Among others, these include the iterative flipping algorithm for PTS (IPTS) [4] and the gradient descent method (GD) [10]. In the iterative flipping algorithm [4], each input data block is divided into M subblocks to form partial transmit sequences as in the ordinary PTS technique. We assume that b_m = 1 for all m and compute the PAPR of the combined signal. Then invert the first phase factor to b₁ = –1 and recompute the resulting PAPR. If it is lower than the previous value then keep this value for b₁, otherwise change b₁ back to its initial value. The algorithm continues that way until all other phase factors have been explored. The name “flipping” came from the fact that flipping the signs of the phase factors occurs. The search complexity of this technique is proportional to (M – 1)W.

The gradient descent method (GD) starts with a pre-determined vector of phase factors. Next, it finds an updated vector of phase factors in its “neighborhood” that results in the largest reduction in PAPR. A Neighborhood of radius r is defined as the set of vectors with Hamming distance equal to or lesser than r from its origin. The performance and complexity of the technique is dependent on the value of r. The search complexity is given by $C_{M-1}^r{W}^{r}I$, where $C_{M-1}^r$ is the binomial coefficient defined by $C_{M-1}^r=\left(\begin{array}{l}M-1\hfill \\ r\hfill \end{array}\right)$

Artificial Bee Colony (ABC) [46] is a Swarm Intelligence (SI) algorithm, which has been applied to several real-world engineering problems. The ABC algorithm models and simulates the honey bee behavior in food foraging. In ABC algorithm, a potential solution to the optimization problem is represented by the position of a food source while the nectar amount of a food source corresponds to the quality (objective function fitness) of the associated solution. In order to find the best solution the algorithm defines three classes of bees: employed bees, onlooker bees and scout bees. The employed bee searches for the food sources, the onlooker bee makes a decision to choose the food sources by sharing the information of employed bee, and the scout bee is used to determine a new food source if a food source is abandoned by the employed bee and onlooker bee. For each food source there exists only one employed bee (i.e. the number of the employed bees is equal to the number of solutions). The employed bees search for new neighbor food source near to their hive.

Ant colony optimization (ACO) [47, 48] is a meta-heuristic inspired by the ants’ foraging behavior. At the core of this behavior is the indirect communication between the ants by means of chemical pheromone trails, which enables them to find short paths between their nest and food sources. Ants can sense pheromone. When they decide to follow a path, they tend to choose the one with strong pheromone intensities way back to the nest or to the food source. Therefore, shorter paths would accumulate more pheromone than longer ones. This feature of real ant colonies is exploited in ACO algorithms in order to solve combinatorial optimization problems considered to be NP-Hard.

PSO is an evolutionary algorithm that mimics the swarm behavior of bird flocking and fish schooling [49]. The most common PSO algorithms include the classical Inertia Weight PSO (IWPSO) and Constriction Factor PSO (CFPSO) [50]. PSO is an easy to implement algorithm with computational efficiency. In PSO, the particles move in the search space, where each particle position is updated by two optimum values. The first one is the best solution (fitness) that has been achieved so far. This value is called pbest. The other one is the global best value obtained so far by any particle in the swarm. This best value is called gbest. After finding the pbest and gbest, the most commonly used velocity update rule of each particle for every problem dimension is given by:

$$u_{G+1,ni}=w{u}_{G,ni}+c_{1}ran{d}_{1\left(0,1\right)}\left(pbes{t}_{G+1,ni}-x_{G,ni}\right)+c_{2}ran{d}_{2\left(0,1\right)}\left(gbes{t}_{G+1,ni}-x_{G,ni}\right)$$(1)

where u_{G + 1,ni} is the i-th particle velocity in the n-th dimension, G + 1 denotes the current iteration and G the previous, x_{G, ni} is the particle position in the n-th dimension, rand_1(0,1), rand_2(0,1) are uniformly distributed random numbers in (0,1), w is a parameter known as the inertia weight, and c₁ and c₂ are the learning factors.

The authors in [51], introduced the shuffled frog leaping algorithm (SFLA), which is inspired by the natural behavior of the frog. SFLA uses a population-based cooperative search metaphor inspired by natural memetics. The basic idea in SFLA is to divide the population into different frog groups. Each group consists of a fixed frog number. In SFLA, the information is carried by a meme, groups of memes are called meme complexes, or "memeplexes". The authors in [37] propose a modified shuffled frog leaping algorithm (MSFLA) using PTS for PAPR reduction.

In a binary-coded GA (BGA) each chromosome encodes a binary string [52]. The most commonly used operators are crossover, mutation, and selection. The selection operator selects two parent chromosomes from the current population according to a selection strategy. The crossover operator combines the two parent chromosomes in order to produce one new child chromosome. The mutation operator is applied with a predefined mutation probability to a new child chromosome. We have selected for this paper the BGA with the same features used in [35].

