A method is presented for function optimisation that generalises the Simulated Annealing algorithm by applying convolutions of the target function with smooth, infinitely differentiable kernels. Hence the search for a global optimum is performed over a sequence of functions that preserve the structure of the original one and converge to it pointwise. From an experimental point of view, the purpose of this paper was to compare the efficiency of this approach with that of the conventional Simulated Annealing. To do this, the proposed technique was tested both on complex combinatorial (discrete) problems (e.g. the Travelling Salesman Problem) and on the search of global minima for continuous functions. In some cases, performance was improved in terms of final results, while in other ones, even if no improvements were attained over the usual Simulated Annealing algorithm, the proposed method shows interesting abilities to provide fairly good approximations in relatively few iterations, i.e. at early stages of the search processRed de Universidades con Carreras en Informática (RedUNCI

Segura, Enrique C.

SEDICI - Repositorio de la UNLP

Optimisation with Simulated Annealing
through Regularisation of the Target Function
   Enrique C. Segura
Departamento de Computación,
Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires
Ciudad Universitaria, Pab. I, (1428) Buenos Aires, Argentina
esegura@dc.uba.ar
Abstract
A method is presented for function optimisation that generalises the Simulated Annealing algorithm
by applying convolutions of the target function with smooth, infinitely differentiable kernels. Hence
the search for a global optimum is performed over a sequence of functions that preserve the
structure of the original one and converge to it pointwise. From an experimental point of view, the
purpose of this paper was to compare the efficiency of this approach with that of the conventional
Simulated Annealing. To do this, the proposed technique was tested both on complex combinatorial
(discrete) problems (e.g. the Travelling Salesman Problem) and on the search of global minima for
continuous functions. In some cases, performance was improved in terms of final results, while in
other ones, even if no improvements were attained over the usual Simulated Annealing algorithm,
the proposed method shows interesting abilities to provide fairly good approximations in relatively
few iterations, i.e. at early stages of the search process.
Keywords: simulated annealing, stochastic optimisation, function regularisation, smooth kernels.
1 INTRODUCTION
The difficulties presented by stochastic optimisation techniques and, in particular, Simulated
Annealing (SA) [4][6][9] from the point of view of their applicability for general cases, is a
recurrent field of study and experimentation in the area of stochastic optimisation. One of these
problems is its high sensitivity regarding the behaviour of the function to be minimised
(maximised): if this function is not smooth enough (i.e. for the continuous case, differentiable
including its derivatives up to a certain order), the algorithm tends to get stuck on a valley (a deep
local minimum), remaining there for a long time, which degrades the efficiency of the method.
More specifically, the difficulty arises when local minima appear satisfying the following condition:
with
|)()(|),( jfifjidf -=
where f is the function to minimise, i is a local minimum, N(i) is the set of states neighbour to i
according to the connection graph and L is the Lipschitz constant defined as:
}),(max{max
)(
lkdfL
kNlSk ÎÎ
=
min df i j L
j N iÎ
@
( )
( , )
1301
being S the space of configurations of the problem. For the continuous case, the neighbourhood of a
given point in the space of configurations is defined as the set of elements at a certain distance 
apart from it, i.e.
}||/{)( r=-Î= yxSyxN .
and the notion of connection graph is similar to that for the discrete case.
   In other words, the function presents deep valleys, as deep as the biggest jump in the whole state
space, which prevents control of the algorithm by means of its parameter (temperature), since this
parameter is adjusted according to the information provided by L.
2 THEORETICAL GROUNDING
From this motivation, the idea arises of applying some technique for smoothing the function to
minimise, so as to preserve its structure, that is, if f is the original function and F is the transformed
(smoothed) one, then, for every pair of states i and j in S with jÎ N(i) it holds:
),(),( jidfjidF £
More specifically, we look for a sequence of smooth kernels TF  such that
 
