Applications for Benchmark Problems

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

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Multi-Modal Function

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Artificial Neural Network - XOR Sum of Errors in BP = 0.010 Sum of Errors in HS = 0.003 TTF TFT FTT FFF Bias

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HS Applications for Real-World Problems

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Sudoku Puzzle 614253879 295784613 378196254 439627581 781549326 526318947 952461738 843972165 167835492

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Music Composition – Medieval Organum IntervalRankIntervalRank Fourth1Fifth2 Unison3Octave3 Third4Sixth4 Second5Seventh5

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Project Scheduling (TCTP)

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University Timetabling

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Internet Routing

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

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Truss Structure Design GA = 546.01, HS = 484.85

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School Bus Routing Problem GA = $409,597, HS = $399,870 Depot School 1 2 3 4 5 6 7 8 9 10 7 5 8 54 5 3 456 8 5 7 4 5 4 5 155 10 15 20 10 15 10 20 Min C1 (# of Buses) + C2 (Travel Time) s.t. Time Window & Bus Capacity

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Generalized Orienteering Problem Case1Case2Case3Case4Case5 ANN12.3813.0512.5112.7812.36 HS12.3813.0812.5612.7812.40 Max. Multi-Objectives 1. Natural Beauty 2. Historical Significance 3. Cultural Attraction 4. Business Opportunity

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Water Distribution Network Design 1 2 3 4 5 6 7 8 9 15 14 11 1812 13 17 10 19 16 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2021 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

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

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

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

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

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Satellite Heat Pipe Design

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Maximize Conductance Minimize Mass BFGS HS BFGS: Mass =25.9 kg, Conductance = 0.3808 W/K HS: Mass = 25.8 kg, Conductance = 0.3945 W/K

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Oceanic Oil Structure Mooring

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RNA Structure Prediction

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Medical Imaging

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Radiation Oncology

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Astronomical Data Analysis

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

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

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Stochastic Partial Derivative of HS

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Stochastic Co-Derivative of HS

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Parameter-Setting-Free HS Overcoming Existing Drawbacks 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 Drawbacks of Meta-Heuristic Algorithms Requirement of Algorithm Parameters

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Wikipedia (Web Encyclopedia)

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Books on Harmony Search

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Visitor Clustering (As of Nov. 2010)

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

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What is Your Contribution?

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Question for Harmony Search? Visit the Website HarmonySearch.info

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