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Egret Swarm Optimization Algorithm: An Evolutionary Computation Approach for Model Free Optimization

Zuyan Chen, Adam Francis, Shuai Li, Bolin Liao, Dunhui Xiao

arXiv:2207.14667v1cs.NE

TL;DR

Non-convex engineering problems can challenge gradient methods, while no metaheuristic suits every optimization problem. ESOA models Great and Snowy Egret hunting strategies, and the paper reports benchmark and engineering results demonstrating its search behavior and robustness.

Abstract

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A novel meta-heuristic algorithm, Egret Swarm Optimization Algorithm (ESOA), is proposed in this paper, which is inspired by two egret species' (Great Egret and Snowy Egret) hunting behavior. ESOA consists of three primary components: Sit-And-Wait Strategy, Aggressive Strategy as well as Discriminant Conditions. The performance of ESOA on 36 benchmark functions as well as 2 engineering problems are compared with Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Differential Evolution (DE), Grey Wolf Optimizer (GWO), and Harris Hawks Optimization (HHO). The result proves the superior effectiveness and robustness of ESOA. The source code used in this work can be retrieved from https://github.com/Knightsll/Egret_Swarm_Optimization_Algorithm; https://ww2.mathworks.cn/matlabcentral/fileexchange/115595-egret-swarm-optimization-algorithm-esoa.

I. INTRODUCTION

Engineering optimization can be difficult for gradient methods, and no metaheuristic is suited to every problem. The section introduces the egret hunting inspiration and the algorithm’s mathematical-model focus.

I. INTRODUCTION

  • Non-convex engineering problems can make conventional gradient approaches difficult to apply and prone to local optima.The cited applications include manipulator control, path planning, and fault diagnosis.Source: §I · p. 1
  • Metaheuristics draw on evolution, physical principles, and biological behavior, with GA/DE, SA/GSA/BH/MVO, and PSO/ACO among the examples.Later examples include GWO, SSA, and BAS applications.Source: §I · p. 1
  • No metaheuristic suits every optimization problem, and improving search efficiency requires some computational cost.The No Free Lunch theorem motivates continued development of new metaheuristic structures.Source: §I · p. 2

II. EGRET SWARM OPTIMIZATION ALGORITHM

  • The algorithm section presents egret migration behavior as inspiration and then discusses ESOA’s mathematical model.Source: §II · p. 2

A. Inspiration

  • Great Egrets trade higher energy expenditure for potentially greater prey returns, while Snowy Egrets’ sit-and-wait approach uses less energy for more reliable profits.Snowy Egrets also sample more sites and observe locations where other birds have found food.Source: A. Inspiration · pp. 2–3

B. Mathematical Model And Algorithm

ESOA models three egret roles in a parallel squad. Its Sit-And-Wait strategy estimates prey-related gradients and uses them to select a next sampling location.

B. Mathematical Model And Algorithm

  • ESOA is a parallel algorithm with Sit-And-Wait, Aggressive, and Discriminant components, assigning forward guidance, random walking, and encircling to Egrets A, B, and C.Source: B. Mathematical Model And Algorithm · p. 3

1) Sit-And-Wait Strategy:

  • The Sit-And-Wait observation estimates prey presence at the squad’s current position, with a weight assigned to the estimate method.The position is x_i and the estimated prey value is ŷ.Source: 1) Sit-And-Wait Strategy · p. 3
  • The strategy derives a practical gradient from the observation error and combines its direction with corrections toward squad-best and swarm-best locations.The integrated gradient uses r_h and r_g, each in [0, 0.5).Source: 1) Sit-And-Wait Strategy · p. 3
  • Egret A’s next sampling location moves along the integrated gradient, with a step scaled by the solution-space span and decaying over iterations.The decay uses current iteration t and maximum iteration t_max.Source: 1) Sit-And-Wait Strategy · p. 3

2) Aggressive Strategy:

The Aggressive strategy uses random search and encircling behaviors to generate candidate locations for the egret squad.

2) Aggressive Strategy:

  • Egret B searches randomly for prey, with its random angle drawn from (−π/2, π/2).Source: 2) Aggressive Strategy · p. 3
  • Egret C uses an encircling update to generate its expected next location.Source: 2) Aggressive Strategy · p. 3
  • Egret C’s update combines the current location with gaps to the squad-best and swarm-best positions, weighted by random coefficients.D_h is the gap to the squad’s best position, while D_g compares against the best position of all squads.Source: 2) Aggressive Strategy · p. 3

3) Discriminant Condition:

The discriminant condition chooses whether the squad adopts its proposed solution or retains its current position.

3) Discriminant Condition:

C. Pseudo Code

ESOA’s pseudocode alternates sit-and-wait and aggressive position proposals, then selects the next position through a discriminant condition. The section also includes benchmark comparison tables for the listed algorithms.

