In the real world, there are a number of optimization problems whose search space is restricted to take binary values; however, there are many continuous metaheuristics with good results in continuous search spaces. These algorithms must be adapted to solve binary problems. This paper surveys articles focused on the binarization of metaheuristics designed for continuous optimization.
In practically any activity that is performed, the resources are scarce; thus, we must properly utilize such resources. To this end, we can use technical optimization. Such problems are common in engineering, economics, machine learning, agriculture and, others areas. We found applications in learning automata in dynamic environments [
Some models are logical only if the variables take on values from a discrete set, often a subset of integers, whereas other models contain variables that can take on any real value. Models with discrete variables are discrete optimization problems, and models with continuous variables are continuous optimization problems. In general, continuous optimization problems tend to be easier to solve than discrete optimization problems; the smoothness of the functions means that the objective function and constraint function values at a point
Discrete optimization, that is, the identification of the best arrangement or selection of a finite number of discrete possibilities [
The difficulty level of such optimization problems is conceptualized by the theory of computational complexity [
A nondeterministic machine has two stages: the first one is the guessing stage and the second one is the checking stage. In the case of class NP, this checking stage is computable in polynomial time. A subclass of problems in NP is called NPcomplete. A problem
Many important discrete optimization problems are known to be NPhard; that is, in the worst case, the time required to solve a problem instance to optimality increases exponentially with its size; hence, these problems are easy to describe and understand but are difficult to solve. Even for problems of moderate size, it is practically impossible to determine all possibilities to identify the optimum. Consequently, heuristic approaches, that is, approximate solution algorithms, are considered to be the only reasonable way to solve difficult discrete optimization problems. Accordingly, there is a vast and still growing body of research on metaheuristics for discrete optimization that aim at balancing the tradeoff between computation time and solution quality [
Metaheuristics provide general frameworks for the creation of heuristic algorithms based on principles borrowed from classical heuristics, artificial intelligence, biological evolution, nervous systems, mathematical and physical sciences, and statistical mechanics. Although metaheuristics have proven their potential to identify highquality solutions for many complex reallife discrete optimization problems from different domains, the effectiveness of any heuristic strongly depends on its specific design [
However, many of the wellknown metaheuristics originally worked on continuous spaces because these can be formulated naturally in a real domain; examples of these metaheuristics are particle swarm optimization (PSO) [
In contrast to the continuous binary approaches, we also found in the literature the inverse transformation, that is, from discrete techniques to continuous [
This article is a review of the main binarization methods used when we are putting continuous metaheuristics to work in binary search spaces. The remainder of this paper is organized as follows. In Section
This section establishes the definitions and notations required for understanding the discretization and binarization techniques. For this purpose, we need to define some basic concepts.
The main goal of optimization metaheuristics is to resolve an optimization problem. An optimization problem corresponds to the pair of search space
A search space,
Let
Let
There are several optimization techniques, as shown in the overview in Figure
Optimization techniques.
In general terms, a metaheuristic attempts to find values for the variables that will provide an optimum value of the objective function.
As our focus is on the continuous, discrete, and binary searches spaces (see Section
Discrete variables arise in many optimization problems, for example, in manufacturing [
For example, in the knapsack problem, the
Then, we want to maximize
Another situation where the use of discrete variables is appropriate is when we need to manage constraints that involve logical conditions. For example, suppose that we want
Another common situation that requires integer variables is when the problem involves setup costs. As an example, consider a generator that supplies electricity to a local region with
If we let
A metaheuristic is formally defined as an iterative generation process that guides a subordinate heuristic by combining intelligently different concepts for exploring and exploiting the search space, and learning strategies are used to structure information to efficiently find nearoptimal solutions [
Metaheuristics are higher level strategies that guide the search process.
The goal is to efficiently explore the search space to find (quasi)optimal solutions.
Metaheuristic algorithms are approximate and generally nondeterministic.
The basic concepts of metaheuristics permit an abstract level of description.
Metaheuristics are not problem specific.
Metaheuristics may utilize domainspecific knowledge in the form of heuristics that are controlled by the upper level strategy.
Today, more advanced metaheuristics use search experience (embodied in some form of memory) to guide the search.
Many of the metaheuristic techniques are motivated in an
There are many problems that require discrete search spaces [
Rounding off generic technique.
Priority position techniques associated with scheduling problems.
Specific techniques associated with metaheuristic discretizations.
This approach is one of the most commonly used approaches for managing discrete variables due to its simplicity and low computational cost. It is based on the strategy of rounding to the nearest integer. It was first used in [
The rounding off operator transforms
There are multiple problems that use this method, for example, optimization of transport aircraft wing [
The randomkey encoding scheme is used to transform a position in a continuous space into a position in a discrete space. The randomkey was first used in a scheduling problem in [
Let us start with a solution
Example of randomkey encoding scheme.

