A Quantum Field Evolution Strategy

An Adaptive Surrogate Approach

org Bremer

and Sebastian Lehnhoff

Department of Computing Science, University of Oldenburg, Uhlhornsweg, Oldenburg, Germany

R&D Division Energy, OFFIS – Institute for Information Technology, Escherweg, Oldenburg, Germany

Keywords:

Evolution Strategies, Global Optimization, Surrogate Optimization, Quantum Potential.

Abstract:

Evolution strategies have been successfully applied to optimization problems with rugged, multi-modal ﬁtness

landscapes, to non linear problems, and to derivative free optimization. Usually evolution is performed by

exploiting the structure of the objective function. In this paper, we present an approach that harnesses the

adapting quantum potential ﬁeld determined by the spatial distribution of elitist solutions as guidance for the

next generation. The potential ﬁeld evolves to a smoother surface leveling local optima but keeping the global

structure what in turn allows for a faster convergence of the solution set. We demonstrate the applicability

and the competitiveness of our approach compared with particle swarm optimization and the well established

evolution strategy CMA-ES.

1 INTRODUCTION

Evolution Strategies have shown excellent perfor-

mance in global optimization especially when it

comes to complex multi-modal, high dimensional,

real valued problems (Kramer, 2010; Ulmer et al.,

2003). A major drawback of population based algo-

rithms is the large number of objective function evalu-

ations. Real world problems often face computational

efforts for ﬁtness evaluations; e. g. in Smart Grid load

planning scenarios, ﬁtness evaluation involves simu-

lating a large number of energy resources and their

behaviour (Bremer and Sonnenschein, 2014).

We propose a surrogate approach that harnesses

a continuously updating quantum potential ﬁeld de-

termined by elitist solutions. On the one hand side,

the quantum ﬁeld exploits global information by ag-

gregating over scattered ﬁtness information similar to

scale space approaches (Horn and Gottlieb, 2001; Le-

ung et al., 2000), on the other side by continuously

adapting to elitist solutions, the quantum ﬁeld surface

quickly ﬂattens to a smooth surrogate for guiding fur-

ther sampling directions. To achieve this, the surro-

gate is a result of Schr

odinger’s equation of which a

probability function is derived that determines the po-

tential function after a clustering approach (cf. (Horn

and Gottlieb, 2001)). We associate minima of the po-

tential ﬁeld, created in denser regions of good solu-

tions’ positions with areas of interest for further inves-

tigation. Thus, offspring solutions are generated with

a trend in descending the potential ﬁeld. By harness-

ing the quantum potential as a surrogate, we achieve

a faster convergence with less objective function calls

compared with using the objective function alone. In

lieu thereof the potential ﬁeld has to be evaluated at

selected point. Although a ﬁne grained computation

of the potential ﬁeld would be a computationally hard

task in higher dimensions (Horn and Gottlieb, 2001),

we achieve a better overall performance because we

need to calculate the ﬁeld only at isolated data points.

The paper starts with a review of using quan-

tum mechanics in computational intelligence and in

particular in evolutionary algorithms; we brieﬂy re-

cap the quantum potential approach for clustering

and present our adaption for integration into evolu-

tion strategies. We conclude with an evaluation of the

approach with the help of several well-known bench-

mark test functions and demonstrate the competitive-

ness to two competitive algorithms: particle swarm

optimization (PSO) and co-variance matrix adaption

evolution strategy (CMA-ES).

2 RELATED WORK

Several evolutionary algorithms have been introduced

to solve nonlinear complex optimization problems

with multi-modal, rugged ﬁtness landscapes. Each

Bremer, J. and Lehnhoff, S.

A Quantum Field Evolution Strategy - An Adaptive Surrogate Approach.

DOI: 10.5220/0006037000210029

In Proceedings of the 8th International Joint Conference on Computational Intelligence (IJCCI 2016) - Volume 1: ECTA, pages 21-29

ISBN: 978-989-758-201-1

of these method has its own characteristics, strengths

and weaknesses. A common characteristics in all EAs

is the generation of an offspring solution set in order

to explore the characteristics of the objective function

in the neighbourhood existing solutions. When the

solution space is hard to explore or objective evalu-

ations are costly, computational effort is a common

drawback for all population-based schemes. Many

efforts have been already spent to accelerate conver-

gence of these methods. Example techniques are: im-

proved population initialization (Rahnamayan et al.,

2007), adaptive populations sizes (Ahrari and Shariat-

Panahi, 2013) or exploiting sub-populations (Rigling

and Moore, 1999).

