**1. Introduction**

68 Simulated Annealing – Single and Multiple Objective Problems

problem, *IEEE Trans. on Veh. Tech*., Vol. 49 (2000) 1265−1272.

Smith, D.H., Hurley, S. & Allen, S.M. (2000). A new lower bound for the channel assignment

Ursem, R.K. (2002). Diversity-guided Evolutionary Algorithms, *Proc. Int. Conf. on Parallel* 

*Problem Solving from Nature VII (PPSN VII),* (2002) Granada, Spain, 462–471.

Simulated annealing (SA) [16], [17], [14]), belongs among those algorithms which allow steps after which the value of the objective function will deteriorate. It can thus be seen again as the local extensions of classical methods of searching. The SA is similar to hill climbing, but differs in the fact that the individual is able to overcome local extremes. However, inspiration for this formulation of statistical mechanics had been found in the description of the physical annealing process of a rigid body. In this analogy, namely during annealing of metals with unstable crystal lattice, there is a stabilization of loose particles in an optimal state, i.e. the formation of a stable crystal lattice. Such a metal has much better properties. The process is carried out by heating the metal at high temperatures to the melting point and then very slowly cooling it. Cooling is done slowly enough to eliminate unstable particles and the metal has acquired the requisite optimal quality. In the early 1980s, Kirkpatrick, Gelatt and Vecchi (Watson Research Center of IBM, USA) and independently Cerny (Department of Theoretical Physics, Comenius University in Bratislava, former Czechoslovakia) proposed solutions to the problem of finding the global minimum of combinatorial optimization analogous to the procedure of the annealing rigid body.

This chapter introduces simulated annealing in special applications focused on deterministic chaos control, synthesis and identification. The first one discusses the use of SA on evolutionary identification of bifurcations, i.e. positions of control parameters of the investigated system related to that event.

The second application discusses the possibility of using SA for the synthesis of chaotic systems. The systems synthesized here were based on the structure of well-known logistic equations. For each algorithm and its version, repeated simulations were conducted and then averaged to guarantee the reliability and robustness of the proposed method. The third and last application is focused on deterministic spatiotemporal chaos realtime control by means of selected evolutionary techniques, with SA. Realtime-like behavior is specially defined and simulated with a spatiotemporal chaos model based on mutually nonlinear joined *n* equations, so-called Coupled Map Lattices (CML). Investigation consists of different case studies with increasing simulation complexity. For all algorithms each simulation was repeatedly evaluated in order to show and check the robustness of the methods used. All

©2012 Zelinka and Skanderova , licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Zelinka and Skanderova , licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Name of the Book 70 Simulated Annealing – Single and Multiple Objective Problems Simulated Annealing in Research and Applications <sup>3</sup>

data were processed and used in order to obtain summarized results and graphs. The most significant results are carefully selected, visualized and commented on in this chapter.

In the real world any object consists of particles. Physical state can be described by vector **x** = (*x*1, *x*2, ..., *xn*,) describing particle position for example. This state is related to energy *y* = *f*(**x**). If such system is on the same temperature *T* long enough, then the probability of existence of such states is given by Boltzmann distribution. The probability that the system is

*e*−*f*(x)/*<sup>T</sup>*

*e*

where summarization is going over all states *x*. For a sufficiently small *T* the probability that the system will be in state *xmin* with minimal energy *f*(**xmin**) is almost 1. In the 50's simulation of annealing was suggested by means of Monte Carlo method with a new decision function 4.

*<sup>P</sup>*(*<sup>x</sup>* <sup>→</sup> *<sup>x</sup>*0) = 1, *f orf*(*x*) <sup>&</sup>lt; *<sup>f</sup>*(*x*0)

The function of this decision is whether new state *x* (when for example one particle will change its position) is accepted or not. In the case that *x* is related to lower energy then the old state it is replaced by a new one. On the contrary, *x* is accepted with probability 0 < P(**x** → **x**0) < 1. If *r* is random number from [0, 1], then new state is accepted only if *r* < P(**x** → **x**0). In (4) T has an important influence on probability *P*(*x* → *x*0) when *f*(*x*) ≥ *f*(*x*0); for big *T* any new state (solution) is basically accepted, for a low *T* states with higher energy are only rarely accepted. If this algorithm (Metropolis algorithm) is repeated for one state in a sufficient number of repetitions, then the observed distribution of generated states is basically Boltzmann distribution. This makes it possible to execute SA on a PC. The SA, repeating Metropolis algorithm for decreasing temperature then uses the final state *Tn* like initial state for the next iteration *xm* with *Tm* = *Tn* − *ε*. Variable *ε* is arbitrary small number.

Randomly selected initial solution *x*<sup>0</sup> from all possible solutions M ;

**begin** randomly select *x* from set of all possible neighbor *N*(*x*0) ;

**then** *x*∗ := *x* update the best solution

**then begin** *x*<sup>0</sup> := *x* ; move to a better solution is always accepted

**else begin** randomly select *r* from uniform distribution on the interval (0,1);

**then** *x*<sup>0</sup> := *x* moving to a worse solution otherwise the current solution

Select decrement function *α*(*t*) and final temperature *Tf* ;

*<sup>Q</sup>*(*T*) = <sup>∑</sup>*<sup>x</sup>*

*<sup>Q</sup>*(*T*) (2)

*<sup>e</sup>*−(*f*(*x*)−*f*(*x*0))/*<sup>T</sup> f orf*(*x*) <sup>≥</sup> *<sup>f</sup>*(*x*0) (4)

<sup>−</sup>*f*(x)/*<sup>T</sup>* (3)

Simulated Annealing in Research and Applications 71

in state **x** is then given by

Pseudocode for SA is

Set initial temperature *T*<sup>0</sup> > 0 ;

Δ*f* := *f*(*x*) − *f*(*x*0) ;

**if** *f*(*x*) < *f*(*x*∗)

**if** *r* < *e*−Δ*<sup>f</sup>* /*T*

**for i := 1 to** *nT* **do**

**if** Δ*f* < 0

**end**

remains unchanged

*x*∗ := *x*<sup>0</sup> ;

*T* := *T*<sup>0</sup> ;

**repeat**

with