    )()( xfxF TT ¾¾ ®¾
¥®                                                           (1)
for any x in S, being T a discrete control parameter (i.e. a temperature taking only on positive
integer values).
   The method consists essentially of convolving the target function with smooth kernels of the
form:
)/(1)( Txg
T
xgT =
where           
                       ò =
S
dxxg 1)(      (2)
and
             )(0 SCg
¥Î  (3)
It is a well known fact in functional analysis [10] that under these assumptions the functions
ò -==
S
TT dyyxgyfgfxF )()(*)(
are infinitely differentiable and satisfy condition (1).
1302
3 THE ALGORITHM
Based on the preceding considerations, an algorithm was conceived to solve optimisation problems
for functions with many local minima. The method consists of a SA where each element in the
space of configurations is a “macrostate” formed with a fixed number of neighbour states. A cost
associated to such a configuration results from the local convolution of the function with a kernel of
the form:
}2/)/(exp{1)( 2Tx
T
xgT -=
where T = T(t) is the temperature at time t. It is easy to check that such function, properly
normalised, satisfies conditions (2) and (3).
   The convolution is discretised as:
å
=
--=
N
i
NT TiNxfT
xF
1
2 }2/)/)]2/(([exp{)(1)(
where ),...,( 1 Nxxx =  stands for the N-tuple of states that form the configuration. This means that
the largest weight is assigned to the state equidistant to both ends of the “macrostate”, defining the
distance between two states as the number of edges from each other in the graph, according to the
path followed when generating them.
   In general terms, the method proceeds in a similar way as conventional SA. An initial
configuration is generated as a sequence of N neighbour states (i.e. a path of length N in the graph).
At each iteration one of both ends of the sequence is chosen (with equal probability or taking that of
lowest individual cost) and a new neighbour is concatenated to it, eliminating at the same time the
state corresponding to the end that was not chosen. The new value of TF  is computed convoluting
the new sequence with the new Tg . If the cost of the new configuration )1( +tx  is less than the
previous one, it is accepted; otherwise, it is accepted only with probability }/exp{ Tdf- , being
))(())1(( )()1( txFtxFdf tTtT -+= + .
4 EXPERIMENTS
The proposed algorithm was tried on both a) discrete and b) continuous problems, as follows.
a) Two instances of the classical Traveling Salesman Problem [1][3][5] (in 2Â , Euclidean and with
triangle inequality), both considerably complex and with known solution:
i) Map of 100 cities in a squared 10 x 10 array, with distance between neighbours equal to unity.
ii) Map of 120 cities with known minimum cost (example taken from [5]).
   The reason to choose the TSP for the experiments in the discrete case is that it is, due to its
complexity (a paradigmatic case of NP-complete problem), widely accepted as one of the more
representative benchmarks of its type.
1303
         Table 1: average performance (over 10 runs for each value of N) for the TSP
         on problem a)i) (quadrangular lattice of 100 cities). Optimal solution f =100.
N t=10000 T=50000 t=100000
 1 145.1 111.2 105.3
10 183.1 116.4 104.8
20 189.6 112.7 104.2
30 196.5 109.8 103.9
               Table 2: average performance (over 10 runs for each value of N)
               for the TSP on problem a)ii) (from [5]). Optimal solution f = 83.21.
 N  t=10000  t=50000  t=100000
  1 114.8 92.0 88.1
10 119.3 93.2 87.9
20 124.8 92.6 86.7
30 126.4 91.5 86.4
   Tables 1 and 2 show results obtained in both cases for different values of N (case N=1
corresponds to the behavior of usual SA) and for different number of iterations.
   First of all, it must be said that the initial hypothesis in the sense that the complexity of the
problem could be substantially reduced as a function of the dimension (parameter N) was not
confirmed by the experiments. The performance is slightly better than that of the conventional SA,
but better solutions are obtained only after many iterations. Anyway, the efficiency grows
monotonically, although slowly, with the order of the method (parameter N). The algorithm is still
under evaluation; more experiments are currently in progress.
b) Minimisation of the function
dcydxcybxayxf ++--+= )cos()cos(),( 22 dg
for 0 £ x £ 5, 0 £ y £ 5  with  a = 1, b = 2, c = 0.3, d = 0.4, g = 3p, d = 4p.
This function has a unique global minimum at  f(0,0) = 0.
