Thresholding – a Selection Operator for Noisy ES Sandor Markon1 , Dirk V. Arnold2, Thomas B¨ack3, Thomas Beielstein2 , and Hans–Georg Beyer2 1
FUJITEC Co.Ltd. World Headquarters, 28-10, Shoh 1-chome, Ibaraki, Osaka, Japan.
[email protected] 2 Department of Computing Science XI, University of Dortmund, D-44221 Dortmund, Germany. farnold,tom,
[email protected] 3 NuTech Solutions GmbH, Martin Schmeisser-Weg 15, D-44227 Dortmund, Germany.
[email protected] Abstract- The starting point for the analysis and experiments presented in this paper is a simplified elevator control problem, called ’S-ring’. As in many other real-world optimization problems, the exact fitness function evaluation is disturbed by noise. Evolution Strategies (ES) can generally cope with noisy fitness function values. It has been proposed that the ’plus’-strategy can find better solutions by keeping over-valued function values, thus preventing inferior offspring with fitness inflated by noise from being accepted. The ’plus’-strategy builds an implicit barrier around the current best population. We propose to make this barrier building process explicit and to employ a threshold value to be used in a selection operator for noisy fitness functions. ’Thresholding’ accepts a new individual if its apparent fitness is better than that of the parent by at least the margin . First analytical investigations and empirical results from tests on the spheremodel and on ’S-ring’ are presented in this paper.
1 Introduction The elevator group control task is a real-time optimization problem of allocating elevator cars to passengers requesting service. Although it has been investigated for many decades, it is still an open research problem. The main difficulties lie in the stochastic nature of passenger arrivals, and in the combinatorial explosion of system states. There are many proposed control methods, but because of the many differences in buildings, traffic patterns, and elevator movements, the published results cannot be compared directly, and their critical evaluation is difficult. Here we propose a closely related artificial problem, the ’S-ring model’. The S-ring is based on the Kac ring, see e.g. [Gou95], and has properties which make it useful for an analysis of an elevator group control simulation:
it can be solved exactly for small problem sizes, while still exhibiting non-trivial dynamics it retains interesting properties of the elevator system it has a problem complexity similar in order to the elevator problem.
Since the S-ring is easily reproducible, other researchers
should be able to compare their work with the results given in the present paper. Comparing different optimization techniques (stochastic approximation, Q-learning and evolution strategies, details to be reported elsewhere), it has been experimentally found that a (1+1)-ES outperforms the other methods. One fundamental problem during an optimization of the Sring, in common with the elevator group control problem, are noisy fitness function evaluations. Frequently, the probability of generating a real improvement is very small. However, a considerable proportion of generated solutions may appear to be an improvement due to noisy fitness function evaluation. This effect can lead to a decrease in the convergence speed, or even to divergence. A recent study [AB00] of the -ES on a noisy sphere has shown that in the course of the search, candidate solutions that are retained tend to be overvalued in the mean if the fitness of a candidate solution is evaluated and stored only upon its creation rather than every time its numerical value is used. This is intuitively clear as those candidate solutions that are perceived to be fitter than they actually are due to noise are more likely to survive in comparisons with other candidate solutions than those that are undervalued. The systematic overvaluation of the parental fitness implicitly introduces a threshold into the selection process. Only those candidate solutions that exceed the perceived parental fitness by at least a certain amount are retained. For the noisy sphere, it has also been shown that this overvaluation can in fact be useful: while it decreases success probabilities, it increases the expected fitness gain q per generation as it mostly leads to the rejection of candidate solutions that would otherwise be accepted merely due to noise rather than to a real advantage in fitness. The present paper investigates the effect of such a threshold for a real-world optimization problem. However, instead of implicitly implementing the threshold simply by not reevaluating parental fitness, we choose to gain greater control over it by explicitly setting its value and reevaluating the parental fitness in every generation. This becomes especially important for real-world optimization problems, when the target system is not necessarily stationary. In such a case re-evaluation is needed for tracking and thresholding may be recommended. The resulting algorithm can be sketched in the notation of
(1+1)
Rudolph ([Rud97]) as follows: 1 A LGORITHM ((1+1)-ES
WITH THRESHOLDING)
yg
:=
yg
:=
( +1)
Step 1: choose one initial parent y(g)
( +1)
Step 2: repeat Step 3: generate a mutation vector z(g)
~(
Step 4: determine the (noisy) fitness values Q y(g) z(g) and Q y(g)
)
~(
+
)
+
Step 5: accept y(g) z(g) as y(g+1) if its noisy fitness exceeds that of y(g) by at least a margin of ; otherwise retain y(g) . until some stopping criterion is satisfied A similar selection operator was introduced (with some errors, see [DH67]) by Maty´asˇ [Mat65] for a (1+1)-Evolution Algorithm (EA) without noise: He analyzed a method that accepts a new point if its fitness function value is better than the old value minus a threshold > . Compared to threshold acceptance algorithms (see e.g. [DS90] and [Rud97]) thresholding as considered here may be called ’threshold rejection’. After introducing mathematical notations for the fitness model and for thresholding a theoretical analysis and the derivation of a formula for the optimal threshold are given for the sphere (Section 2). Familiarity with the ES and the progress rate theory on the sphere model is advantageous to understand this analysis. In Subsection 2.4 the progress rate formula is evaluated numerically to illustrate the influence of thresholding. Section 3 introduces the S-ring system as a model of an elevator system. Subsection 3.6 defines the quality gain which is required to investigate experimentally the influence of thresholding on the S-ring optimization. Section 4 concludes with a brief summary and directions for further research.
