Music-Inspired Optimization Algorithms: From Past to Present

Abstract

This chapter illustrates the definition of music with regard to its historical roots and then denotes the different interpretations of music from the standpoint of well-known philosophers and scientists. A concise history of music is presented through a review of archaeological evidence. Besides these initial topics, Chap. 3 deals with the music-inspired meta-heuristic optimization algorithms from past to present: the single-stage computational single-dimensional harmony search algorithm (SS-HSA); the single-stage computational single-dimensional improved harmony search algorithm (SS-IHSA); and the continuous two-stage computational, multidimensional, single-homogeneous melody search algorithm (TMS-MSA). This chapter also helps readers to identify the enhancements applied on the original SS-HSA in the form of a structural classification, including (1) the enhanced versions of the original SS-HSA, based on parameter adjustments; (2) enhanced versions of the original SS-HSA, according to a combination of this algorithm with other meta-heuristic optimization algorithms; and (3) enhanced versions of the original SS-HSA, in accordance with architectural principles. Finally, the chapter elaborates on reasonability and applicability of the music-inspired meta-heuristic optimization algorithms from past to present for solving complicated, real-world, large-scale, non-convex, non-smooth optimization problems and, subsequently, outlines a valuable background for elucidating innovative versions of the music-inspired meta-heuristic optimization algorithms in Chap. 4.

Appendices

Appendix 1: List of Abbreviations and Acronyms

AIP	Alternative improvisation procedure
BW	Bandwidth
CDVs	Continuous decision-making variables
DDVs	Discrete decision-making variables
GA	Genetic algorithm
HM	Harmony memory
HMCR	Harmony memory considering rate
HMS	Harmony memory size
HSA	Harmony search algorithm
IHSA	Improved harmony search algorithm
MM	Melody memory
MNI	Maximum number of improvisations/iterations
MNI-PGIS	Maximum number of improvisations/iterations of the pseudo-group improvisation stage
MNI-SIS	Maximum number of improvisations/iterations of the single improvisation stage
MSA	Melody search algorithm
NCDV	Number of continuous decision-making variables
NDDV	Number of discrete decision-making variables
NDV	Number of decision-making variables including continuous and discrete decision-making variable
PAR	Pitch adjusting rate
PGIS	Pseudo-group improvisation stage
PMCR	Player memory considering rate
PMs	Player memories
PMS	Player memory size
PN	Player number
SIS	Single improvisation stage
SOSA	Symphony orchestra search algorithm
SS-HSA	Single-stage computational, single-dimensional harmony search algorithm
SS-IHSA	Single-stage computational, single-dimensional improved harmony search algorithm
TMS-EMSA	Two-stage computational, multidi-mensional, single-homogeneous enhanced melody search algorithm
TMS-MSA	Two-stage computational, multidi-mensional, single-homogeneous melody search algorithm