C. Pseudo Code

  • The algorithm takes an initial swarm-position matrix and returns the best position and its fitness.The pseudocode labels these outputs xbest and ybest, respectively.Source: C. Pseudo Code · p. 4
  • The loop computes sit-and-wait and aggressive proposals, then uses a discriminant condition to choose the next position.Sit-and-wait updates an integrated gradient and observation weight to obtain Egret A’s expected position; the aggressive function obtains positions for Egrets B and C.Source: C. Pseudo Code · p. 4

F ESOA PSO [9] GA [3] DE [4] GWO [11] HHO [32]

III. EXPERIMENTAL RESULTS AND DISCUSSIONS

The experimental evaluation covers benchmark-function families with different optimization challenges and compares ESOA with five established algorithms. Experiments use fixed population, dimension, and iteration settings.

III. EXPERIMENTAL RESULTS AND DISCUSSIONS

  • The experiments use 30 dimensions, populations of 50, and a maximum of 500 iterations.The algorithms start from random matrices; the passage also specifies PSO parameters w=0.8, c1=0.5, c2=0.5, and a GA mutation value of 0.001.Source: §III · p. 6
  • The evaluation covers unimodal, simple multimodal, hybrid, and composition functions, which differ in local-optimum structure and complexity.Unimodal functions have one global minimum; multimodal and composition functions contain local traps, while hybrid functions combine different test functions.Source: §III · p. 6
  • ESOA is compared with PSO, GA, DE, GWO, and HHO across the benchmark functions.The paper groups PSO, GA, and DE as traditional algorithms and GWO and HHO as newer methods.Source: §III · p. 6

A. Evaluation Of Exploitation Ability

The section evaluates exploitation on unimodal functions, exploration on multimodal and hybrid functions, and overall performance on composition functions. ESOA is reported to perform competitively across these tests, with specific strengths and qualifications by function range.

A. Evaluation Of Exploitation Ability

  • On unimodal F1–F4, ESOA performs strongly; on F5–F7, it trails GWO and HHO but remains superior to PSO, GA, and DE.The paper uses unimodal functions to assess convergence speed and exploitation because they have one global optimum.Source: A. Evaluation Of Exploitation Ability · p. 6

B. Evaluation Of Exploration Ability(F3-F19)

  • On F3–F19, the results are described as evidence of strong exploration, with ESOA superior to the other five algorithms on F18 average fitness.The authors link ESOA’s ability to overcome interference from local optima to its Aggressive Strategy.Source: B. Evaluation Of Exploration… · p. 6

C. Comprehensive Performance Assessment(F20-F29)

  • 2346 average fitness for ESOA on F22 versus 3129 for DE; ESOA is also described as extremely competitive across F20–F29.The authors attribute performance in these composition scenarios to the combination of Sit-And-Wait and Aggressive strategies regulated by a discriminant condition.Source: C. Comprehensive Performance… · p. 7

D. Algorithm Stability

ESOA’s reported stability is supported by standard-deviation results, while its convergence behavior differs across benchmark classes. The section also describes applying ESOA to engineering problems using a penalty function for inequality constraints.

D. Algorithm Stability

  • ESOA’s standard deviation ranks among the top results in both tables in most situations, supporting the authors’ claim of stability.It is much ahead of the second position on certain test functions.Source: D. Algorithm Stability · p. 7

E. Analysis Of Convergence Behavior

  • On unimodal functions, ESOA reaches near the global optimum in fewer than 10 iterations; on multimodal and hybrid functions, it surpasses almost all methods in most cases and continues searching.The authors associate this unimodal convergence speed with potential use in some online optimization problems; composition-function results are described as excellent.Source: E. Analysis Of Convergence Behavior · pp. 7–8

IV. TYPICAL APPLICATION

  • For engineering applications, a penalty function integrates inequality constraints into the objective function.The transformed objective is denoted f̂(x); φ is the penalty parameter, while f(x) and g_j(x) are the original objective and inequality constraints.Source: §IV · p. 8

A. Himmelblau’s Nonlinear Optimization Problem

The section evaluates ESOA on Himmelblau’s nonlinear optimization problem and reports the best optimal result among the compared algorithms, alongside a standard-deviation comparison.

A. Himmelblau’s Nonlinear Optimization Problem

B. Tension/Compression Spring Design

The section tests ESOA on a spring-weight minimization problem subject to engineering constraints and reports its average fitness and standard-deviation performance against five other algorithms.

B. Tension/Compression Spring Design

V. CONCLUSION

The conclusion presents ESOA as an egret-hunting-inspired algorithm and reports its evaluation on benchmark functions and practical engineering problems. The authors note that additional mathematical forms of its strategies and discriminant condition remain under development.

V. CONCLUSION

  • ESOA combines the Snowy Egret’s Sit-And-Wait Strategy, the Great Egret’s Aggressive Strategy, and a Discriminant Condition.The paper compares it with PSO, GA, DE, GWO, and HHO on 36 test functions spanning four function classes, and also evaluates two engineering problems.Source: §V · p. 9
  • Other mathematical forms of ESOA’s strategies and discriminant condition are under development to accommodate more applications and optimization scenarios.Source: §V · p. 9
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