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A hybrid bat algorithm

Iztok Fister, Dušan Fister, Xin-She Yang

arXiv:1303.6310v3cs.NE

TL;DR

The paper addresses improving the bat algorithm’s behaviour for higher-dimensional optimization problems. It hybridizes the bat algorithm with differential-evolution strategies and reports significant performance improvements over the original bat algorithm on standard benchmark functions.

  • Problem

    The paper addresses improving the bat algorithm’s behaviour for higher-dimensional problems in optimization.

  • Method

    The proposed hybrid bat algorithm combines the original bat algorithm with differential-evolution strategies.

  • Results

    The hybrid bat algorithm significantly improves the original bat algorithm according to almost all reported measures on standard benchmark functions.

  • Takeaways & Limitations

    The hybridization can be useful for future applications of the bat algorithm.

Abstract

from arXiv · show

Swarm intelligence is a very powerful technique to be used for optimization purposes. In this paper we present a new swarm intelligence algorithm, based on the bat algorithm. The Bat algorithm is hybridized with differential evolution strategies. Besides showing very promising results of the standard benchmark functions, this hybridization also significantly improves the original bat algorithm.

I. INTRODUCTION

The paper situates HBA within nature-inspired and swarm-intelligence optimization, addressing the Bat Algorithm’s difficulty on higher-dimensional problems by hybridizing it with differential-evolution strategies. Experiments on standard benchmark functions report significant improvement over the original BA.

  • Background: Nature-inspired and swarm-intelligence algorithms have been developed for optimization, modeling, simulation, and both continuous and combinatorial problems.Examples include evolutionary algorithms, particle swarm optimization, ant colony optimization, artificial bee colony, firefly, cuckoo search, and bat algorithms.
  • Background: Evolutionary algorithms differ partly in solution representation, supporting binary, real-valued, Lisp-program, or finite-state-automata representations.
  • Motivation: The Bat Algorithm performs well on lower-dimensional optimization problems but may converge too quickly initially on higher-dimensional problems.
  • Contribution: The proposed Hybridized Bat Algorithm combines the original BA with differential-evolution strategies to improve its behavior on higher-dimensional problems.
  • Results: Experiments on a standard literature benchmark set show that HBA can significantly improve the performance of the original BA.

II. BAT ALGORITHM

The Bat Algorithm models bat echolocation through population-based movement, local search, random exploration, and adaptive pulse-rate and loudness parameters. Its iterative procedure initializes bats, generates and evaluates solutions, accepts improvements, and tracks the best solution.

  • Biological inspiration: The Bat Algorithm is inspired by echolocation, in which bats emit sound pulses and use returning echoes while flying.
  • Bat behavior: Each bat searches with a position, velocity, frequency, wavelength, loudness, and pulse-emission rate that can adjust according to target proximity.
  • Bat behavior: The algorithm assumes loudness decreases from a positive initial value A0 toward a minimum constant Amin.
  • Optimization procedure: Each iteration generates solutions by adjusting frequency, velocity, and location, then performs local search around a best solution or random flight.
  • Optimization procedure: Candidate solutions are accepted when a random condition and improved fitness are satisfied, after which pulse rates increase and loudness decreases.
  • Optimization procedure: The complete procedure initializes the population, generates solutions, conducts local search and random exploration, and ranks bats to find the current best.

III. DIFFERENTIAL EVOLUTION

Differential evolution optimizes real-valued candidate solutions by combining population members through mutation, crossover, and selection. Its notation describes the choices used in these operators.

  • Differential evolution maintains candidate solutions and creates new ones by combining existing solutions, retaining the candidate with the better fitness.
  • Differential selection compares candidate solutions and keeps the one with the better optimization score.
  • Differential mutation adds a scaled difference between two randomly selected solutions to a third solution.The scaling factor F is a positive real number in [0.1,1.0].
  • Differential crossover builds a trial vector by copying parameter values from different solutions, with CR ∈[0.0,1.0] controlling the copied fraction.The j = jrand relation ensures that the trial vector differs from the original solution.
  • DE notation identifies the base-vector choice, number of vector differences, and crossover distribution; for example, DE/rand/1/bin uses a randomly selected base vector and one difference vector.

IV. HYBRID BAT ALGORITHM

The Hybrid Bat Algorithm combines the original bat algorithm with differential-evolution strategies. Its key modification applies DE/rand/1/bin when modifying a solution within the bat-algorithm procedure.

