Secretary bird optimization algorithm

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Secretary bird optimization algorithm (SBOA) is a metaheuristic optimization algorithm introduced in 2024. It is a population-based method inspired by the hunting and predator-evasion behaviour of the secretarybird (Sagittarius serpentarius). In the published formulation, the search process is divided into two main stages: an exploration phase based on hunting behaviour and an exploitation phase based on escape behaviour.^[1]

A 2025 review described SBOA as a recent bio-inspired optimizer and surveyed its variants and application areas.^[2]

Metaheuristic optimization algorithms are widely used for solving nonlinear, multimodal, and high-dimensional problems for which exact mathematical optimization methods may be difficult or expensive to apply.^[1] SBOA was proposed within the broader family of nature-inspired algorithms and bio-inspired computing methods.^[1]

In the original paper, SBOA was presented as a population-based optimizer in which each search agent represents a candidate solution. The algorithm was designed around two broad behavioural themes attributed to secretary birds in nature: hunting prey, especially snakes, and avoiding predators.^[1]

The secretarybird (Sagittarius serpentarius) is a large terrestrial bird of prey native to sub-Saharan Africa and associated with open grasslands and savannas.^[3] It is noted for ground-based hunting, especially against reptiles and other small prey, and for its distinctive kicking and stamping behaviour when attacking prey.^[4]

In the SBOA paper, this biological inspiration is mapped to the search process as follows: preparatory behaviour corresponds to initialization, hunting behaviour corresponds to exploration, and predator-evasion behaviour corresponds to exploitation.^[1]

SBOA is a population-based optimization method. Each secretary bird in the population corresponds to one candidate solution in the search space, and the position of a bird determines the values of the decision variables.^[1]

The algorithm begins with a randomly initialized population bounded by lower and upper search limits. Candidate solutions are then iteratively updated through two major stages:

• Exploration phase, modeled on hunting behaviour;
• Exploitation phase, modeled on escape behaviour.^[1]

At each iteration, the objective function is evaluated for all candidate solutions, and the current best solution is retained.^[1]

In the original formulation, the position of each search agent is initialized within problem bounds as

$${\displaystyle X_{i,j}=lb_{j}+r\,(ub_{j}-lb_{j}),\qquad i=1,2,\ldots ,N,\quad j=1,2,\ldots ,\mathrm {Dim} }$$

where ${\displaystyle lb_{j}}$ and ${\displaystyle ub_{j}}$ are the lower and upper bounds of the ${\displaystyle j}$-th variable, ${\displaystyle r}$ is a random number in ${\displaystyle [0,1]}$, ${\displaystyle N}$ is the population size, and ${\displaystyle \mathrm {Dim} }$ is the dimension of the problem.^[1]

The population matrix is expressed as

$${\displaystyle X={\begin{bmatrix}x_{1,1}&x_{1,2}&\cdots &x_{1,\mathrm {Dim} }\\x_{2,1}&x_{2,2}&\cdots &x_{2,\mathrm {Dim} }\\\vdots &\vdots &\ddots &\vdots \\x_{N,1}&x_{N,2}&\cdots &x_{N,\mathrm {Dim} }\end{bmatrix}}}$$

and the vector of objective-function values is written as

$${\displaystyle F={\begin{bmatrix}F(X_{1})\\F(X_{2})\\\vdots \\F(X_{N})\end{bmatrix}}.}$$

^[1]

In the SBOA paper, hunting behaviour is divided into three stages: searching for prey, consuming prey, and attacking prey.^[1]

The first stage is associated with broad search and is modeled using a differential-update mechanism:

$${\displaystyle x_{i,j}^{\mathrm {new} ,P1}=x_{i,j}+\left(x_{\mathrm {random\_1} }-x_{\mathrm {random\_2} }\right)\times R_{1},\qquad t<{\frac {1}{3}}T}$$

where ${\displaystyle T}$ is the maximum number of iterations and ${\displaystyle R_{1}}$ is a random vector.^[1]

The second stage introduces a Brownian-motion-based local search around the best solution found so far:

$${\displaystyle RB=\mathrm {randn} (1,\mathrm {Dim} )}$$

$${\displaystyle x_{i,j}^{\mathrm {new} ,P1}=x_{\mathrm {best} }+\exp \!\left(\left({\frac {t}{T}}\right)^{4}\right)(RB-0.5)\,(x_{\mathrm {best} }-x_{i,j}),\qquad {\frac {1}{3}}T<t<{\frac {2}{3}}T}$$

where ${\displaystyle x_{\mathrm {best} }}$ denotes the current best solution.^[1]

The third stage uses a Lévy flight-based perturbation to improve global search and reduce premature convergence:

$${\displaystyle x_{i,j}^{\mathrm {new} ,P1}=x_{\mathrm {best} }+\left(1-{\frac {t}{T}}\right)^{\left(2{\frac {t}{T}}\right)}x_{i,j}\,RL,\qquad t>{\frac {2}{3}}T}$$

where

$${\displaystyle RL=0.5\times \mathrm {Levy} (\mathrm {Dim} ).}$$

^[1]

The exploitation stage models two predator-evasion strategies: camouflage and escape by running or flying away.^[1]

These alternatives are represented as two cases:

$${\displaystyle x_{i,j}^{\mathrm {new} ,P2}={\begin{cases}x_{\mathrm {best} }+(2RB-1)\left(1-{\frac {t}{T}}\right)^{2}x_{i,j},&{\text{if }}r<0.5\\[6pt]x_{i,j}+R_{2}\,(x_{\mathrm {random} }-Kx_{i,j}),&{\text{otherwise}}\end{cases}}}$$

where ${\displaystyle R_{2}}$ is a normally distributed random vector and ${\displaystyle K}$ is a randomly selected integer defined by

$${\displaystyle K=\mathrm {round} (1+\mathrm {rand} (1,1)).}$$

^[1]

The overall SBOA workflow consists of initialization, iterative position updating, fitness evaluation, and retention of the best solution found so far.^[1] In the original article, the search process is presented through both a flowchart and pseudocode, with the exploration stage occupying the earlier part of the update cycle and the exploitation stage refining candidate solutions afterward.^[1]

Like many population-based optimizers, SBOA is organized around the balance between exploration and exploitation.^[1] In the original article, exploration is associated with broader search over the solution space, while exploitation is associated with local refinement near promising solutions.^[1]

The paper also introduced diversity-based expressions for reporting exploration and exploitation percentages during the run of the algorithm.^[1]

The original paper analyzed SBOA using Big O notation. Let ${\displaystyle N}$ denote the population size, ${\displaystyle \mathrm {Dim} }$ the number of decision variables, and ${\displaystyle T}$ the maximum number of iterations. The time complexity of initialization is given as ${\displaystyle O(N)}$, while the solution-update process is described as ${\displaystyle O(T\times N)+O(T\times N\times \mathrm {Dim} )}$. The total computational complexity is summarized as

$${\displaystyle O\!\left(N\times (T\times \mathrm {Dim} +1)\right).}$$

^[1]

In the original publication, SBOA was evaluated on benchmark suites including CEC 2017 and CEC 2022, and was also applied to constrained engineering design problems and three-dimensional path planning for unmanned aerial vehicles.^[1] Later work and reviews have described additional variants and applications of the method.^[2]

SBOA has also been used as an optimizer for training multilayer perceptron models by encoding network weights and biases as candidate solutions and minimizing mean squared error.^[5]

• Metaheuristic
• Nature-inspired algorithm
• Swarm intelligence
• Lévy flight
• Differential evolution
• Multilayer perceptron