Sometimes a surrogate model is used in case

of computational expensive objective functions

(Loshchilov et al., 2012) to substitute a share of objec-

tive function evaluations with cheap surrogate model

evaluations. The surrogate model represents a learned

model of the original objective function. Recent ap-

proaches use Radial Basis Functions, Polynomial Re-

gression, Support Vector Regression, Artiﬁcial Neu-

ral Network or Kriging (Gano et al., 2006); each ap-

proach with individual advantages and drawbacks.

At the same time, quantum mechanics has in-

spired several ﬁelds of computational intelligence

such as data mining, pattern recognition or optimiza-

tion. In (Horn and Gottlieb, 2001) a quantum me-

chanics based method for clustering has been in-

troduced. Quantum clustering extends the ideas of

Scale Space algorithms and Support Vector Cluster-

ing (Ben-Hur et al., 2001; Bremer et al., 2010) by

representing an operator in Hilbert space by a scaled

Schr

odinger equation that yields a probability func-

tion as result. The inherent potential function of the

equation that can be analytically derived from the

probability function is used to identify barycenters of

data cluster by associating minima with centers. In

(Weinstein and Horn, 2009a) this approach has been

extended to a dynamic approach that uses the fully

ﬂedged time dependant variant of the Schr

odinger

equation to allow for a interactive visual data mining

especially for large data sets (Weinstein et al., 2013).

We adapted and extended the quantum ﬁeld part

of the clustering approach to optimization and use

the potential function to associate already found so-

lutions from the objective domain with feature vec-

tors in Hilbert space; but with keeping an emphasis of

the total sum (cf. (Horn and Gottlieb, 2001)) and thus

with keeping in mind all improvements of the ongo-

ing optimum search.

(Rapp and Bremer, 2012) used a quantum po-

tential approach derived from quantum clustering

to detect abnormal events in multidimensional data

streams. (Yu et al., 2010) used quantum clustering

for weighing linear programming support vector re-

gression. In this work, we derive a sampling method

for a pµ`λq-ES from the quantum potential approach

originally used for clustering by (Horn and Gottlieb,

2002).

A quantum mechanical extension to particle

swarm optimization has been presented e.g. in (Sun

et al., 2004; Feng and Xu, 2004). Here particles move

according to quantum mechanical behavior in contrast

to the classical mechanics ruled movement of parti-

cles in standard PSO. Usually a harmonic oscillator

is used. In (Loo and Mastorakis, 2007) both meth-

ods quantum clustering and quantum PSO have been

combined by deriving good particle starting positions

from the clustering method ﬁrst. For the simulated

Annealing (SA) approach also a quantum extension

has been developed (Suzuki and Nishimori, 2007).

Whereas in classical SA local minima are escaped by

leaping over the barrier with a thermal jump, quantum

SA introduces the quantum mechanical tunneling ef-

fect for such escapes.

We integrated the quantum concept into evolution

strategies; but by using a different approach: we har-

ness the information in the quantum ﬁeld about the

so far gained success as a surrogate for generating the

offspring generation. By using the potential ﬁeld, in-

formation from all samples at the same time is con-

densed into a directed generation of the next genera-

tion.

3 THE SCHR

ODINGER

POTENTIAL

We start with a brief recap of the Schr

odinger po-

tential and describe the concept following (Horn and

Gottlieb, 2002; Horn and Gottlieb, 2001; Weinstein

and Horn, 2009b). Let

Hψ ” p´

pot

∇

`V px

xqqψpx

xq “ Eψpx

xq (1)

be the Schr

odinger equation rescaled to a single free

parameter σ

pot

and eigenstate ψpx

xq. H denotes the

Hamiltonian operator corresponding to the total en-

ergy E of the system. ψ is the wave function of the

quantum system and ∇

denotes the Laplacian differ-

ential operator. V corresponds to the potential energy

in the system. In case of a single point at x

Eq. (1)

results in

V “

pot

x ´x

(2)

E “

(3)

ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

Figure 1: Function (left column) and exemplary evolution (after 1, 3 and 8 iterations) of the quantum potential that guides the

search for minima for the test functions: Alpine, Goldstein-Price and Himmelblau (from top to bottom).

with d denoting the dimension of the ﬁeld. In this case

ψ is the ground state of a harmonic oscillator. Given

an arbitrary set of points, the potential at a point x

x can

be expressed by

V px

xq “ E `

pot

∇

“ E ´

2σ

pot

x ´x

x´x

2σ

pot

(4)

In Eq. (4) the Gaussian wave function

ψpx

xq “

x ´x

x´x

2σ

pot

(5)

is associated to each point and summed up. Please

note that the bandwidth parameter (usually named σ,

cf. (Horn and Gottlieb, 2002)) has been denoted σ

pot

to discriminate the bandwidth of the wave function

and the variance σ in the mutation used in the evo-

lutions strategy later. In quantum mechanics, usually

the potential V px

xqis given and solutions or eigenfunc-

tions ψpx

xq are sought. In our approach we are already

given ψpx

xq determined by a set of data points. The

set of data points is given by elitist solutions. We then

look for the potential V pxq whose solution is ψpx

xq.