0
^
In this section we analyze theoretically the asymptotic behavior of the ES algorithm with thresholding on the noisy optimization problem of a spherical objective function. We derive an analytical expression for the optimal threshold level, and we show that thresholding is needed in a wide range of parameter values to obtain convergence.
,
( )
( )
( )
( )
( )
( )
( )
() IR (0 )
Minimization of a function Q y , y 2 N , disturbed by additive Gaussian noise N ; 2 , is considered. Let Q y be the actually measured fitness (noisy fitness), i.e., (1)
) Q~(y g ) ) > Q~(y g )
g
( )
( )
g;
( )
( )
9 > > = > > ;
(2)
where
N (0; E)
zg
( )
(3)
is the mutation vector with i.i.d. normally distributed random components of standard deviation and is the threshold value. At each generation the parental fitness is measured anew (even when the offspring of generation g has not replaced the parent at g ). A spherically symmetric fitness function Q y is assumed with the location of the minimum at y, i.e.,
0
+1
()
^
Q(y) = Q(ky y^ k) = Q(R):
(4)
ky y^ k =: R is the distance to the optimum. Without loss of
generality, the optimum is assumed at 0. As a special fitness model, the case
Q(R) = Q
0
+ R
(5)
is considered. 2.2 Progress Rate
=
Let R kR(g) k be the distance to the optimum at generation g and let r kR(g+1)k be the distance to the optimum at generation g . The progress rate ' is defined as
= +1
' := E[R r℄ =
1
Z 0
(R
r)p(r)dr:
(6)
The standard asymptotical analysis (N ! 1) shows that, given isotropic Gaussian mutations z N 0; E , the random difference R R kR zk can be expressed as
(
+
)
R := R kR + zk ' x 2 NR ; (7) where x is an N (0; ) normally distributed random variate 2
2
(see e.g. [Bey93]). Thus, the progress rate integral (6) can be expressed as
' = E[R r℄ '
2.1 Algorithm and Fitness Model
Q~ (y) := Q(y) + N (0; 1):
,
y g +z g Q~ (y g + z g yg Q~ (y g + z g
:=
2 Thresholding: An Analysis
~( )
(1 + 1)-ES with thresholding
The selection scheme of the reads
Z
1 1
x
N 2 R p(x)P (x)dx; 2
a
(8)
with
1 p(x) = p e 2 ( ) 2 and the acceptance probability P (x)
1 x 2
(9)
a
P (x) := Pr(Q~ (R + z) Q~ (R) ) a
(10)
expressing the probability of offspring’s survival in the first line of (2). Using the noise model (1), Pa x becomes
() P (x) = Pr(Q(R + z) + N (0; 1) Q(R) + N (0; 1) ): a
1
(11)
and
(0 1)
(0 1)
2
(0 1)
( )= ( + ) P (x) = Pr(
p
2N (0; 1) Q(R) Q(R + z) = Q(R) p2Q(R + z) p2 ;
a
)
N ' := ' : (18) R the structure of the ' -formula allows for an
2
N R
and
Furthermore, additional normalization
N := 0 QR
(13)
2
' ' p
2
2
1 + 2 exp 12
p
2
''
1
N 1 e 21 ( x )2 p 2 1 2 R Q0 N Q0 p2 x 2p2 R p2 dx:
2
2
exp
6 6 4
r
2
1 +2
0
1 2
B Br
N Q0 R
+ Q N0R + 2 Q 0NR
(14)
2
2
2
2
P
s1+1
=1
B Br
+ +2
2 2
2
N Q0 R
C C 2 A
(20)
(21)
^ max ' ,
' =0 2 + ^ = 21 + 2 ^ = :
!
