Evolution strategies with exclusion-based selection operators and a Fourier series auxiliary function
Introduction
Evolutionary algorithms (EAs) are global search procedures based on the evolution of a set of solutions viewed as a population of interacting individuals. These algorithms include genetic algorithms (GAs), genetic programming (GP), evolution strategies (ES) and evolution programming (EP) [9], [18], [19], [20]. EAs have broad applications and successes as tools for search and optimization. But for solving large scale and complex optimization problems, EAs have not demonstrated themselves to be very efficient. Particularly their efficiencies have been criticized [17], [27]. We believe the main factor which causes low efficiency of the current EAs is the convergence toward undesired attractions. This phenomenon occurs when the objective function has some local optima with normal attractions or its global optimum is located in a narrow attraction in a minimization case. The relationship between the convergence to a global minimum and the geometry (landscape) of the minimization problem is very important. If the population of EAs gets trapped in the suboptimal states, which are located in comparative large attractions, then it is difficult for the variation operators to produce an offspring which outperforms its parents. In the second case, if the global optima are located in relatively narrow attractions, and the individuals of EAs have not found these attractions yet, the possibility of the variation operators to produce offspring which locate in these narrow attractions is quite low. In both cases, the stochastic mechanism of EAs also yields unavoidable resampling, which increases the algorithm’s complexity and decreases the search efficiency. Many previous studies have investigated such issues for genetic algorithms [23], [28], their parallel versions [13] and parameter control [1], evolutionary programming [22], evolution strategies [10], and evolutionary hybrid approaches [11], [14], [15].
In our previous research work [26], we have proposed highly efficient and practical strategies to establish accelerated EAs for solving global optimization problems, called the exclusion-based selection operators. These strategies could carry out computationally verifiable tests on “prospects” (or non-existence) of global optimum solutions in the cells. Any less-prospective cells could be excluded and EAs could concentrate on the highly prospective solution space in the search process. The exclusion-based selection operators could effectively prevent the individuals of EAs from resampling and getting into the attractions of local optima, therefore, could accelerate the convergent speed of EAs. Through the space shrinking strategies, EAs with the exclusion-based selection operators could significantly enhance the efficiency and the precision of the solutions. However, when a global optimum of a minimization problem is located in an extremely narrow attraction, the exclusion-based selection operators may not be able to find this narrow attraction and delete this global optimum accidentally, which causes the algorithm to be unreliable.
In this paper, we propose a new complimentary efficient and practical strategy—a Fourier series auxiliary function, that can enlarge the narrow attractions of the optima and flatten the large attractions. This auxiliary function can guide an algorithm to search the optima with narrow attractions more efficient, while such optima are difficult to be found in the objective function by EAs. Furthermore, this strategy runs in parallel with the exclusion-based selection operators and compensates the deficiency of the exclusion-based selection operators on the algorithm’s risk of missing optima with very narrow attractions.
Integrating these two complementary strategies can be considered as searching in two scales in parallel, one for optima in normal attractions and one for optima in very narrow attractions respectively. In the case study, the integrated strategies are incorporated into evolution strategies (ES), yielding a new type of accelerated exclusion and Fourier series auxiliary function ES: the EFES. Simulation examples all demonstrate that the new ES consistently and significantly outperforms the previous ES in efficiency and solution quality, particularly for the complex problems with optima in narrow attractions.
The paper is organized as follows: we will briefly introduce the set of “exclusion-based” selection operators in Section 2. Then the Fourier series auxiliary function is proposed in Section 3. Incorporation of the integrated strategies proposed with any known EA can lead to an accelerated version of the algorithm. We present in Section 4, as a case study, the accelerated version of evolution strategies—the evolution strategies with exclusion-based selection operators and a Fourier series auxiliary function (EFES). The EFES is then experimentally examined, analyzed and compared in Section 5 with a suite of typical and difficult, multimodal function optimization problems. The paper concludes in Section 6 with some useful remarks on future research related to the present work.
Section snippets
Exclusion-based selection operators
This section introduces a somewhat different selection mechanism—the exclusion-based selection operators [26].
Consider any EA solving an optimization problem, say,where f:Ω ⊂ Rn → R is a function. For simplicity, consider the problem (P) with the domain Ω specified by Ω = [u1,v1]∗[u2, v2]∗⋯∗[un, vn]. The new scheme is based on the cellular partition methodology. Given an integer d, let , we defineand let the
A Fourier series auxiliary function
The exclusion-based selection operators are the efficient accelerating operators based on the interval arithmetic and the cell mapping methods. However, if the function optimization problem is very complex, say there are many narrow optimal attractions, the optima may be excluded by mistake using the above operators. In any case, searching an optimum with a narrow attraction is difficult for evolutionary algorithms. Furthermore, it is the main reason that renders the exclusion-based selection
A case study: EFES
The exclusion-based selection operators could prevent the population of evolutionary algorithms (EAs) getting into the large attractions of local optima and speed up converge through the space shrinking strategy. But the global optimum with a narrow attraction might be deleted by mistake. The Fourier series auxiliary function g(x) guides the population of EAs to find the optima with narrow attractions more efficiently through enlarging narrow attractions and flattening large attractions of f(x
Simulations and comparisons
We experimentally evaluate the performance of the EFES, and compare it with the (μ,λ) standard evolution strategies (SES) [9] and the fast evolutionary strategies (FES) [21]. All experiments are conducted on a Pentium IV 1.4G computer.
The EFES has been implemented with N = 1000 and ε1 = ε2 = 10−8. The maximum number M of the ES evolution was taken uniformly to be 500 in each epoch of the EFES. In the experiments, all the five exclusion operators (E1)–(E5) were applied and the “Orange-peeling”
Conclusion
This paper has suggested, analyzed, and explored a new complementary efficient strategy for evolutionary computation—the Fourier series auxiliary function strategy. Integrating this strategy with the exclusion-based selection operators, which are very fast strategies, forms a unique hybridized optimization approach searching in two scales in parallel for optima in normal and very narrow attractions respectively. Any known evolutionary algorithm incorporated with these integrated strategies may
Acknowledgment
This research was partially supported by RGC Earmarked Grant 4192/03E of Hong Kong SAR and RGC Research Grant Direct Allocation of the Chinese University of Hong Kong.
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