The mathematical models of Biogeography are based on the work of Robert MacArthur and Edward Wilson in the early 1960s. Using this model, they have been able to predict the number of species in a habitat. The habitat is an area that is geographically isolated from other habitats. The geographical areas that are well suited as residences for biological species are said to have a high habitat suitability index (HSI). Therefore, every habitat is characterized by the HSI which depends on factors like rainfall, diversity of vegetation, diversity of topographic features, land area, and temperature. Each of the features that characterize habitability is known as suitability index variables (SIV). The SIVs are the independent variables while HSI is the dependent variable.

In BBO-PTS a solution to the M-dimensional problem can be represented as a vector of SIV variables [SIV₁, SIV₂,…………,SIV_M]^T, which is a habitat or island. The SIV variables represent the phase vector b = [b₁, b₂, …, b_M]^T . The value of HSI of a habitat is the value of the PTS objective function that corresponds to that solution and it is found by:

$$HSI=F\left(habitat\right)=F\left(SI{V}_1,SI{V}_2,\mathrm{.....}SI{V}_M\right)=F\left(b\right)$$(2)

Habitats with a low PAPR value are good solutions of the objective function, while poor solutions are those habitats with high PAPR value. The Habitats with low PAPR are those that have large population and high emigration rate μ. For these habitats, the immigration rate λ is low. The poor solutions are those that have high PAPR, which means they have high immigration rate λ and low emigration rate μ. The immigration and emigration rates are functions of the number of species in the habitats. These are given by:

$${\mu }_k=E\left(\frac{k}{S_{\max }}\right)$$(3)

$${\lambda }_k=I\left(1-\frac{k}{S_{\max }}\right)$$(4)

where I is the maximum possible immigration rate, E is the maximum possible emigration rate, k is the rank of the given candidate solution, and S_max is the maximum number of species (e.g. population size). Therefore, the best candidate solution has a rank of S_max and the worst candidate solution has a rank of one.

BBO-PTS uses both mutation and migration operators. The application of these operators to each SIV in each solution is decided probabilistically. For each generation, there is a probability P_mod ∈ [0, 1] that each candidate solution will be modified by migration. P_mod is a user-defined parameter that is typically set to a value close to one, and is analogous to crossover probability in GAs. The migration for NP habitats can be described in Algorithm 1.

The X_i in the above algorithm is habitat i. The information sharing between habitats is accomplished using the immigration and emigration rate. The λ_i is proportional to the probability that SIVs from neighbouring habitats will migrate into habitat X_i. The μ_i is proportional to the probability that SIVs from habitat X_i will migrate into neighboring habitats. The mutation rate m of a solution S is defined to be inversely proportional to the solution probability and it is given by:

$$m_S=m_{\max }\left(\frac{1-P_s}{P_{\max }}\right)$$(5)

where P_s is the probability that a habitat contains S species, P_max is the maximum P_s value over all s ∈ [1, S_max], and m_max is a user-defined parameter. Simon in [35] described how P_s changes from time t to time t + ∆t as:

$$P_s\left(t+\text{Δ}t\right)=P_s\left(t\right)\left(1-{\lambda }_s\text{Δ}t-{\mu }_s\text{Δ}t\right)+P_{s-1}\left(t\right){\lambda }_{s-1}\text{Δ}t+P_{s+1}\left(t\right){\mu }_{s+1}\text{Δ}t.$$(6)

In this paper, we use the binary mutation operator suggested in [53], which uses the inverse operation to update the habitat. The binary Mutation procedure is described in Algorithm 2.

The m_i in Algorithm 2 is the mutation rate of solution i. As with other evolutionary algorithms, BBO also incorporates elitism. This is implemented with a user-selected elitism parameter p. This means that the p best phase vectors remain from one generation to the other.

The basic concept of OBL was originally introduced by Tizhoosh in [54]. The basic idea of OBL is to calculate the fitness not only of the current individual but also to calculate the fitness of the opposite individual. Then the algorithm selects the individual with the lower (higher) fitness value. At first we give the definitions for the basic concepts of OBL [54–56].

Definition (Opposite Number) let x ∈ [a, b] be any real number. The opposite number is defined by$$x_O=a+b-x$$(7)

Definition (Opposite Point). Similarly if we extend the above definition to D-dimensional space then let P(x₁, x₂, …x_D) be a point where x₁, x₂,… x_D ∈ 𝕽 and x_j ∈ [a_j, b_j] ∀ j ∈ {1, 2, … D}. The opposite point P_O(x_O1, x_O2, …x_OD) is defined by its components$$x_{Oj}=a_j+b_j-x_j$$(8)

Definition (Semi-opposite Point) [57]. If we change the components of a point by its opposites only in some components and the other remain unchanged then the new point is a semi-opposite point. This is defined by$$P_{SO}\left(x_{SO1},x_{SO2},\mathrm{..}{x}_{SOj}\mathrm{..},x_{SOD}\right)where\forall j\in \left\{1,2,\mathrm{..},D\right\}{x}_{SOj}=\{x_{j}or{x}_{Oj}$$(9)For example in a two-dimensional space where each dimension can be either 0 or 1 we consider the point P1(0, 1). Then the two semi-opposite points are P2(0, 0) and P3(1, 1), while the opposite point is P4(1, 0).