=^
(22)
2
2
2
2
Therefore,
P
s1+1
(15)
1 + 2 exp 18 ( + 2 ) 1 2 1 2 + 2
max ' = p 2 2
p
2
2
2
N Q0 R
:
2
Considering ' as a function of the normalized threshold , the optimal value can be determined
1
2
R q = Q0 ' : N
where Ps1+1 is the success probability 0
Note that the applicability of the ' formula as an approximation for the case N < 1 is restricted by the condition N . While the progress rate ' is a theoretical measure that can be determined for simple fitness functions only, there is a second progress measure, called the quality gain q, that can be determined in any case by simulation experiments. q is defined as the expected fitness gain (without noise). For the sphere model there is a connection to the progress rate that reads1 (see [Bey93], p. 173, Eq. 33):
) 123
2
p
!#
+ + 2
2
2 1
)
C 7 C 7 2 A 5
5
2
2
2
2
2
"
Integration (longer calculation, not presented here) and using normalizations given below, yields 2
2
!2 3
+ + 2
2.3 Optimal Normalized Threshold
2
' ' p
2
p
2
x
(19)
2
2
Using (9), (12), and (13) the progress rate integral (8) reads Z
N := 0 QR
and
such that one obtains
(12)
( ) ( ) ( + )
Q(R) Q(R + z) = Q(R) Q(kR + zk) = Q(R) Q(R R) = ddRQ R + O(R) ' Q0 (R) x 2 NR :
:=
4
: is the cumulative distribution function of the where standard normal variate. In order to express the difference Q R Q R z as a function of x asymptotically, the Q difference is expanded in a Taylor series breaking off after the linear term and taking (7) into account
(17)
The normalizations are
2
Taking into account that due to parental resampling the two noise sources N1 ; and N2 ; are independent of p N ; each other, the difference of both produces a random variate (it has implicitly been assumed that is only a function of the parental state but not of the mutations, i.e., R R z . This is a reasonable assumption provided that the mutations are sufficiently small). Thus, rearranging (11) one gets
dQ Q0 := : dR
2
(16) 1q
is defined in[Bey93] as
Q:
2
p
2
2
(23)
0.25
normalized progress rate '
normalized progress rate '
0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2
0.2
0.15
0.1
0.05
0 0
1
2
3
4
5
6
normalized mutation strength
0
Figure 1: Normalized progress rate ' as a function of normalized
mutation strength . The solid curves represent results for optimally chosen threshold parameter = 2 , the dashed curves for = 0. The respective curves correspond to, from top to bottom, normalized noise strengths = 0:0, 0:4, 0:8, 1:2, 1:6, and 2:0.
1
2
3
4
normalized threshold
5
6
Figure 2: Normalized progress rate ' as a function of the nor-
malized threshold parameter for optimally adjusted mutation strength . The curves correspond to, from top to bottom, normalized noise strengths = 0:0, 0:4, 0:8, 1:2, 1:6, and 2:0.
The crosses visualize data measured in ES runs described in detail in the text.
and the optimal success probability reads
P
= 1 12
s1+1
p
2
+ 2
2
:
(24)
Given a normalized noise level , both functions still depend on . Finding their optimal values must be done numerically. Reversing the normalization of the optimum condition (22) using (19) and the special fitness model (5) yields
^
= =
Q0R N N = Q0R 2
2
N : (Q Q ) 2
(25)
0
That is, if one is able to estimate and the fitness distance (to the optimal fitness value Q0 ), the optimal threshold can be used in the -ES.
