Cooling Scheme

For SA cooling methods have been developed, which guarantee a global mini-mum for an infinite long calculation time. There the temperature has to be like following:

Tk = a

b+ log(k) (2.29)

a and b are positive and system-dependant constants; k is the number of already executed temperature steps. The decisive disadvantage of this method is that the calculation time is longer than the complete enumeration of all configurations.

Another problem is that it’s not clear, whether the real optimum has been found.

So this cooling strategy is of no use in practice; instead empirical curves are used, which converge faster.

The first empirical method to mention is linear cooling. The temperature is reduced constantly with ∆T:

Tk =Tstart−k·∆T with 0.01≤∆T ≤0.5 (2.30)

Tstart is the start temperature, which has to be determined for each optimisation run. Besides Tk mustn’t be smaller than zero; the optimisation run has to be stopped before.

The logarithmic or exponential cooling uses a repeated multiplication of the start temperature with the factor α:

Tk=α^k·Tstart with 0.8≤α ≤0.999 (2.31)

The best cooling method depends on the optimisation problem. Therefore a test run has to be made, in order to estimate the curves of the physical variables.

Especially from the heat capacity it can be derived, how the system behaves. If the system freezes very fast, the linear cooling method is chosen; in the opposite case the logarithmic method is better.

Start and End Temperatures

The correct start temperature is important for the optimisation run. If it is too high, calculation time is wasted at the beginning; if it is too low, the solutions are bad. The start temperature cannot be given directly, because it depends on the single optimisation problems. For SA a good start temperature can be found in the following way: for the temperature Tstart the system shall be able to move more or less freely in the phase space. At the beginning, transitions shall be accepted which raise the energy of the system. The acceptance rate Pacc for this transitions can be set freely. Then a random walk through the phase space is made and the number n of transitions measured, which raise the energy level of the system. The number of accepted transitions for simulated annealing can be approximated as follows: where ∆ ¯H⁺ is the expectation value of the transitions raising the energy level of the system. The acceptance level Pacc is given by

Pacc= nacc

Mostly the acceptance rate is chosen between 80 and 90 %. Of course this is a very rough estimation of Tstart, but the order of the temperature can easily be found with this method. A similiar consideration can be made for TA and GDA:

Tstart≈∆ ¯H+ threshold accepting (2.35)

TSstart ≈ H^max great deluge algorithm (2.36)

The end temperature Tend shall be determined in such a way that the system is mostly frozen; the acceptance rate of all transitions shall tend to zero. But in degenerated systems there can be transistions with no effect on the energy.

Those don’t need to be considered, when the acceptance rate is calculated. It is also good to make several steps atT = 0. If the acceptance rate of all non-trivial transitions is zero over a long period of time, the optimisation run can be stopped.

Of course there is no security, whether the global optimum is reached, but the probability is very high to find a local optimum near the global one. Especially for systems with very broad energy valleys in the area of the global minimum the acceptance rate could be clearly above zero, in spite of a constant energy level. In this case it is better to take the Hamilton function as a criterion for the closeness to the optimum: if the energy doesn’t change over several temperature steps, the system can assumed to be frozen.

Chapter 3

Different Metaheuristics

Physical optimisation is a global optimisation technique which traverses the search space by generating neighbouring solutions of the current solution. A superior neighbour is always accepted. An inferior neighbour is accepted probabilistically based on the difference in quality and a temperature parameter. The temper-ature parameter is modified as the algorithm progresses to alter the ntemper-ature of the search. In order to have a contrast to physical optimisation, some related algorithms are presented in the following:

At first genetic algorithms (GA) which maintain a pool of solutions. New solutions are generated not only by ”mutation” as in simulated annealing (SA), but also by ”combination” of two solutions from the pool. Probabilistic criteria, similar to those used in SA, are used to select the candidates for mutation or combination, and for discarding excess solutions from the pool.

Secondly, evolution strategies (ES) which evolve individuals by means of mutation as well as intermediate and discrete recombination. Thus ES are very similiar to genetic algorithms; they are designed particularly to solve problems in the real-value domain; they use self-adaptation to adjust control parameters of the search.

The metaheuristic tabu search (TS) is similar to SA: both traverse the so-lution space by testing mutations of an individual soso-lution. While SA generates only one mutated solution, tabu search generates many mutated solutions and moves to the solution with the lowest energy of those generated. In order to pre-vent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions. It is forbidden to move to a solution that contains elements of the tabu list which is updated as the solution traverses the solution space.

At last ant colony algorithms are presented. They use many ants to tra-verse the solution space and find locally productive areas. While usually inferior to genetic algorithms and other forms of local search, it is able to produce results

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in problems where no global or up-to-date perspective can be obtained, and thus the other methods cannot be applied.

In this dissertation tabu search and ant colony algorithms are introduced for the sake of completeness. Only genetic and physical optimisation algorithms are applied. Thereby the focus is on physical optimisation; for evaluation and comparison of the results also a genetic algorithm was implemented.