  • The paper proposes a new bat algorithm called the Hybrid Bat Algorithm (HBA).
  • HBA hybridizes the original bat algorithm using differential-evolution strategies.
  • The algorithm initializes bat positions, velocities, pulse rates, and loudness, then iterates while adjusting frequency, velocities, and locations.
  • During each iteration, HBA may modify a solution with DE/rand/1/bin, generate a randomly flying solution, accept qualifying solutions, update pulse rates and loudness, and rank bats.
  • HBA differs from the original BA at line 9, where solution modification uses the DE/rand/1/bin strategy.

V. EXPERIMENTS AND RESULTS

Experiments compare the original and hybrid bat algorithms on optimization benchmarks under matched parameters. The study varies problem dimension and evaluates repeated runs using several summary statistics.

  • The experiments aim to show that HBA significantly improves the original BA using the same parameters for both algorithms.
  • The experiment tests the impact of problem dimension using D = 10, D = 20, and D = 30.
  • Generation limits increase with dimension: 1,000 for D = 10, twice as much for D = 20, and 3,000 for D = 30.
  • The initial loudness and pulse rate are both 0.5, while frequency is sampled from Qi ∈[0.0,2.0].
  • The algorithms optimize benchmark functions 25 times, with best, worst, mean, median, and standard-deviation results recorded.

A. Test suite

The test suite contains five standard optimization functions drawn from the literature. These functions are presented as the benchmark problems for the experiments.

  • The test suite consists of five standard functions taken from the literature.
  • The functions serve as the optimization benchmarks used in the study.
  • The test-suite functions are represented below in the paper.

1. The Griewangk’s function

The Griewangk’s function is designed to test optimization when variables cannot be optimized independently. It is multimodal at lower dimensions, but appears unimodal after n > 30.

  • The function tests optimization failures that arise when each variable is optimized independently.
  • The function is multimodal because its number of local optima increases with dimensionality.
  • For dimensionalities above 30, multimodality seems to disappear and the problem becomes unimodal.
  • The variables are constrained by −600 ≤xi ≤600, and the global minimum is at 0.

2. The Rosenbrock’s function

The Rosenbrock’s function has a global optimum inside a narrow, flat, parabolic valley. Its strongly dependent variables make convergence to the global optimum difficult.

  • The Rosenbrock’s function has value 0 at the global minimum.
  • Its global optimum lies inside a narrow-shaped, flat, parabolic valley.
  • The variables are strongly dependent on one another, making convergence to the global optimum difficult.
  • The function’s variables are bounded by −15.00 ≤xi ≤15.00.

5. The Ackley’s function

The experiments compare BA and HBA on benchmark functions across dimensions, using mean values and broader summary statistics. HBA significantly outperforms BA overall, especially on f1, f3, and f5, although standard-deviation differences are not consistently improved and no statistical analysis was performed.

  • Experimental design: The benchmark suite contains five functions, f1 through f5, evaluated at dimensions D = 10, 20, and 30 using best, worst, mean, median, and standard deviation values.
  • Algorithm comparison: HBA significantly improves BA according to almost all measures, except standard deviation in some cases such as Ackley’s function.
  • Evaluation caveat: No statistical analysis was performed because the authors considered HBA’s results evidently better than BA’s.
  • Experimental design: Figures 1–3 show mean values for functions f1, f3, and f5 across dimensions, using a logarithmic y-axis where higher means indicate greater difficulty.
  • Function-specific results: For f1, D = 20 gives the best result, whereas the highest dimension gives the worst result.
  • Function-specific results: For f5 under HBA, D = 10 is most difficult and D = 20 is easiest, whereas BA worsens as dimensionality increases.
  • Function-specific results: For f3, solving becomes more efficient as dimensionality increases, but D = 30 remains the most difficult case.
  • Algorithm comparison: HBA significantly outperforms the original BA on functions f1, f3, and f5, while its improvements on f2 and f4 are not significant.

VI. CONCLUSION

The paper introduces HBA, a hybrid of the bat algorithm and differential-evolution strategies, and reports significant improvement over the original BA. Future work will test HBA on larger-scale and more diverse optimization settings while deepening the parametric study.

  • HBA is a new variant of the bat algorithm that combines BA with differential-evolution strategies.
  • Experiments show that HBA significantly improves the original BA.
  • The authors plan to test HBA on large-scale global optimization problems.
  • Future testing will use more diverse function sets and deepen the parametric study.
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