   For this continuous case, generation of a new neighbour from a given point (x,y) at each step was
performed by randomly selecting a point (x´,y´) on the circumference defined
by r=¢-+¢- 22 )()( yyxx . A value r = 0.005 was used for this experiment.
   Table 3 displays results obtained for different values of N and for different number of iterations.
Note that in this case the necessary number of steps for the algorithm either to find a good solution
or being stopped is more than one order of magnitude lower that in the discrete cases.
   It is easy to observe that the best results with the proposed technique are obtained only for a
parameter N = 20, and these are just slightly better than those reached with the SA in its
conventional form. Moreover, for higher values of N the performance gets worse. This might
suggest that the proposed technique is not quite convenient in this case, given its higher
computational complexity with respect to the classical SA (N = 1).
1304
             Table 3: average performance (over 10 runs for each value of N) for
    minimisation of the continuous function given in b). Optimal solution f (0,0) = 0.
 N   t=500   t=1000   t=2000
  1 0.0038 0.0024 0.00016
10 0.0017 0.0009 0.00020
20 0.0013 0.00073 0.00014
30 0.0015 0.0022 0.00016
   However, the values in the first and second columns of the table (t = 500 and t = 1000) could be
indicating, in its turn, that for short runs of the algorithm (e.g. when not much computational time
can be taken and/or a rough approximation to the optimal solution is considered good enough for
certain particular purpose) the technique introduced here might be useful.
5 DISCUSSION
An algorithm for stochastic optimisation was presented, featuring a generalised formulation of the
classical Simulated Annealing by applying convolutions of the target function with smooth,
infinitely differentiable kernels. The search for a global optimum is performed over a sequence of
functions that preserve the structure of the original one and converge to it pointwise.
   Convergence to a global minimum (maximum) is, the same as in the conventional SA, guaranteed
from a theoretical point of view [2][7][8].
   As for practical concerns, the aim of this work was to compare the capabilities of the proposed
method with those of the usual Simulated Annealing. In this sense, the results confirm that it avoids
long permanence in local minima with deep valleys, thus providing optimal or sub-optimal
solutions in lower times than the conventional SA algorithm.
   On the other hand, from the point of view of absolute efficiency (i.e. the best solutions obtained
regardless of the time elapsed), some difference must be remarked between the discrete and the
continuous cases . While in the former this efficiency grows monotonically, although slowly, with
the order of the method (parameter N), in the latter (continuous) the results get worse beyond a
certain value of N. In this case, however, faster convergence to coarse, approximate solutions is
observed.
   All these considerations are still subject to more extensive experimentation.
REFERENCES
[1] Cerny, V.; Thermodynamical approach to the traveling salesman problem: an efficient
simulation algorithm; J. Opt. Th. and Appl., 45(1):41-51, 1985.
[2] Geman, S. and Geman, D.; Stochastic relaxation, Gibbs distributions and the Bayesian
restoration of images; IEEE Trans. PAMI 6(6):721-741, 1984.
[3] Gutin, G. and Punnen, A.P. (eds.); The Traveling Salesman Problem and its Variations,
Springer, 2002.
1305
[4] Kirkpatrick S., Gelatt C. D. and Vecchi M.P.; Optimization by Simulated Annealing; Science,
220: 671-680, 1983.
[5] Lawler E.L., Lenstra J.K., Rinooy Kan A. H. G. and Shmoys D.B. (eds.); The traveling
salesman problem. A guided tour of combinatorial optimization; J. Wiley and Sons, 1985.
[6] Metropolis N., Rosenbluth A., Rosenbluth M., Teller A.H. and Teller E.; Equation of state
calculations by fast computing machines; J. Chem. Phys., 21(6):1087-1091, 1953.
[7] Mitra D., Romeo F. and Sangiovanni-Vincentelli A.; Convergence and finite-time behavior of
simulated annealing; Adv. Appl. Prob. 18: 747-771, 1986.
[8] Otten R.H.J.M. and Ginneken L.P.P.P. van; The Annealing Algorithm. The International Series
in Engineering and Computer Science, Springer, 1989.
[9] Salamon P., Paolo Sibani P. and Frost R.; Facts, Conjectures, and Improvements for
Simulated Annealing. SIAM Monographs on Mathematical Modeling and Computation, 2002.
[10] Yosida K.; Functional Analysis; Berlin, Springer-Verlag, 2003.
1306


