A novel metaheuristic inspired by horned lizard defense tactics

Optimization determines the optimal values of a problem’s variables to minimize (or maximize) an objective function. It can be represented in the following way:

Given a function f to minimize, where \(f:\mathbb {R}^D \rightarrow \mathbb {R}\), with D as the dimension of the problem (number of variables), sought a vector \(x_{o} \in \mathbb {R}^D\) such that \(f({x_o}) \leqslant f(x) \,\, \forall \, x \in \mathbb {R}^D\), while \(x_{o}\) satisfying inequality and equality constraints, \({g_p}({x_0}) \leqslant 0\) and, \({h_q}({x_0}) = 0\), with \(p = 1,2,..,P\) and, \(q = 1,2,..,Q\), where P and Q, are number of inequality and equality constraints, respectively. Additionally, it must be met that \(x_i^l \leqslant {x_i} \leqslant x_i^u\), where \(i=1,2,..,D\) and i-th variable varies in the interval \([x_i^l,x_i^u]\).

Optimization problem-solving strategies are classified as Deterministic or Stochastic. Deterministic methods are grouped as gradient-based or not gradient-based, where both classifications show good performance for linear, convex, and simple optimization tasks. Nevertheless, these methods don’t work in certain cases that involve complex problems, objective functions that can’t be differentiated, search spaces that aren’t linear, problems that aren’t convex, and NP-hard problems. Given that many real-world optimization problems are NP-hard, the scientific community frequently uses stochastic methodologies such as metaheuristics instead of deterministic methods.

Some of the metaheuristics and their classification are depicted in Fig. 1 and described as follows:Since no algorithm can effectively and efficiently solve any optimization or instances of the same problem (Joyce and Herrmann 2017), the scientific community keeps developing new algorithms to solve challenging optimization problems that outperform or vie with those already described in the literature.

A generic metaheuristic framework consists of four phases described below and depicted in Fig. 2.

Phase 4: The vector with the best fitness is then compared to the fitness of the previously recombined vectors in each iteration. If the number of iterations has not been met, stop condition, it returns to phase 2. Otherwise, the best fitness value reached is reported. The number of iterations, known as generations, is a value defined before starting the algorithm that influences the algorithm’s performance in this evolutionary process.

Finally, every metaheuristic should have a good balance between the ability to explore (diversify) and exploit (intensify) the search solution space. In other words, it should have global and local search strategies to improve its performance.

This paper proposes a Horned Lizard Optimization Algorithm (HLOA) as a novel swarm-based algorithm. This algorithm was inspired by how the horned lizard reptile conceals and defends itself from predators. The contributions of this work can be briefed as follows:The remaining sections of this paper are structured as follows: The second section illustrates the HLOA bio-inspiration, a detailed mathematical formulation, the time complexity, and the pseudo-code. Then, the performance of the proposed approach is benchmarked with several testbench functions, and their comparison with ten recent bio-inspired algorithms is presented in the third section. In the fourth section, the algorithm’s results and discussion are presented. The fifth section describes the application of HLOA to real-world optimization problems and the constraint-handling technique employed. Finally, the paper summarizes the conclusions and future work.

The numerical efficiency and stability of the HLOA algorithm were evaluated by solving 63 classical benchmark optimization functions reported in the literature. Each of these functions is described in Appendix A, Tables 30, 31, and 32, where Dim represents the function’s dimension, Interval is the boundary of the search space for the function and, the optimum value is \(f_{min}\). The HLOA algorithm was compared with ten recent bio-inspired algorithms as described below:Each algorithm’s benchmark function was executed 30 times, with the population size and number of iterations set to 30 and 200, respectively. Furthermore, the Wilcoxon signed-rank test was used to compare their performance, and the ranking of each algorithm was obtained by the Friedman test. It is to be noted that the three best-ranked algorithms determined by this selection are then evaluated on IEEE CEC 2017 and CEC 2019, as described below.

IEEE CEC 2017 Testbech Functions: Experimental studies are performed on 28 10-dimensional, 30-dimensional, 50-dimensional, and 100-dimensional benchmark problems from the IEEE CEC 2017 “Constrained Real-Parameter Optimization”, as outlined in Table 33.

