Enhanced sooty tern optimization algorithm using multiple search guidance strategies and multiple position update modes for solving optimization problems

Abstract

The Sooty Tern Optimization Algorithm (STOA) is a newly proposed bio-inspired algorithm that mimics the migration and attacking behaviors of the sea bird sooty tern in nature. STOA has several excellent advantages, including fewer parameters, a simple structure, a fast convergence rate, and high exploitation. Nevertheless, it is difficult to find the global optimal solution and prone to losing population diversity when dealing with complex optimization problems due to its single search guidance strategy and position update method. An enhanced STOA (ESTOA) is proposed to address these shortcomings that incorporates multiple search guidance strategies and position update modes. In terms of search guidance, in addition to the best individual in the original STOA, the mean individual and a randomly selected individual are also designed to guide the search. Six position update modes are proposed in conjunction with the guidance strategies, including one improved scaling mode with an extended spiral radius and five other modes based on offset operations. Due to their distinct design objectives, these guidance strategies and position update modes exhibit varying levels of search intensity and optimization effect. However, they complement one another and work cooperatively to achieve a good balance of global exploration and local exploitation. Several widely used sets of benchmark functions with a wide range of dimensions and varying degrees of complexity are used to validate ESTOA’s performance. The obtained results are compared to those of other state-of-the-art optimization algorithms in terms of convergence accuracy and a variety of numerical performance evaluation parameters. A significant improvement in solution quality demonstrates that ESTOA can increase population diversity and maintain a good balance between global exploring and local exploiting abilities.

Appendix.: Basic, CEC2015 computationally expensive, and CEC2011 real-world benchmark test functions

1.1 A.1. Basic benchmark test functions

• (1) Sphere

  ${f_{1}}(\mathbf {X}) = \sum \limits _{i = 1}^{D} {{x_{i}^{2}}}$, x_i ∈ [− 100, 100]^D, Unimodal and Separable (US)

• (2) Elliptic

  ${f_{2}}(\mathbf {X}) = \sum \limits _{i = 1}^{D} {{({10}^{6})^{(i - 1)/D - 1}}{x_{i}^{2}}}$, x_i ∈ [− 100, 100]^D, Unimodal and Non-separable (UN)

• (3) SumSquares

  ${f_{3}}(\mathbf {X}) = \sum \limits _{i = 1}^{D} {i{x_{i}^{2}}}$, x_i ∈ [− 10, 10]^D, Unimodal and Separable (US)

• (4) SumPower

  ${f_{4}}(\mathbf {X}) = \sum \limits _{i = 1}^{D} {{\left | {x_{i}} \right |}^{(i + 1)}}$, x_i ∈ [− 10, 10]^D, Multimodal and Separable (MS)

• (5) Schwefel2.22

  ${f_{5}}(\mathbf {X}) = \sum \limits _{i = 1}^{D} {\left | {x_{i}} \right |{ + }\prod \limits _{i = 1}^{D} {\left | {x_{i}} \right |}}$, x_i ∈ [− 10, 10]^D, Unimodal and Non-separable (UN)

• (6) Schwefel2.21

  ${f_{6}}(\mathbf {X}) = {\max \limits _{i}}\left \{\left | {x_{i}} \right |,1 \le i \le D\right \}$, x_i ∈ [− 100, 100]^D, Unimodal and Non-separable (UN)

• (7) Step

  ${f_{7}}(\mathbf {X}) = \sum \limits _{i = 1}^{D} {{\left (\left \lfloor {{x_{i}} + 0.5} \right \rfloor \right )}^{2}}$, x_i ∈ [− 100, 100]^D, Unimodal and Separable (US)

• (8) Quartic

  ${f_{8}}(\mathbf {X}) = \sum \limits _{i = 1}^{D} {i{x^{4}}}$, x_i ∈ [− 1.28, 1.28]^D, Unimodal and Separable (US)

• (9) QuarticWN

  ${f_{9}}(\mathbf {X}) = \sum \limits _{i = 1}^{D} {i{x_{i}^{4}} + random[0,1)}$, x_i ∈ [− 1.28, 1.28]^D, Unimodal and Separable (US)