English

Optimisation with simulated annealing through regularisation of the target function

A method is presented for function optimisation that generalises the Simulated Annealing algorithm by applying convolutions of the target function with smooth, infinitely differentiable kernels. Hence the search for a global optimum is performed over a sequence of functions that preserve the structure of the original one and converge to it pointwise. From an experimental point of view, the purpose of this paper was to compare the efficiency of this approach with that of the conventional Simulated Annealing. To do this, the proposed technique was tested both on complex combinatorial (discrete) problems (e.g. the Travelling Salesman Problem) and on the search of global minima for continuous functions. In some cases, performance was improved in terms of final results, while in other ones, even if no improvements were attained over the usual Simulated Annealing algorithm, the proposed method shows interesting abilities to provide fairly good approximations in relatively few iterations, i.e. at early stages of the search process.Red de Universidades con Carreras en Informátic

Segura, Enrique Carlos

Servicio de Difusión de la Creación Intelectual

Optimisation with Simulated Annealingthrough Regularisation of the Target Function   Enrique C. SeguraDepartamento de Computación,Facultad de Ciencias Exactas y Naturales, Universidad de Buenos AiresCiudad Universitaria, Pab. I, (1428) Buenos Aires, Argentinaesegura@dc.uba.arAbstractA method is presented for function optimisation that generalises the Simulated Annealing algorithmby applying convolutions of the target function with smooth, infinitely differentiable kernels. Hencethe search for a global optimum is performed over a sequence of functions that preserve thestructure of the original one and converge to it pointwise. From an experimental point of view, thepurpose of this paper was to compare the efficiency of this approach with that of the conventionalSimulated Annealing. To do this, the proposed technique was tested both on complex combinatorial(discrete) problems (e.g. the Travelling Salesman Problem) and on the search of global minima forcontinuous functions. In some cases, performance was improved in terms of final results, while inother ones, even if no improvements were attained over the usual Simulated Annealing algorithm,the proposed method shows interesting abilities to provide fairly good approximations in relativelyfew iterations, i.e. at early stages of the search process.Keywords: simulated annealing, stochastic optimisation, function regularisation, smooth kernels.1 INTRODUCTIONThe difficulties presented by stochastic optimisation techniques and, in particular, SimulatedAnnealing (SA) [4][6][9] from the point of view of their applicability for general cases, is arecurrent field of study and experimentation in the area of stochastic optimisation. One of theseproblems is its high sensitivity regarding the behaviour of the function to be minimised(maximised): if this function is not smooth enough (i.e. for the continuous case, differentiableincluding its derivatives up to a certain order), the algorithm tends to get stuck on a valley (a deeplocal minimum), remaining there for a long time, which degrades the efficiency of the method.More specifically, the difficulty arises when local minima appear satisfying the following condition:with|)()(|),( jfifjidf -=where f is the function to minimise, i is a local minimum, N(i) is the set of states neighbour to iaccording to the connection graph and L is the Lipschitz constant defined as:}),(max{max)(lkdfLkNlSk ÎÎ=min df i j Lj N iÎ@( )( , )1301being S the space of configurations of the problem. For the continuous case, the neighbourhood of agiven point in the space of configurations is defined as the set of elements at a certain distance apart from it, i.e.}||/{)( r=-Î= yxSyxN .and the notion of connection graph is similar to that for the discrete case.   In other words, the function presents deep valleys, as deep as the biggest jump in the whole statespace, which prevents control of the algorithm by means of its parameter (temperature), since thisparameter is adjusted according to the information provided by L.2 THEORETICAL GROUNDINGFrom this motivation, the idea arises of applying some technique for smoothing the function tominimise, so as to preserve its structure, that is, if f is the original function and F is the transformed(smoothed) one, then, for every pair of states i and j in S with jÎ N(i) it holds:),(),( jidfjidF £More specifically, we look for a sequence of smooth kernels TF  such that     )()( xfxF TT ¾¾ ®¾¥®                                                           (1)for any x in S, being T a discrete control parameter (i.e. a temperature taking only on positiveinteger values).   