0.15  0.56  0.99  0.12  0.45  0.76  0.73  0.87  0.95 

2  4  9  1  3  6  5  7  8 
This method has been used with the gravitational search algorithm [
The same method, but called small value position (SVP), was used for the first time in [
This method has been used in [
Let us start with a continuous metaheuristic that produces values in
Let
This procedure allows us to map to discrete values between 1 and
In our study and the conceptualization of binarization techniques, we found two main groups of binarization techniques. The first group of techniques we call twostep binarization; these techniques allow working with the continuous metaheuristics without operator modifications and include two steps after the original continuous iteration; these two steps make the binarization of the continuous solution. The second group of techniques is called continuousbinary operator transformation; it redefines the algebra of the search space, thereby reformulating the operators.
This technique works with the continuous operators without modifications. To perform the binarization, two additional steps are applied. The first step corresponds to introducing operators that transform the solution from
A general scheme for twostep binarization methods.
In our revision, we found that two types of functions have been used in research: the Sshape [
(a) Sshape and (b) Vshape and transfer functions.
Let
Elitist roulette.
In particle swarm optimization, this approach was first used in [
Let us start with the solution
The vector

6.2  7.3  2.4  7.8  9.1  2.5  6.9  5 

5  3  8  2  1  7  4  6 

1  0  1  1  0  1  0  1 
This technique has been used in other types of binary problems; for example, in [
In binary heuristic optimization, this method was first applied in PSO using a set of benchmark functions [
Consider an
In the first step from the fourdimensional space, we obtain a function in a function space. Specifically, from every solution
Then, for each initial 4dimensional solution
The angle modulation technique was applied to network reconfiguration problems in [
These methods are characterized to redefine the operators of the metaheuristic, and there are two main groups. We call the first group modified algebraic operations. In this group, the algebraic operations of the search space are modified, and examples include the Boolean approach and set approach. The second group is called promising regions, and the operators are restructured in terms of selected promising regions in the search spaces. This selection is performed using a probability vector. Examples of this group include the quantumbased binary approach and binary method based on the estimation of distribution.
In Figure
Example of Boolean approach.
This method has been applied to different binary optimization problems using the particle swarm method [
In the literature, there are numerous set frameworks applied to PSO algorithms. Reference [
In a general framework, it is necessary to define some operations; let
The addition of two velocities:
The difference between two positions:
Multiplication of a velocity by a scalar:
Addition of velocity and position:
Using these operations, our equations must be modified:
This method modifies the operators velocity and position, and the construction is not simple. There are many variations of (
In the quantum approach, each feasible solution has a position
Then, the new
The quantum method has been applied to a swarm optimization algorithm in combinatorial optimization [
The new solutions are obtained by sampling the search space using EDA. After each iteration, the distribution is reestimated using the new candidates.
In the case of binary optimization, [
Let
We want to obtain a particle
With the particle
If random()
Else
With this rule, we obtain
This method was constructed for a particle swarm optimization technique. However, it is easy to adapt for other metaheuristics. The advantage of this procedure is its adaptation on each iteration; however, it needs to adjust the parameters
This section aims to summarize the techniques and problems recently addressed. Additionally with the information obtained from the articles along with our experience in the area, we want to capture our vision of what are the trends in binarization. This last point is very difficult to answer and is not intended to be a quantitative analysis but rather our vision regarding the area.
From 65 papers, we have summarized, reviewed, and classified techniques that allow transforming continuous metaheuristics into discrete or binary metaheuristics. Figure
Number of papers by technique.
Another technique that appeared quite often was the quantum binary approach (QBA). From the articles read it is observed that the implementations are particular for each metaheuristic, with quite good results. From our point of view, there is a line of research associated with designing a general quantum mechanism that allows binarizing any continuous metaheuristic. Another important point to work on is the development of a methodology for the selection of parameters associated with binarization.
For the case of angle modulation, there is space to perform binarizations on new metaheuristics, where different variations of angle modulation can be explored. This exploration of new binarizations is very powerful if it is accompanied by analysis of positions and velocities of particles of the system to understand the conditions in which angle modulation works properly. As a suggestion we propose reading the work done in [
From the point of view of problems, the greater number of problems addressed corresponds to classic problems such as the knapsack (KS), set covering problem (SCP), and traveling salesman problem (TSP). The summary is shown in Figure
Number of papers by problem.
This work surveyed important discretization and binarization methods of continuous metaheuristics. Inside the binarization conglomerate, we propose two main group classifications. The first group we call twostep binarization methods, which use an intermediate space from where the binarization is mapped. The second group we call continuousbinary operator transformation, where the metaheuristic operator is adapted to a binary problem. When we analyze the operator adaptation, we found methods that transform the algebraic operations and methods that use a probability for performing the transition in the search space. Table
Classification summary of binary approach.
Discretization  (i) Rounding off  
(ii) Randomkey or small value position  
(iii) Metaheuristic discretization  