(1 + 1)
2.4 Numerical Evaluations A number of things can be learned from Equation (20) regarding the influence on the performance of the -ES of the parameters and . For simplicity, we restrict ourselves to the quadratic sphere ( in Equation (5)) in this section. Figure 1 displays the normalized progress rate ' as a function of normalized mutation strength for a number of noise levels and both for the -ES with ideally chosen 2 and for . The curves threshold parameter represent results from Equation (20). The crosses correspond to measurements of the progress rate of the -ES on the quadratic sphere with parameter space dimension N and have been averaged over ; generations each. The
(1 + 1)
=2
=
(1+1)
400 000
=0 (1+1)
= 40
agreement between theoretical predictions and empirical observations is quite good and would improve further with increasing parameter space dimension. It can be seen from Figure 1 that the performance gain achieved by introducing the threshold can be considerable. While for low noise strengths ( < : ) the progress rates of the two strategies hardly differ at all, for larger noise strengths ( > : ) the strategy with zero threshold is hardly capable of positive progress and in fact exhibits negative progress rates for a wide range of mutation strengths. The strategy with non-zero threshold on the other hand achieves non-negative progress rates for any mutation strength. We consider this observation a strong point for the use of a non-zero threshold. The correct choice of the threshold parameter remains as an important challenge. Figure 2 shows the normalized progress rate ' as a function of the normalized threshold parameter in case of optimally adjusted normalized mutation strength . It has been obtained by numerically solving Equation (20) for the optimal normalized mutation strength. It can be seen that for > the optimal choice for is greater than zero. For normalized noise strengths greater than about : , non-zero is necessary for achieving a positive progress rate. With increasing noise strength the choice of the threshold parameter becomes less critical as the maxima of the curves in Figure 2 become less pronounced. Another important parameter that needs to be adapted to both the noise level and to the current location in search space is the mutation strength . A simple mutation strength adaptation rule for the -ES is Rechenberg’s 1/5-success rule (see [Rec73]): keep track of the proportion of the mutations that have been accepted in a number of most recent generations; increase the mutation strength if this proportion is above one fifth, decrease it if it is below. This rule has been found to work reasonably well in a variety of fitness environments and is capable of handling not too high fitness noise. However, Figure 3 shows that this may no longer be the case
04
10
0
14
(1 + 1)
optimal success probability Ps1+1
(t) P rob (x) (t + 1) 000 1 p 000 p 100
0.3
0.25
100 1 010 1 p
0.2
p
0.15
0.1
0.05
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
normalized noise strength
1.8
2
Figure 3: Optimal success probability Ps1+1 as a function of the normalized noise strength for = 2 (solid line) and for = 0 (dashed line). if a non-zero threshold is used. The optimal success probability in case of optimally chosen initially drops faster than and can be seen to be substantially below that in case of 20% for noise strengths for which the optimal success probability in case of still does not deviate too much from one fifth. Therefore, when using thresholding one should aim for lower success probabilities when using a success probability based mutation strength adaptation scheme.
=0 =0
3 The S-ring system We introduce an artificial problem, called ‘S-ring’, to serve as a benchmark for developing efficient algorithms for noisy optimization problems. First we define formally the problem, and describe its relation to the elevator group control problem; then we show how it is solved by the ES algorithm with thresholding. 3.1 Definition Consider the following automaton[CD98]: The state at time t is given as
one-dimensional
cellular
[s (t); (t); : : : ; sn (t); n (t)℄ x(t) 2 X = f0; 1g n: There are n sites, each with a 2-bit state (si ; i), and with 0
0
1
2
1
periodic boundary conditions at the ends. However, instead of using synchronous updating at all sites independently, one updating cycle is decomposed into n steps as follows: The state evolution is sequential, scanning the sites from n to , then again around from n . At each time step, one triplet i ; si ; si+1 is updated, the updating being governed by the stochastic state transition rules in Table 1, and by the ’policy’ X ! f ; g. The last column shows ; the reward, i.e., the change in the number of sites with i this will be needed later.
1 0
(
1
)
:
01
=0
110 1 001 1 p p 101 1 011 1 111 1
100 001 101 010 101 010 001 101 101 011 011
0 1 0 1
r 0 1 0 0 1 0 0 +1 0 1 0 0 +1
Table 1: State transitions of the S-ring. The entries can be read as in the following example: The server has to make a decision (to ’take’ or to ’pass’ the customer) if there is a customer waiting (1xx), and if there is a server present on the same floor (11x) but no server on the next floor (110). It is obvious that there are states where no decision is required resp. possible.