The method consists essentially of convolving the target function with smooth kernels of theform:)/(1)( TxgTxgT =where                                  ò =Sdxxg 1)(      (2)and             )(0 SCg¥Î  (3)It is a well known fact in functional analysis [10] that under these assumptions the functionsò -==STT dyyxgyfgfxF )()(*)(are infinitely differentiable and satisfy condition (1).13023 THE ALGORITHMBased on the preceding considerations, an algorithm was conceived to solve optimisation problemsfor functions with many local minima. The method consists of a SA where each element in thespace of configurations is a “macrostate” formed with a fixed number of neighbour states. A costassociated to such a configuration results from the local convolution of the function with a kernel ofthe form:}2/)/(exp{1)( 2TxTxgT -=where T = T(t) is the temperature at time t. It is easy to check that such function, properlynormalised, satisfies conditions (2) and (3).   The convolution is discretised as:å=--=NiNT TiNxfTxF12 }2/)/)]2/(([exp{)(1)(where ),...,( 1 Nxxx =  stands for the N-tuple of states that form the configuration. This means thatthe largest weight is assigned to the state equidistant to both ends of the “macrostate”, defining thedistance between two states as the number of edges from each other in the graph, according to thepath followed when generating them.   In general terms, the method proceeds in a similar way as conventional SA. An initialconfiguration is generated as a sequence of N neighbour states (i.e. a path of length N in the graph).At each iteration one of both ends of the sequence is chosen (with equal probability or taking that oflowest individual cost) and a new neighbour is concatenated to it, eliminating at the same time thestate corresponding to the end that was not chosen. The new value of TF  is computed convolutingthe new sequence with the new Tg . If the cost of the new configuration )1( +tx  is less than theprevious one, it is accepted; otherwise, it is accepted only with probability }/exp{ Tdf- , being))(())1(( )()1( txFtxFdf tTtT -+= + .4 EXPERIMENTSThe proposed algorithm was tried on both a) discrete and b) continuous problems, as follows.a) Two instances of the classical Traveling Salesman Problem [1][3][5] (in 2Â , Euclidean and withtriangle inequality), both considerably complex and with known solution:i) Map of 100 cities in a squared 10 x 10 array, with distance between neighbours equal to unity.ii) Map of 120 cities with known minimum cost (example taken from [5]).   The reason to choose the TSP for the experiments in the discrete case is that it is, due to itscomplexity (a paradigmatic case of NP-complete problem), widely accepted as one of the morerepresentative benchmarks of its type.1303         Table 1: average performance (over 10 runs for each value of N) for the TSP         on problem a)i) (quadrangular lattice of 100 cities). Optimal solution f =100.N t=10000 T=50000 t=100000 1 145.1 111.2 105.310 183.1 116.4 104.820 189.6 112.7 104.230 196.5 109.8 103.9               Table 2: average performance (over 10 runs for each value of N)               for the TSP on problem a)ii) (from [5]). Optimal solution f = 83.21. N  t=10000  t=50000  t=100000  1 114.8 92.0 88.110 119.3 93.2 87.920 124.8 92.6 86.730 126.4 91.5 86.4   Tables 1 and 2 show results obtained in both cases for different values of N (case N=1corresponds to the behavior of usual SA) and for different number of iterations.   First of all, it must be said that the initial hypothesis in the sense that the complexity of theproblem could be substantially reduced as a function of the dimension (parameter N) was notconfirmed by the experiments. The performance is slightly better than that of the conventional SA,but better solutions are obtained only after many iterations. Anyway, the efficiency growsmonotonically, although slowly, with the order of the method (parameter N). The algorithm is stillunder evaluation; more experiments are currently in progress.b) Minimisation of the functiondcydxcybxayxf ++--+= )cos()cos(),( 22 dgfor 0 £ x £ 5, 0 £ y £ 5  with  a = 1, b = 2, c = 0.3, d = 0.4, g = 3p, d = 4p.This function has a unique global minimum at  f(0,0) = 0.   