Binarization  (i) Twostep binarization  (i) Transfer function and binarization 
(ii) Great value priority (GVP) and mapping  
(iii) Angle modulation and rule  
(ii) Continuousbinary operator transformation  (i) Modified algebraic operations: setbased approach  
(ii) Promising regions: quantum binary approach  
(iii) Promising regions: binary method based on estimation of distribution 
Moreover, we provide a summary of the main discretization techniques, indicating the metaheuristic that was used and what problem was resolved. This summary is shown in Table
Summary of discretization methods.
Binarization techniques  Metaheuristic  Problem  References 

Rounding off  Particle swarm  Voltage control  [ 
Transport aircraft wing  [  
Task assignment  [  
Ant colony  Distribution system Reconfiguration  [  
Firefly  Integer programming  [  


Randomkey or small value position  Memetic  Shop scheduling  [ 
Hybrid gravitationalannealing  Traveling salesman  [  
Particle swarm  Permutation flowshop sequencing  [  
Firefly  Scheduling jobs on grid computing  [  
Cuckoo search  Reliable embedded system reconfiguration  [  


Metaheuristic discretizations  Teachinglearning  Distribution system  [ 
Additionally, we investigate what specific metaheuristics use these binarization techniques. The conclusion is that the most frequently used method is the transfer function, belonging to twostep binarization. Furthermore, we searched for what type of optimizations problems were resolved by the different techniques. The summary is shown in Tables
Summary of twostep binarization methods.
Binarization techniques  Metaheuristic  Problem  References 

Transfer function  Firefly  Set covering problem  [ 
Synthesis of thinned planar antenna array  [  
Nonlinear binary optimization  [  
Network and reliability constrained unit commitment problem  [  
Permutation flowshop scheduling problem  [  
Algae  Knapsack problem  [  
Artificial bee colony  Set covering problem  [  
Thermal unit commitment  [  
Cuckoo search  Bulk power system  [  
Differential evolutionary  Multiagent systems  [  
Knapsack problems  [  
Binary bat  Unimodal, multimodal  [  
Traveling Salesman  [  
Gravitational search  Unimodal, multimodal  [  
Open source development model  Combinatorial problems  [  
Particle swarm  Optimize sizing of capacitor banks  [  
Bulk power system  [  
Network reconfiguration  [  
Unit commitment problem  [  
Knapsack problems  [  
Teachinglearning based  Designing plasmonic nanobipyramids based on absorption coefficient  [  
Electromagnetismlike method  Traveling sales  [  
Catfish  Feature selection  [  


Great value priority  Binary Bat  Antenna positioning problem  [ 
Particle swarm  Quadratic assignment problem  [  


Angle modulation  Particle swarm 

[ 
Binary problems  [  
Finding defensive islands of largescale power systems  [  
Differential evolution  Knapsack problems  [  
Binary problems  [  
Artificial bee colony  Binary problems  [  
Feature selection  [  
Binary bat  Graph coloring  [  
Antenna positioning problem  [ 
Summary of continuousbinary operator transformation.
Binarization techniques  Metaheuristic  Problem  Reference 

Boolean approach  Particle swarm  Antenna design problem  [ 
Binary problems  [  
Binary artificial bee colony  Binary problems  [  


Setbased approach  Particle swarm  Traveling salesman problem  [ 
Multidimensional knapsack problem  
Vehicle routing problem  [  
Feature selection  [  
Jumping frogs  Combinatorial problems  [  
Water cycle  Truss structure  [  
Mine blast  Truss structure  [  
Gravitational  Traveling Salesman  [  
Imperialist competition  Transmission expansion Planning  [  
Invasive weed  Typical benchmark functions (Sphere, Rosenbrock, Rastrigin, Griewank)  [  
Social impact theory  Pattern recognition  [  


Quantum binary approach  Particle swarm  Competitive facility  [ 
Location problems  
Knapsack problem  [  
Power quality monitor placement method  [  
Differential evolution  Knapsack problem  [  
Combinatorial problems  [  
Image thresholding  [  
Cuckoo search  01 knapsack problem  [  
Bin packing problem  [  
Ant colony optimization  Image thresholding  [  
Harmony search  01 knapsack problem  [  
Monkey  01 knapsack problem  [  


Binary method based on estimation of distribution  Particle swarm  Knapsack problem  [ 
Differential Evolution  Optimization problems  [  
Genetic  Economic dispatch problem  [  
Local search  Probabilistic traveling salesman problem  [  
Memetic  Probabilistic traveling salesman problem  [ 
The principal research in this area is to try to understand in a general way how exploration and exploitation properties are mapped from continuous metaheuristics to discrete or binary metaheuristics. This allows improving the result of the metaheuristics and enlarging the spectrum of discrete or binary problems to solve. This compilation work of discretization and binarization techniques allows us to conclude that no general technique exists that allows for efficient discretization.
The authors declare that they have no conflicts of interest.
Broderick Crawford is supported by Grant CONICYT/FONDECYT/REGULAR/1171243, Ricardo Soto is supported by Grant CONICYT/FONDECYT/REGULAR/1160455, José García is supported by INFPUCV 2016, and Gino Astorga is supported by Postgraduate Grant, Pontificia Universidad Católica de Valparaíso, 2015.