Results

A visualisation of an in-progress meta-optimiser is shown in Figure3.7. Lighter grey particles and the lighter grey bounding box represents the realm of the SRO: the axes of the parametersω,φandc. As seen here, several of the super-particles are towards the centre of the cube, indicating that smaller values of each parameter are perhaps more suitable. The darker red particles represent the best averaged fitness obtained from the sub-optimisers. The super-particles and the sub-particles are superimposed within the same region.

In the author’s experience [124], it was necessary to make small differences inωand large differences inc

in order to optimise the MOLPSO for different test functions. Only one function was used to measure fitness in sub-optimisers at a time, however.

3.5. CALIBRATING METAHEURISTIC OPTIMISERS 61

(a) First timestep. (b) Several timesteps later.

FIGURE3.7: A snapshot of the GPU-based SRO (grey particles) and the end-resulting particle distribution of sub-optimisers (dark-red particles) after several frames of the SRO.

Rosenbrock 8D Schwefel 8D Griewangk 16D Michalewicz 5D Ackley 16D Meta-PSO

Mean Result 4.05 −1941 2.3 −3.69 0.0

Std. Dev. 0.14 32.4 0.13 0.011 0.0

TABLE3.5: The mean results generated by the MOLPSO meta optimiser for several test functions. Low values denote higher quality solutions. Function boundaries are given in Tab.3.1.

Rosenbrock 8D Schwefel 8D Griewangk 16D Michalewicz 5D Ackley 16D

ω 0.2557 3.8690 1.7098 −0.3540 0.6784

φg 2.1863 3.0114 4.0 0.3332 0.4677

c 0.7154 4.000 3.5909 3.8098 0.3988

F(x) 3.7471 1330.7 1.9462 −3.6953 0.0000

62 3. CONTINUOUS GLOBAL OPTIMISATION

Table3.5shows convergence results for the test functions used, which were obtained by averagingS(p)across

40independent runs of300time steps of the SRO. This table is discussed below.

Of the test functions used, the Ackley function in16dimensions appears to have been the easiest to optimise. The ability of the meta-optimiser to generate parameters (given in Table3.6) which allow a20-particle optimiser to consistently optimise the Ackley function is impressive, as it requires a considerable amount of computing power to obtain these parameters.

In previous experiments (see [124] and Section3.3), acceptable values forφg were typically in the range

[0.01,0.1]. As shown in Table3.6,φgandωare quite large. Larger step sizes were expected for the Schwefel and

Griewangk functions due to the large boundaries involved, however. This behaviour is made clearer in Figure3.8.

The excessive freedom allowed in the bounds of the parametersω,φg andcallowed the SRO to seek out

unconventional parameters as well. Thecvalue for the Schwefel function is one example. For ease of reference,c

is a scaling parameter of the L´evy distribution of step sizes. Larger values corresponds to generally larger steps, which in the case of the Scwhefel function in Table3.5is very large, perhaps indicating a need for larger step sizes. It can also, however, be a symptom that the sub-optimiser’s population is simply not substantial enough to effectively search the landscape of the vast search space of the Schwefel function.

During testing, the average time taken to compute one timestep of the super-optimiser was approximately

150msec. Larger numbers of particles quickly increased this number, however. The purpose of this short study was to indicate that it is indeed possible to effectively accomplish meta-optimisation using GPUs, rather than to demonstrate the precise speedup over a CPU implementation.

3.5.3

Discussion

Good results were obtained by using meta-optimisation to find appropriate parameters for optimising certain test functions. However, the process in doing so is still computationally expensive. Due to CUDA block size limitations, it was not possible to extend the sub-optimiser dimension count to more than16without a total redesign of the algorithm.

The resulting parameters also appeared in some cases to take advantage of the individual characteristics of the test functions which they were assigned. This is usually an undesirable effect, but could be mitigated somewhat by testing all functions at the same time. Though, this would again increase the computational cost tremendously. This problem is otherwise known as “overfitting”.

An auxiliary advantage of meta-optimisation is that it facilitates comparison [217] between optimisers.

As shown earlier, certain agent-based models such as flocking models share the inspiration behind optimisers such as thePSO. In much the same way, both the PSO and most agent-based models require some form of calibration. This short study has indicated that meta-optimisation is effectively equivalent to the calibration of an agent-based model. The definition of the fitness function is perhaps more difficult in an agent-based model, but the precise process is demonstrated very well by meta-optimisation of the MOLPSO using another MOLPSO.

Having discussed optimisers which involve arbitrary numbers of dimensions, and also noting that it is important to have a qualitative sense of how an optimiser operates, it is therefore relevant to discuss visualisation. Higher dimensions above 3D are less trivial to visualise, and require special pre-processing. The next section demonstrates some simple techniques for visualising these.

3.5. CALIBRATING METAHEURISTIC OPTIMISERS 63

(a) Meta-optimiser on Rosenbrock’s function. (b) Meta-optimiser on Michalewicz’s function.

(c) Meta-optimiser on Griewangk’s function. (d) Meta-optimiser on Schwefel’s function.

(e) Meta-optimiser on Ackley’s Path function.

FIGURE3.8: Plots of the mean fitness obtained from the super-optimiser across all40separate runs, including error bars representing the average standard deviations of each point (sub-optimiser) across the separate runs.

64 3. CONTINUOUS GLOBAL OPTIMISATION