• (10) Rosenbrock

  ${f_{10}}(\mathbf {X}) = \sum \limits _{i = 1}^{D{ - }1} {\left [100{{\left ({x_{i + 1}} - {x_{i}^{2}}\right )}^{2}} + {{({x_{i}} - 1)}^{2}}\right ]}$, x_i ∈ [− 10, 10]^D, Unimodal and Non-separable (UN)

• (11) Rastrigin

  ${f_{11}}(\mathbf {X}) = \sum \limits _{i = 1}^{D} {\left [{x_{i}^{2}} - 10{\cos \limits } (2\pi {x_{i}}) + 10\right ]}$, x_i ∈ [− 5.12, 5.12]^D, Multimodal and Separable (MS)

• (12) Non-Continuous Rastrigin

  ${f_{12}}(\mathbf {X}) = \sum \limits _{i = 1}^{D} {\left [{y_{i}^{2}} - 10{\cos \limits } (2\pi {y_{i}}) + 10\right ]}$

  ${y_{i}} = \left \{ \begin {array}{l} {x_{i}} \qquad \qquad \left | {{x_{i}}} \right | < \frac {1}{2} \\ \frac {round(2{x_{i}})}{2}\quad \left | {{x_{i}}} \right | \ge \frac {1}{2} \end {array} \right .$, x_i ∈ [− 5.12, 5.12]^D, Multimodal and Separable (MS)

• (13) Griewank

  ${f_{13}}(\mathbf {X}) = \frac {1}{4000}\sum \limits _{i = 1}^{D} {{x_{i}^{2}} - \prod \limits _{i = 1}^{D} {{\cos \limits } \left (\frac {x_{i}}{\sqrt {i}}\right )} + 1}$, x_i ∈ [− 600, 600]^D, Multimodal and Non-separable (MN)

• (14) Schwefel2.26

  ${f_{14}}(\mathbf {X}) = 418.98{\text {*}}D - \sum \limits _{i = 1}^{D} {x_{i}^{}{\sin \limits } \left (\sqrt {\left | {{x_{i}}} \right |} \right )}$, x_i ∈ [− 500, 500]^D, Unimodal and Non-separable (UN)

• (15) Ackley

  ${f_{15}}(\mathbf {X}) = { - }20\exp \left \{ - 0.2\sqrt {\frac {1}{D}\sum \limits _{i = 1}^{D} {{x_{i}^{2}}} } \right \} - \exp \left \{ \frac {1}{D}\sum \limits _{i = 1}^{D} {{\cos \limits } (2\pi {x_{i}})} \right \} + 20 + e$, x_i ∈ [− 32, 32]^D, Multimodal and Non-separable (MN)

• (16) Penalized1

  ${f_{16}}(\mathbf {X}) = \frac {\pi }{D}\left \{ 10{\sin \limits ^{2}}(\pi {y_{1}}) + \sum \limits _{i = 1}^{D - 1} {{({y_{i}} - 1)}^{2}} \left [1\vphantom {+ 10{\sin \limits ^{2}}(\pi {y_{i + 1}})}\right .\right .$ $\left .\left .+ 10{\sin \limits ^{2}}(\pi {y_{i + 1}})\right ] + {\left ({y_{D}} - 1\right )^{2}}\right \} + \sum \limits _{i = 1}^{D} {u({x_{i}},10,100,4)}$

  ${y_{i}} = 1 + \frac {1}{4}({x_{i}} + 1), {u_{{x_{i}},a,k,m = }}\left \{ \begin {array}{l} k{({x_{i}} - a)^{m}}\quad {x_{i}} > a \\ 0\qquad \qquad \quad - a \le {x_{i}} \le a \\ k{({x_{i}} - a)^{m}}\quad {x_{i}} < - a \end {array} \right .$, x_i ∈ [− 50, 50]^D, Multimodal and Non-separable (MN)