Appendix 2: List of Mathematical Symbols

Index:
b	Index for equality constraints running from 1 to B
e	Index for inequality constraints running from 1 to E
m	Index for improvisations/iterations running from 1 to MNI in the SS-HSA and also running from 1 to (MNI‐SIS) + (MNI‐PGIS) in the TMS-MSA
p	Index for existing players in a music group running from 1 to PN
s, s∗	Index for harmony vectors stored in the HM running from 1 to HMS in the SS-HSA and also an index for melody vectors stored in each PM running from 1 to PMS in the TMS-MSA
v	Index for decision-making variables, including the continuous and discrete decision-making variables, running from 1 to NDV in the SS-HSA and also an index for continuous decision-making variables running from 1 to NCDV in the TMS-MSA
w v	Index for candidate permissible values of discrete decision-making variable v running from 1 to Wv in the SS-HSA
Set:
ΨB	Set of indices of equality constraints
ΨE	Set of indices of inequality constraints
ΨHMS	Set of indices of harmony vectors stored in the HM
ΨMNI	Set of indices of improvisations/iterations in the SS-HSA
Ψ(MNI‐SIS) + (MNI‐PGIS)	Set of indices of improvisations /iterations in the TMS-MSA
ΨNDV	Set of indices of decision-making variables, including the continuous and discrete decision-making variables
ΨNCDV	Set of indices of continuous decision-making variables
ΨNDDV	Set of indices of discrete decision-making variables
ΨPMS	Set of indices of melody vectors stored in each PM
ΨPN	Set of indices of existing players in a music group
W v	Set of indices of candidate permissible values of discrete decision-making variable v
Parameters:
BW	Bandwidth
BWmax	Maximum bandwidth
BWmin	Minimum bandwidth
HMCR	Harmony memory considering rate
HMS	Harmony memory size
MNI	Maximum number of improvisations/iterations in the SS-HSA
MNI‐SIS	Maximum number of iterations of the SIS in the TMS-MSA
MNI‐PGIS	Maximum number of iterations of the PGIS in the TMS-MSA
PAR	Pitch adjusting rate
PARmax	Maximum pitch adjusting rate
PARmin	Minimum pitch adjusting rate
PMCR	Player memory considering rate
PMS	Player memory size
${x}_v^{\mathrm{max}}$	Upper bound on the decision-making variable v
${x}_v^{\mathrm{min}}$	Lower bound on the decision-making variable v
X	Nonempty feasible decision-making space
Z	Feasible objective space
Variables:
BW m	Bandwidth in improvisation/iteration m of the SS-HSA or bandwidth in improvisation/iteration m of the TMS-MSA
f(x)	Objective function of the optimization problem
f(xs)	Value of the objective function—Fitness function—Derived from the harmony vector s stored in the HM matrix
$f\left({\mathrm{x}}_p^s\right)$	Value of the objective function—Fitness function—Derived from the melody vector s stored in memory submatrix relevant to existing player p in the musical group
$f\left({\mathrm{x}}_m^{\mathrm{new}}\right)$	Value of the objective function—Fitness function—Derived from the new harmony vector in improvisation/iteration m of the SS-HSA
$f\left({\mathrm{x}}_{m,p}^{\mathrm{new}}\right)$	Value of the objective function—Fitness function—Derived from the new melody vector played by existing player p in the musical group in improvisation/iteration m of the TMS-MSA
F(x)	Vector of objective function of the optimization problem
gb(x)	Equality constraint b of the optimization problem or component b of the vector of equality constraints
G(x)	Vector of equality constraints of the optimization problem
he(x)	Inequality constraint e of the optimization problem or component e of the vector of inequality constraints
H(x)	Vector of inequality constraints of the optimization problem
HM	Harmony memory matrix
k	Random integer with a uniform distribution through the set {1, 2,  … , NCDV} in the TMS-MSA
MM	Melody memory matrix
PAR m	Pitch adjusting rate in improvisation/iteration m of the SS-HSA or pitch adjusting rate in improvisation/iteration m of the TMS-MSA
PM p	Memory submatrix relevant to existing player p in the musical group
r	Random integer with a uniform distribution through the set {1, 2,  … , HMS} in the SS-HSA and random integer with a uniform distribution through the set {1, 2,  … , PMS} in the TMS-MSA
t	Random integer with a uniform distribution through the set {−1, +1}
U(0, 1)	Random number with a uniform distribution between 0 and 1
x v	Decision-making variable v or component v of the vector of decision-making variable
${x}_{m,v}^{\mathrm{new}}$	Element v of the new harmony vector in improvisation/iteration m of the SS-HSA
${x}_{m,p,v}^{\mathrm{best}}$	Element v of the best melody vector stored in the memory submatrix relevant to existing player p in the musical group in improvisation/iteration m of the TMS-MSA
${x}_{m,p,v}^{\mathrm{new}}$	Element v of the new melody vector played by existing player p in the musical group in improvisation/iteration m of the TMS-MSA
${x}_v^s$	Element v of the harmony vector s stored in the HM matrix
${x}_{p,v}^s$	Element v of the melody vector s stored in the memory submatrix relevant to existing player p in the musical group
xv(wv)	Candidate permissible value w of discrete decision-making variable v
x	Vector of decision-making variables
${\mathrm{x}}_m^{\mathrm{new}}$	New harmony vector in improvisation/iteration m of the SS-HSA
${\mathrm{x}}_{m,p}^{\mathrm{new}}$	New melody vector played by existing player p in the musical group in improvisation/iteration m of the TMS-MSA
xbest	Best harmony vector stored in the HM matrix in the SS-HSA and also best melody vector stored in the MM matrix in the TMS-MSA
${\mathrm{x}}_p^{\mathrm{best}}$	Best melody vector stored in the memory submatrix relevant to existing player p in the musical group
xs	Harmony vector s stored in the HM matrix
${\mathrm{x}}_p^s$	Melody vector s stored in the memory submatrix relevant to existing player p in the musical group
xworst	Worst harmony vector stored in the HM matrix
${\mathrm{x}}_p^{\mathrm{worst}}$	Worst melody vector stored in the memory submatrix relevant to existing player p in the musical group
y	Random integer with a uniform distribution through the set {xv(1),  … , xv(wv),  … , xv(Wv)}
z	Vector of the objective function