IEEE CEC-06 2019 Testbech Functions: The “100-Digit Challenge” testbench functions from IEEE CEC-06 2019 are described in Table 3.

The entire parameter settings for each algorithm are shown in Table 2. All experiments were conducted on a standard desktop computer with the following specifications: Intel core i7-10750 H CPU, 2.60 GHz, 32 GB RAM, Linux Kubuntu 20.04 LTS operating system, and implemented in MATLAB R2021b.

This section presents the computational results of HLOA on benchmark optimization problems. The comparison results are shown in Table 4, displaying the best, mean, and standard deviation values. Moreover, Figs. 9, 10, and 11 summarize the convergence graphs of all functions versus all algorithms chosen for this investigation. To analyze the significant differences between the results of the proposed HLOA and the other algorithms, a non-parametric Wilcoxon Signed-rank test with a significance level of \(\%5\) was conducted. This trial determines the significance level of two algorithms. An algorithm is statistically significant if the calculated p-value is less than 0.05. Table 5 summarizes the result of this test. Furthermore, the eleven algorithms were ranked by computing the Friedman test for 63 benchmark functions. Friedman test ranks the algorithms according to their average performance and generates a ranking score, where a lower value indicates a better performance. The results are shown in Table 6. According to the statistical data presented in Table 4, the Horned Lizard Optimization Algorithm (HLOA) can achieve exceptional outcomes. Among the 63 benchmark functions, there are unimodal and multimodal functions to test the algorithm’s capabilities related to the exploitation and exploration in the solution space. A more reliable way to compare the performance of algorithms when solving benchmark functions is through statistical tests. The results of the Wilcoxon rank-sum test, in Table 5, indicate that HLOA outperformed the following algorithms: Black Widow Optimization Algorithm (BWOA), Dingo Optimization Algorithm (DOA), Coot Bird Algorithm (COOT), Crystal Structure Algorithm (CSA), Enhanced Jaya Algorithm (EJAYA), Rat Swarm Optimizer (RSO), Smell Agent Optimization (SAO) and, Tunicate Swarm Algorithm (TSA). Meanwhile, there are no significant differences between HLOA versus the Jumping Spider Optimization Algorithm (JSOA) and Wild Horse Optimizer (WHO). Moreover, in the Friedman test analysis, the HLOA algorithm is ranked first, whereas the JSOA, WHO, and BWOA are ranked second, third, and fourth, respectively. Based on this analysis, in the rest of the paper, these algorithms are only used in IEEE CEC 2017 ”Constrained Real-Parameter Optimization”, IEEE CEC-06 2019 “100-Digit Challenge”, and real-world applications.

On the other hand, the findings obtained by computational analysis of HLOA on benchmark functions from IEEE CEC 2017 “Constrained Real-Parameter Optimization” problems for dimensions 10, 30, 50, and 100 are shown in Tables 7, 8, 9, and 10. Say tables display the best, mean, and standard deviation computed. To examine the disparities among the algorithms, a Wilcoxon signed-rank test was used with a significance level of \(5\%\). The ranking between the algorithms HLOA, BWOA, JSOA, and WHO was determined using the Friedman test.

Table 11 summarizes the Wilcoxon rank-sum test for 10-dimensional problems and indicates that HLOA outperformed all the algorithms. Nevertheless, in Table 12 "Friedman test", it is ranked in second place.

For 30-dimensional problems, the Wilcoxon rank-sum test in Table 13 shows that HLOA outperformed the JSOA and BWOA algorithms. Meanwhile, there are no significant differences with WHO. Whereas, in Table 14 "Friedman test" ranks HLOA in second place.

According to the results presented in Table 15, it can be observed that in the case of 50-dimensional problems, the HLOA algorithm demonstrated superior performance compared to the JSOA and BWOA algorithms, as indicated by the Wilcoxon rank-sum test. In contrast, there are no substantial disparities with the WHO. In comparison, the Friedman test assigns the highest rank to HLOA. See Table 16.

In the context of 100-dimensional problems, the Wilcoxon rank-sum test in Table 17 shows that HLOA outperformed all algorithms. Meanwhile, the Friedman test has determined that HLOA holds the highest rank, as seen in Table 18.