For this continuous case, generation of a new neighbour from a given point (x,y) at each step wasperformed by randomly selecting a point (x´,y´) on the circumference definedby r=¢-+¢- 22 )()( yyxx . A value r = 0.005 was used for this experiment.   Table 3 displays results obtained for different values of N and for different number of iterations.Note that in this case the necessary number of steps for the algorithm either to find a good solutionor being stopped is more than one order of magnitude lower that in the discrete cases.   It is easy to observe that the best results with the proposed technique are obtained only for aparameter N = 20, and these are just slightly better than those reached with the SA in itsconventional form. Moreover, for higher values of N the performance gets worse. This mightsuggest that the proposed technique is not quite convenient in this case, given its highercomputational complexity with respect to the classical SA (N = 1).1304             Table 3: average performance (over 10 runs for each value of N) for    minimisation of the continuous function given in b). Optimal solution f (0,0) = 0. N   t=500   t=1000   t=2000  1 0.0038 0.0024 0.0001610 0.0017 0.0009 0.0002020 0.0013 0.00073 0.0001430 0.0015 0.0022 0.00016   However, the values in the first and second columns of the table (t = 500 and t = 1000) could beindicating, in its turn, that for short runs of the algorithm (e.g. when not much computational timecan be taken and/or a rough approximation to the optimal solution is considered good enough forcertain particular purpose) the technique introduced here might be useful.5 DISCUSSIONAn algorithm for stochastic optimisation was presented, featuring a generalised formulation of theclassical Simulated Annealing by applying convolutions of the target function with smooth,infinitely differentiable kernels. The search for a global optimum is performed over a sequence offunctions that preserve the structure of the original one and converge to it pointwise.   Convergence to a global minimum (maximum) is, the same as in the conventional SA, guaranteedfrom a theoretical point of view [2][7][8].   As for practical concerns, the aim of this work was to compare the capabilities of the proposedmethod with those of the usual Simulated Annealing. In this sense, the results confirm that it avoidslong permanence in local minima with deep valleys, thus providing optimal or sub-optimalsolutions in lower times than the conventional SA algorithm.   On the other hand, from the point of view of absolute efficiency (i.e. the best solutions obtainedregardless of the time elapsed), some difference must be remarked between the discrete and thecontinuous cases . While in the former this efficiency grows monotonically, although slowly, withthe order of the method (parameter N), in the latter (continuous) the results get worse beyond acertain value of N. In this case, however, faster convergence to coarse, approximate solutions isobserved.   All these considerations are still subject to more extensive experimentation.REFERENCES[1] Cerny, V.; Thermodynamical approach to the traveling salesman problem: an efficientsimulation algorithm; J. Opt. Th. and Appl., 45(1):41-51, 1985.[2] Geman, S. and Geman, D.; Stochastic relaxation, Gibbs distributions and the Bayesianrestoration of images; IEEE Trans. PAMI 6(6):721-741, 1984.[3] Gutin, G. and Punnen, A.P. (eds.); The Traveling Salesman Problem and its Variations,Springer, 2002.1305[4] Kirkpatrick S., Gelatt C. D. and Vecchi M.P.; Optimization by Simulated Annealing; Science,220: 671-680, 1983.[5] Lawler E.L., Lenstra J.K., Rinooy Kan A. H. G. and Shmoys D.B. (eds.); The travelingsalesman problem. A guided tour of combinatorial optimization; J. Wiley and Sons, 1985.[6] Metropolis N., Rosenbluth A., Rosenbluth M., Teller A.H. and Teller E.; Equation of statecalculations by fast computing machines; J. Chem. Phys., 21(6):1087-1091, 1953.[7] Mitra D., Romeo F. and Sangiovanni-Vincentelli A.; Convergence and finite-time behavior ofsimulated annealing; Adv. Appl. Prob. 18: 747-771, 1986.[8] Otten R.H.J.M. and Ginneken L.P.P.P. van; The Annealing Algorithm. The International Seriesin Engineering and Computer Science, Springer, 1989.[9] Salamon P., Paolo Sibani P. and Frost R.; Facts, Conjectures, and Improvements forSimulated Annealing. SIAM Monographs on Mathematical Modeling and Computation, 2002.[10] Yosida K.; Functional Analysis; Berlin, Springer-Verlag, 2003.1306

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Optimisation with simulated annealing through regularisation of the target function

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