Conclusion

7 . 1 Sumlnary of thesis

Two challenges are addressed in this thesis. The first of these is the analysis of backtracking adaptive search, the logical extension of pure adaptive search and hesitant adaptive search algorithms that have already received attention in optimisation literature. The second, more far-reaching goal is to provide a framework whereby the theoretical analysis now available for backtracking adaptive Eearch can be used to estimate the convergence time of a general stochastic global optimisation algoriL1m.

The first of these aims is achieved in Chapters 4 and 5. A moment generating function for back­ tracking adaptive search on discrete, continuous or mixed domains has been presented. The mean and variance of convergence times for finite backtracking adaptive search are derived explicitly in Corollaries 4.3.1 and 5.3.1. The former of these has been published in [1]; the analogous result for a continuous range function 1f is given in [9] .

In Section 4.4 a closed expression for the expected convergence time for finite backtracking adap­ tive search with arbitrary initial distribution is presented. Several special cases are illustrated in Section 4.5, including the derivation of results discussed in [31 , 5 1 , 56] .

Chapter 5 contains a definition of backtracking adaptive search for a general range distribution. Moment generating functions for the number of iterations before convergence are then derived. The moment generating function approach also appears in [50] .

7. 1 . Summary of thesis 127

The second of these aims seeks to apply backtracking adaptive search as a model for a general stochastic global optimisation algorithm. The analysis of the convergence rate of this algorithm, summarised above, is now complete. The challenge is then to model a general optimisation algorithm with backtracking adaptive search and use this model for the prediction of convergence times. Much further research is possible in this direction.

A framework of processes for approximating the convergence rate of an arbltrary Markovian stochastic global optimisation algorithm is presented in Chapter 2 , with several theoretical results proved in Chapter 3. A paper outlining this framework and containing some of the analysis has also been published [49J. A chain of intermediate processes serve to break the approximation into several further stages. Each process in this chain is derived from the previous one, and can be used to approx­ imate its convergence behaviour . The chain culminates in a tractable stochastic process, backtracking adaptive search. Using the results outlined above, convergence rates can be quickly obtained for the backtracking adaptive search model.

The range process is the image of the solutions sampled by an optimisation process in the domain, projected into the range using the objective function. Since the range process is not Markovian,

the averaged range process is introduced as a time-inhomogeneous Markovian approximation to the

range process. Theorem 3.2. 1 shows that this process is identical in marginal di8tribution to the range process. Corollary 3.2.1 then shows that the number of iterations before convergence for both processes is also identical in distribution. An accurate estimate of the convergence time of the averaged range process would therefore also provide an accurate estimate of the convergence time of the original process.

Although the averaged range process is time-inhomogeneous, in many situations the transition matrices tend to a limit very quickly. The Markovian process using this limit tre.nsition matrix is called the asymptotic averaged range process. Theorem 3.3.3 provides a general definition of the asymptotic averaged range process transition matrix pertaining to any optimisation process. The expected convergence time of this process differs from that of the original process; however, under a weak assumption, Theorem 3.4.1 shows that the difference in expected convergence time between the asymptotic averaged range process and the original process it approximates is the sum of a series

128 Chapter 7. Conclusion

whose terms tend to decay geometrically. The required assumption can be ensured by i mposing the weak condition that pure random search be conducted from some level with positive probability. For a range of examples discussed in Chapters 2, 3 and 6, the convergence rate of the asymptotic averaged range process is very similar to that of the optimisation process it approximates.

The asymptotic averaged range process and backtracking adaptive search are both time-homogeneous Markov processes defined on the range. The advantage of approximating the asymptotic averaged range process with backtracking adaptive search is that predictions can easily be made of the conver­ gence rate of a backtracking adaptive search model. Similar predictions on general Markov processes require matrix inversion, which is in general very time-consuming. The special structure of backtrack­ ing adaptive search allows expected convergence times to be calculated using the analytical results of Chapters 4 and 5.

A method is proposed in Section 4.6 for approximating any asymptotic averaged range process with backtracking adaptive search. The complete framework thus provides a sequential procedure by which a backtracking adaptive search algorithm can be chosen as a model for any optimisation process. Prediction of convergence times for optimisation processes can then be made based on this model. Some examples of the complete procedure are given in Chapter 6.

Section 6.2 illustrates the entire framework under the assumption that the structure of the Marko­ vian optimisation algorithm is known exactly. Although asymptotic averaged range process approxi­ mations give very good estimates of the expected convergence time, the simple method of obtaining backtracking adaptive search models suggested in Section 4.6 yields much less accurate estimates. An alternative method using maximum likelihood estimation provides much better results, but this method has impractical computational requirements.

Section 6.3 addresses the issue of estimating parameters for use in the model from a finite number of observations of the optimisation algorithm. Several possible approaches are listed. The preferred method is to estimate parameters for the asymptotic averaged range process directly from observation of the optimisation algorithm and then to proceed to the backtracking adaptive search model. The asymptotic averaged range process is much simpler than the original process in the domain, or the range or averaged range processes; yet Theorem 3.4.1 implies that its expected convergence time is

7.2. Further y.;ork 129

in many cases very close to that of the original algorithm. The technique illustrated in this section is a practical method for estimating convergence time for any optimisation algorithm, based on the analysis developed in this thesis of the approximation framework and backtracking adaptive search model.

7 . 2 Further work

The method of Section 4.6 is very simple; improvement in this step remains an important area for development. Aside from the error inherent in the prediction of parameters from a finite number of observations of a process, this step is the major source of error in the approximation framework. A maximum likelihood technique was much more successful in the examples of Sect.ion 6.2, but this method is too complex to be practicable. Other methods may be able to give improved performance within an efficient time frame.

A second difficulty is encountered in Section 6.3, that the probabilities of transitions to optimal states must in general be estimated before such transitions have been observed. Evidently much uncertainty must inevitably surround such prediction. Future research may focus on restricted problem types for which such probabilities can be estimated with some accuracy.

One remaining challenge is thus to improve the method used to approximate the asymptotic averaged range process with backtracking adaptive search. Another major area to be addressed is the prediction of parameters for model processes based only on the observation of a finite number of iterations of an algorithm, before it converges. Both of these areas to be addressed represent significant difficulties in applying the methodology to estimate convergence rates of optimisation algorithms.

If these challenges can be answered, the prediction method put forward in this thesis stands to fill an important need. The complete strategy for analysis allows prediction of how long a particular stochastic global optimisation algorithm should be run to reach a set level. Quantitative measures of the effectiveness of stochastic global optimisation algorithms can then be made. A method for estimating the time required by an optimisation algorithm would be a valuable addition to the resources of optimisation practitioners. Information of this kind is also useful for tuning algorithms, for example

130 Chapter 7. Conclusion

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