Introduction to Meta-heuristic Optimization Algorithms

Abstract

This chapter begins with a concise definition of the optimization problem and its parameters, along with a mathematical description of an optimization problem with continuous and discrete decision-making variables whose objective functions are employed in a standard form of an optimization problem along with equality and inequality constraints. Subsequently, the authors address the classifications of an optimization problem from different perspectives, which deserve attention and can achieve full knowledge regarding an optimization problem and its parameters. In addition, a succinct overview pertaining to the optimization algorithms with a focus on meta-heuristic optimization algorithms is reported.

Appendices

Appendix 1: List of Abbreviations and Acronyms

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 variables
NR	Newton-Raphson
SI-MHOAs	Swarm intelligence-based meta-heuristic optimization algorithms
BI-MHOAs-NSI	Biologically inspired meta-heuristic optimization algorithms not based on swarm intelligence
P&C-MHOAs	Physics- and chemistry-based meta-heuristic optimization algorithms
H&S-MHOAs	Human behavior- and society-inspired meta-heuristic optimization algorithms

Appendix 2: List of Mathematical Symbols

Index:
a	Index for objective functions running from 1 to A
b	Index for equality constraints running from 1 to B
e	Index for inequality constraints running from 1 to E
v	Index for decision-making variables, including the continuous and discrete decision-making variables, running from 1 to the NDV and an index for continuous decision-making variables running from 1 to the NCDV and also an index for discrete decision-making variables running from 1 to the NDDV
Set:
ΨA	Set of indices of objective functions
ΨB	Set of indices of equality constraints
ΨE	Set of indices of inequality constraints
ΨNCDV	Set of indices of continuous decision-making variables
ΨNDDV	Set of indices of discrete decision-making variables
ΨNDV	Set of indices of decision-making variables, including the continuous and discrete decision-making variables
W v	Set of indices of candidate permissible values of discrete decision-making variable v
ℜ	Set of real numbers
ℜB	B-dimensional set of real numbers
ℜE	E-dimensional set of real numbers
ℜNDV	NDV-dimensional set of real numbers
Parameters:
${x}_v^{\mathrm{max}}$	Upper bound on the continuous decision-making variable v
${x}_v^{\mathrm{min}}$	Lower bound on the continuous decision-making variable v
X	Nonempty feasible decision-making space, including feasible continuous and discrete decision-making spaces
Z	Feasible objective space
Variables:
fa(x)	Objective function a of the optimization problem or component a of the vector of objective functions
F(x)	Vector of objective functions 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
x v	Continuous or discrete decision-making variable v
xv(wv)	Candidate permissible value w of discrete decision-making variable v
x	Vector of decision-making variables
z	Vector of objective functions