3.2 Analogy with the Elevator System The system and the state transition rules can be visualized by viewing them as a caricature model of an elevator system in a building (see Fig. 4). In this view, the i bit stands for the existence of customers (waiting passengers) in each of the queues waiting for up and down service. The si bits represent the presence of one of the m servers (elevator cars) which are running around in an up - down loop. The building has f floors, thus the number of queues is n f . At each time step, one of the floor queues is considered, where passengers may arrive with probability p. If the queue has both a customer and a server present, the server makes a decision to ’take’ ( ) or ’pass’ ( ) the customer, according to the policy . In case of a ’take’ decision, the customer is removed, and the server stays there; in the ’pass’ case, or if there is no customer, the server steps to the next site. The exception is that if the next site is already occupied, the server stays; in such case, if there is a customer, he or she is served unconditionally. This exclusion rule corresponds to disallowing elevators bypassing each other; although strictly speaking unrealistic (it would be more realistic for a circulating type of linear motor elevator systems, where cars keep their relative positions), it is needed to keep the number of the servers constant, preserved through transitions.
=2
1
2
0
3.3 Control of the S-ring The S-ring can be used to define an optimal control problem, by equipping it with an objective function Q (here is the expectation operator):
E
Q(n; m; p; ) = E
X
i :
(26)
c
f-1
s
where is the Heaviside function2 and y is the weight vector. In general there is no guarantee that the and classes of the optimal policy are linearly separable. However, for this special case, the perceptron can realize the optimal policy. In the following the optimal weights y should be found to define an optimal policy.
0
f-1
f-th floor
Customer
Server #2
3.5 Policy Examples
Server #1
Customer
1
The most obvious heuristic policy is the ’greedy’ one: when given the choice, always serve the customer: Customer
Customer
Server #3
g 1:
2nd floor c
1
s
c
1
n-1
s
n-1
Rather counter-intuitively, this policy is not optimal, except in the heavy traffic p > : case. This means that a good policy must bypass some customers occasionally. A quite good heuristic policy is the ’balance’ policy:
(
1st floor c
0
s
0
Figure 4: The S-ring as an elevator system.
b =
For given parameters n, m, and p, the system evolution depends only on the policy , thus this can be written as Q Q . The optimal policy is defined as
=
()
= arg min Q():
(27)
The basic optimal control problem is to find for given parameters n, m, and p. 3.4 Representations of the Policy In a general case, can be realized as a look-up table of the state x. It is assumed that x is normalized by shifting it to start at the site where the decision is taken. An enumeration of x is defined by first regarding the state bits as the binary representation of an integer:
=
nX1 i=0
2i( i + 2nsi):
(28)
Based on this, a compact index can now be defined as the ordinal number of the state when ordered according to P. Only the valid states are counted, i.e., those satisfying si m. In principle, is found by enumerating all possible and selecting the one with the highest Q. Since this count grows exponentially with n, the naive approach would not work even for any but the smallest cases. A more compact representation of the policy is a linear discriminator (perceptron)
=
(x) = (yT x);
(30)
(29)
0 5)
0 1
=1
if sn 1 otherwise.
The intention is to put some distance between servers by ’passing’ when there is another tailing server, letting it serve the customer. By balancing the positioning of the servers, b is significantly better than g for medium values of p. Even for a small S-ring neither of the above is an optimal policy; the real is not obvious, and its difference from heuristic suboptimal policies is non-trivial. 3.6 A prototype S-ring
=6
=2
In the following n queues and m servers are used throughoutly. To determine one single fitness function value, 1,000 state transitions of the S-ring are evaluated. Passengers : . The weight vector y was iniarrive with probability p tialized randomly. This simple simulation model (only 4 floors, 2 elevator cars, and a constant customer arrival rate – a real-world situation is much more complicated) already generates noisy fitness function values Q y . Introducing thresholding should improve the performance of the -ES and decrease the influence of noise.
=03
~( )
(1 + 1)
3.6.1 S-Ring Single-Step Algorithm Generating mutants from the same point many times is called a single-step. The S-ring single-step algorithm was implemented to estimate the influence of parameters like noise , threshold or mutation rate on the local performance of a (1+1)-ES optimizing the S-ring. 2 A LGORITHM (S-R ING S INGLE -S TEP A LGORITHM ) Step 1: choose one initial weight vector y(g) (parent) 2 The
x
0.
()
Heaviside function x has value zero for x
)
2
;
is esti-
(32)
( )=1
+ R (yg
y ); 0
R 2 [0; 1℄
to estimate the influence of depending on the distance to the optimum. Figure 5 shows the influence of thresholding for three different values:
= 0:0 (no thresholding): No gain can be achieved.
