Harmony SearchHarmony Search, HS. TaxonomyHarmony Search belongs to the fields of Computational Intelligence and Metaheuristics. InspirationHarmony Search was inspired by the improvisation of Jazz musicians. Specifically, the process by which the musicians (who may have never played together before) rapidly refine their individual improvisation through variation resulting in an aesthetic harmony. MetaphorEach musician corresponds to an attribute in a candidate solution from a problem domain, and each instrument's pitch and range corresponds to the bounds and constraints on the decision variable. The harmony between the musicians is taken as a complete candidate solution at a given time, and the audiences aesthetic appreciation of the harmony represent the problem specific cost function. The musicians seek harmony over time through small variations and improvisations, which results in an improvement against the cost function. StrategyThe information processing objective of the technique is to use good candidate solutions already discovered to influence the creation of new candidate solutions toward locating the problems optima. This is achieved by stochastically creating candidate solutions in a stepwise manner, where each component is either drawn randomly from a memory of highquality solutions, adjusted from the memory of highquality solutions, or assigned randomly within the bounds of the problem. The memory of candidate solutions is initially random, and a greedy acceptance criteria is used to admit new candidate solutions only if they have an improved objective value, replacing an existing member. ProcedureAlgorithm (below) provides a pseudocode listing of the Harmony Search algorithm for minimizing a cost function. The adjustment of a pitch selected from the harmony memory is typically linear, for example for continuous function optimization: $x' \leftarrow x + range \times \epsilon$where $range$ is a the user parameter (pitch bandwidth) to control the size of the changes, and $\epsilon$ is a uniformly random number $\in [1,1]$. Input :
$Pitch_{num}$, $Pitch_{bounds}$, $Memory_{size}$, $Consolidation_{rate}$, $PitchAdjust_{rate}$, $Improvisation_{max}$
Output :
$Harmony_{best}$
Harmonies $\leftarrow$ InitializeHarmonyMemory ($Pitch_{num}$, $Pitch_{bounds}$, $Memory_{size}$)EvaluateHarmonies (Harmonies )For ($i$ To $Improvisation_{max}$)$Harmony$ $\leftarrow$ $\emptyset$ For ($Pitch_{i}$ $\in$ $Pitch_{num}$)If (Rand () $\leq$ $Consolidation_{rate}$)$RandomHarmony_{pitch}^i$ $\leftarrow$ SelectRandomHarmonyPitch (Harmonies , $Pitch_{i}$)If (Rand () $\leq$ $PitchAdjust_{rate}$)$Harmony_{pitch}^i$ $\leftarrow$ AdjustPitch ($RandomHarmony_{pitch}^i$)Else $Harmony_{pitch}^i$ $\leftarrow$ $RandomHarmony_{pitch}^i$ End Else $Harmony_{pitch}^i$ $\leftarrow$ RandomPitch ($Pitch_{bounds}$)End End EvaluateHarmonies ($Harmony$)If (Cost ($Harmony$) $\leq$ Cost (Worst (Harmonies )))Worst (Harmonies ) $\leftarrow$ $Harmony$End End Return ($Harmony_{best}$)Heuristics
Code ListingListing (below) provides an example of the Harmony Search algorithm implemented in the Ruby Programming Language. The demonstration problem is an instance of a continuous function optimization that seeks $min f(x)$ where $f=\sum_{i=1}^n x_{i}^2$, $5.0\leq x_i \leq 5.0$ and $n=3$. The optimal solution for this basin function is $(v_0,\ldots,v_{n1})=0.0$. The algorithm implementation and parameterization are based on the description by Yang [Yang2009], with refinement from Geem [Geem2010a]. def objective_function(vector) return vector.inject(0.0) {sum, x sum + (x ** 2.0)} end def rand_in_bounds(min, max) return min + ((maxmin) * rand()) end def random_vector(search_space) return Array.new(search_space.size) do i rand_in_bounds(search_space[i][0], search_space[i][1]) end end def create_random_harmony(search_space) harmony = {} harmony[:vector] = random_vector(search_space) harmony[:fitness] = objective_function(harmony[:vector]) return harmony end def initialize_harmony_memory(search_space, mem_size, factor=3) memory = Array.new(mem_size*factor){create_random_harmony(search_space)} memory.sort!{x,y x[:fitness]<=>y[:fitness]} return memory.first(mem_size) end def create_harmony(search_space, memory, consid_rate, adjust_rate, range) vector = Array.new(search_space.size) search_space.size.times do i if rand() < consid_rate value = memory[rand(memory.size)][:vector][i] value = value + range*rand_in_bounds(1.0, 1.0) if rand()<adjust_rate value = search_space[i][0] if value < search_space[i][0] value = search_space[i][1] if value > search_space[i][1] vector[i] = value else vector[i] = rand_in_bounds(search_space[i][0], search_space[i][1]) end end return {:vector=>vector} end def search(bounds, max_iter, mem_size, consid_rate, adjust_rate, range) memory = initialize_harmony_memory(bounds, mem_size) best = memory.first max_iter.times do iter harm = create_harmony(bounds, memory, consid_rate, adjust_rate, range) harm[:fitness] = objective_function(harm[:vector]) best = harm if harm[:fitness] < best[:fitness] memory << harm memory.sort!{x,y x[:fitness]<=>y[:fitness]} memory.delete_at(memory.size1) puts " > iteration=#{iter}, fitness=#{best[:fitness]}" end return best end if __FILE__ == $0 # problem configuration problem_size = 3 bounds = Array.new(problem_size) {i [5, 5]} # algorithm configuration mem_size = 20 consid_rate = 0.95 adjust_rate = 0.7 range = 0.05 max_iter = 500 # execute the algorithm best = search(bounds, max_iter, mem_size, consid_rate, adjust_rate, range) puts "done! Solution: f=#{best[:fitness]}, s=#{best[:vector].inspect}" end Download: harmony_search.rb.
ReferencesPrimary SourcesGeem et al. proposed the Harmony Search algorithm in 2001, which was applied to a range of optimization problems including a constraint optimization, the Traveling Salesman problem, and the design of a water supply network [Geem2001]. Learn MoreA book on Harmony Search, edited by Geem provides a collection of papers on the technique and its applications [Geem2009], chapter 1 provides a useful summary of the method heuristics for its configuration [Yang2009]. Similarly a second edited volume by Geem focuses on studies that provide more advanced applications of the approach [Geem2010], and chapter 1 provides a detailed walkthrough of the technique itself [Geem2010a]. Geem also provides a treatment of Harmony Search applied to the optimal design of water distribution networks [Geem2009a] and edits yet a third volume on papers related to the application of the technique to structural design optimization problems [Geem2009b]. Bibliography

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