• (17) Penalized2

  ${f_{17}}(\mathbf {X}) = \frac {1}{10}\left \{ {\sin \limits ^{2}}(\pi {x_{1}}) + \sum \limits _{i = 1}^{D - 1} {{({x_{i}} - 1)}^{2}} \left [1\vphantom {10{\sin \limits ^{2}}(3\pi {x_{i + 1}})}\right .\right .$ $\left .\left .+10{\sin \limits ^{2}}(3\pi {x_{i + 1}})\right ] + {({x_{D}} - 1)^{2}}\left [1 + {\sin \limits ^{2}}(2\pi {x_{D}})\right ]\right \} + \sum \limits _{i = 1}^{D} {u({x_{i}},5,100,4)}$

  x_i ∈ [− 50, 50]^D, Multimodal and Non-separable (MN)

• (18) Alpine

  ${f_{18}}(\mathbf {X}) = \sum \limits _{i = 1}^{D} {\left | {{x_{i}}{\sin \limits } ({x_{i}}) + 0.1{x_{i}}} \right |}$, x_i ∈ [− 10, 10]^D, Multimodal and Separable (MS)

• (19) Levy

  ${f_{19}}(\mathbf {X}) = \sum \limits _{i = 1}^{D - 1} {{{({x_{i}} - 1)}^{2}}\left [1 + {{{\sin \limits } }^{2}}(3\pi {x_{i + 1}})\right ]} + {\sin \limits ^{2}}(3\pi {x_{1}}) + \left | {{x_{D}} - 1} \right |\left [1 + {\sin \limits ^{2}}(3\pi {x_{D}})\right ]$, x_i ∈ [− 10, 10]^D, Multimodal and Non-separable (MN)

• (20) Weierstrass

  ${f_{20}}(\mathbf {X}) = \sum \limits _{i = 1}^{D} \left (\sum \limits _{k = 0}^{k_{{\max \limits } }} {\left [{a^{k}}{\cos \limits } \left (2\pi {b^{k}}({x_{i}} + 0.5)\right )\right ]} \right )$ $- D\sum \limits _{k = 0}^{k_{{\max \limits } }} {\left [{a^{k}}{\cos \limits } (2\pi {b^{k}}0.5)\right ]} a = 0.5, b = 3, {k_{{\max \limits } }} = 20$

  x_i ∈ [− 0.5, 0.5]^D, Multimodal and Non-separable (MN)

• (21) Schaffer

  ${f_{21}}(\mathbf {X}) = 0.5 + \frac {{{{\sin \limits } }^{2}}\left (\sqrt {\sum \limits _{i = 1}^{D} {x_{i}^{2}}} \right ) - 0.5}{{\left (1 + 0.001 * \left [\sum \limits _{i = 1}^{D} {x_{i}^{2}} \right ]\right )}^{2}}$, x_i ∈ [− 100, 100]^D, Multimodal and Non-separable (MN)

• (22) Himmelblau

  ${f_{22}}(\mathbf {X}) = \frac {1}{D}\sum \limits _{i = 1}^{D} {\left ({x_{i}^{4}} - 16{x_{i}^{2}} + 5{x_{i}}\right )}$, x_i ∈ [− 5, 5]^D, Multimodal and Separable (MS)

• (23) Michalewicz

  ${f_{23}}(\mathbf {X}) = { - }\sum \limits _{i = 1}^{D} {{\sin \limits } ({x_{i}}){{{\sin \limits } }^{20}}\left (\frac {i * {x_{i}^{2}}}{\pi }\right )}$, x_i ∈ [0,π]^D, Multimodal and Separable (MS)

• (24) Dixon&Price

  ${f_{24}}(\mathbf {X}) = {({x_{1}} - 1)^{2}} + \sum \limits _{i = 2}^{D} {i{{\left (2{x_{i}^{2}} - {x_{i - 1}}\right )}^{2}}}$, x_i ∈ [− 10, 10]^D, Unimodal and Non-separable (UN)

1.2 A.2. CEC2015 computationally expensive benchmark test functions

Table 9 CEC2015 computationally expensive benchmark test functions

1.3 A.3. CEC2011 real-world benchmark test functions

Table 10 Real-world problems selected from CEC2011