Overvalued points can mislead the ES away from the optimum.
if A is true, where I is a ’success’ indicator ( I A else ). pi is a relative success measure, the expected value of which corresponds to the success probability given by (16).
= 0:3: In a region not too close to the optimum posi-
3. The quality gain q in one step from y(g) is then given by
= 1:0: Obviously was chosen too high, no values
0
()
q( ) :=
t X i=1
pi ( )(Qg Qi):
(33)
This quantity corresponds to the quality gain formula given in [Bey93], p. 173, Eq. 33.
tive gain can be achieved. Noise prevents positive gain close to the optimum.
are accepted, the simulation stagnates. This behavior can be observed in many situations when was chosen too high.
4 Summary and Outlook A (1+1)-ES with thresholding was introduced and analyzed for the sphere. We determined the optimal normalized thresh-
old and performed numerical evaluations that may give reasons for the conjecture that thresholding can improve the performance of ES in noisy environments. Numerical evaluations showed that thresholding requires a modified adaptation rule for the mutation rate. Then the results from the sphere model were transferred to the S-ring. The S-ring was introduced as a simulation model to investigate an elevator group control task – a complex real-world optimization task with noisy fitness function values. We used thresholding to handle the noise because reevaluations of the fitness function were too costly. First experimental results were successful. The analytical results from the sphere are consistent with experimental results on the S-ring, but many details remain unknown, so that the exploration of the following topics will remain for future research:
Robustness of the parameter Finding a suitable -adaption rule Dynamical adaptation of the parameter for real-world optimization problems Performance of multi-membered Evolution Strategy, e.g. (=I ; )-ES
In practical optimization work, useful values of might be determined according to some statistical testing procedure from observed process statistics, somewhat similar to the recommendation of Stagge[Sta98]. Currently, experiments are continued with a (1+1)-ES algorithm on the S-ring to achieve deeper insight into the dynamic behavior of thresholding. Acknowledgments S. Markon gratefully acknowledges helpful discussions with Prof. Y. Nishikawa of OIT and his associates, during the early stages of this research. H.-G. Beyer is Heisenberg Fellow of the Deutsche Forschungsgemeinschaft (DFG) under grant Be 1578/4-2. D. Arnold gratefully acknowledges support by the DFG, grant Be 1578/6-3. T. Beielstein’s research was supported by the DFG as a part of the collaborative research center ’Computational Intelligence’ (531).
Bibliography [AB00] Dirk V. Arnold and Hans-Georg Beyer. Local performance of the (1+1)-ES in a noisy environment. Technical Report CI–80/00, Universit¨at Dortmund, Fachbereich Informatik, 2000. (submitted for publication). [Bey93] Hans-Georg Beyer. Towards a theory of evolution strategies: Some asymptotical results from the =; -theory. Evolutionary Computation, 1(2):165– 188, 1993.
)
(1+
[CD98] B. Chopard and M. Droz. Cellular Automata Modeling of Physical Systems. Cambridge U.P., 1998.
[DH67] M. Driml and O. Hanˇs. On a randomized optimization procedure. In J. Koˇzeˇsnik, editor, Transactions of the 4th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes (held at Prague 1965), pages 273–276, Prague, 1967. Czechoslovak Academy of Sciences. [DS90] G. Dueck and T. Scheuer. Threshold accepting: a general purpose optimization algorithm appearing superior to simulated annealing. Journal of Computational Physics, 90:161–175, 1990. [Gou95] H. Gould. Thermal and Statistical Physics Simulations. John Wiley, 1995. [Mar95] Sandor Markon. Studies on Applications of Neural Networks in the Elevator System. PhD thesis, Kyoto University, 1995. [Mat65] J. Maty´asˇ. Random Optimization. Automation and Remote Control, 26(2):244–251, 1965. [Rec73] Ingo Rechenberg. Evolutionsstrategie. Optimierung technischer Systeme nach Prinzipien der biologischen Evolution. problemata. frommann-holzboog, 1973. [Rud97] G¨unter Rudolph. Convergence Properties of Evolutionary Algorithms. Verlag Dr. Kovaˇc, Hamburg, 1997. [Sta98] Peter Stagge. Averaging efficiently in the presence of noise. In A. Eiben, editor, Parallel Problem Solving from Nature, PPSN V, Lecture Notes in Computer Science, pages 188–197, Berlin, 1998. Springer.