In this paper we propose a OBBO version based on semi-opposite points. We call this algorithm Generalized OBBO (GOBBO). We define a new control parameter named opposition probability p_o ∈ [0, 1]. This parameter controls if a SIV variable in a habitat will be replaced by its opposite or not. Moreover, as in previous opposition-based algorithms [43–45] we use the jumping rate parameter j_r ∈ [0, 1] which controls in each generation if the opposite population is created or not. The opposite based algorithms require two additional parts to the original algorithm code; the opposition-based population initialization and the opposition-based generation jumping [43–45]. The opposition based population initialization for GOBBO is described in Algorithm 3. For this case low_j, upper_j are the lower and upper limits in the j-th dimension respectively.

The opposition-based generation jumping follows a similar approach. The algorithm description is given in Algorithm 4. The min_j, max_j are the minimum and maximum values of the j-th dimension in the current population respectively.

The GOBBO algorithm for PAPR reduction is outlined below:

• 1)

  Initialize the GOBBO control parameters. Map the problem solutions to phase vectors and habitats. Set the habitat modification probability P_mod, the maximum immigration rate I, the maximum emigration rate E, the maximum migration rate m_max and the elitism parameter p (if elitism is desired). Set the jumping rate j_r and the opposition probability p_o.

• 2)

  Initialize a random population of NP habitats (phase vectors) from a uniform distribution. Set the number of generations G to one.

• 3)

  Initialize the opposite population.

• 4)

  Map the PAPR value to the number of species S, the immigration rate λ_k, the emigration rate μ_k for each solution (phase vector) of the population.

• 5)

  Apply the migration operator for each non-elite habitat based on immigration and emigration rates using (3) and (4).

• 6)

  Update the species count probability.

• 7)

  Apply the mutation operator.

• 8)

  Evaluate objective function value.

• 8)

  If rnd [0, 1] < j_r calculate the opposite population.

• 9)

  Sort the population according to the PAPR value from best to worst.

• 10)

  Apply elitism by replacing the p worst habitats of the previous generation with the p best ones.

• 11)

  Repeat step 3 until the maximum number of generations G_max or the maximum number of objective function evaluations is reached.

A flowchart of the GOBBO algorithm is given in Fig. 1. The time complexity of the original BBO algorithm at each iteration is 𝓞(NP²M + NP f), where f is the time complexity of the objective function and M is the problem dimensions. Sorting the population in algorithms 3 and 4 using the quick sort algorithm has time complexity 𝓞(NP²). Algorithms 3 and 4 both have time complexity 𝓞(NP² + NPM + NP f). The time complexity of the GOBBO algorithm at each iteration is therefore 𝓞(NP²(M + 1) + NPM + NP f), which reduces to 𝓞(NP²(M + 1) + NP f).

Fig. 1

GOBBO-PTS flowchart.

Similarly to the other evolutionary algorithms (EAs), such as GAs, ABC, PSO, in the BBO approach there is a way of sharing information between solutions [39]. This feature makes BBO suitable for the same types of problems that the other algorithms are used for, namely high-dimensional data. Additionally, BBO has some unique features that are different from those found in the other evolutionary algorithms. For example, quite differently from GAs, Ant Colony Optimization (ACO) [48] and PSO, from one generation to the next the set of the BBO’s solutions is maintained and improved using the migration model, where the emigration and immigration rates are determined by the fitness of each solution. BBO differs from PSO in the fact that PSO solutions do not change directly; the velocities change. The BBO solutions share directly their attributes using the migration models. The migration operator provides BBO with a good exploitation ability. These differences can make BBO outperform other algorithms [39, 58, 59]. It must be pointed out that if PSO or ABC are constrained to discrete space then the next generation will not necessarily be discrete [59] . However, this is not true for BBO; if BBO is constrained to a discrete space then the next generation will also be discrete to the same space. As the authors in [59] suggest, this indicates that BBO could perform better than other EAs on combinatorial optimization problems, which makes BBO suitable for application to the PTS problem. The main computational cost of EAs is in the evaluation of the objective function. The BBO mechanism is simple, like that of PSO and ABC. Therefore, for most problems, the computational cost of BBO and other EAs will be the same since it will be dominated by objective function evaluation [58]. More details about the BBO algorithm can be found in [39, 58, 59].