The wave function part corresponds with the

Parzen window estimator approach for data clustering

(Parzen, 1962) or with scale-space clustering (Leung

et al., 2000) that interprets this wave function as the

density function that could have generated the under-

lying data set. The maxima of this density function

correspond therefore with data centers.

In quantum clustering and by requiring ψ to be the

ground state of the Hamiltonian H the potential ﬁeld

V establishes a surface that shows more pronounced

minima (Weinstein and Horn, 2009b). V is unique up

to a constant factor. By setting the minimum of V to

zero it follows that

E “ ´min

pot

∇

. (6)

A Quantum Field Evolution Strategy - An Adaptive Surrogate Approach

With this convention V is determined uniquely with

0 ď E ď

. In this case, E is the lowest eigenvalue of

the operator H and thus describes the ground state.

V is expected to exhibit one or more minima

within some region around the data set and grow

quadratically on the outside (Horn and Gottlieb,

2002). In quantum clustering, these minima are as-

sociated with cluster centers. We will interpret them

as balance points or nuclei where the minimum of an

associated function f lies if the set of data points that

deﬁnes V is a selection of good points (in the sense of

a good ﬁtness according to f ).

4 THE ALGORITHM

We start with a general description of the idea. In our

approach we generate the quantum potential ﬁeld of

an elitist selection of samples. Out of a sample of

λ solutions the best µ are selected according to the

objective function. These µ solutions then deﬁne a

quantum potential ﬁeld that exhibits troughs at the

barycenters of good solution taking into account all

good solutions at the same time. In the next step this

potential ﬁeld is used to guide the sampling of the next

generation of λ offspring solutions from which the

next generation is selected that deﬁnes the new ﬁeld.

In this way, the potential ﬁeld continuously adapts in

each iteration to the so far found best solutions.

The advantage of using the potential ﬁeld results

from its good performance in identifying the barycen-

ters of data points. Horn and Gottlieb (Horn and Got-

tlieb, 2001) demonstrated the superior performance

compared with density based approaches like Scale

Space or Parzen Window approaches (Roberts, 1997;

Parzen, 1962). Transfered to optimization this means

the quantum potential allows for a better identiﬁcation

of local optima. As they can be explored faster they

can be neglected earlier which in turn leads to a faster

convergence of the potential ﬁeld towards the global

optimum (cf. Figure 1).

Figure 1 gives an impression of the adaption pro-

cess that transforms the quantum ﬁeld into an easier

searchable function. Each row shows the situation af-

ter 1, 3 and 8 iterations for different 2-dimensional ob-

jective functions. The left column displays the origi-

nal objective function; from left to right the evolving

potential ﬁeld is displayed together with the respec-

tive offspring solutions that represent the so far best.

The minimum of the potential ﬁeld evolves towards

the minimum of the objective function (or towards

more than one optimum if applicable).

Figure 2 shows the approach formally. Starting

from an initially generated sample X equally dis-

X – tx

„ Upx

, x

u, 1 ď i ď n

repeat

S Ð H

repeat

– x

P X , i „ Up1, |X |q

s „ N px

, σ

if p ď e

V px

q´V ps

, p „Up0, 1q then

S ÐS

S Ys

end if

until |S| ““ λ

V ÐV pS, σ

pot

X Ð selectpS, f , µq

σ Ð σ ¨ω

until }f px

best

q´ f px

q} ď ε

Figure 2: Basic scheme for the Quantum Sampling ES Al-

gorithm.

tributed across the whole search domain deﬁned by

a box constraint in each dimension. Next, the off-

spring is generated by sampling λ points normally

distributed around the µ solutions from the old gen-

eration with an each time randomly chosen parent

solution as expectation and with variance σ

that

decreases with each generation. We use a rejec-

tion sampling approach with the metropolis criterion

(Metropolis et al., 1953) for acceptance applied to the

difference in the potential ﬁeld between a new candi-

date solution and the old solution. The new sample is

accepted with probability

“ minp1, e

∆V

q. (7)

∆V “ V px

old

q´V px

new

q denotes the level difference

in quantum ﬁeld. A descent within the potential ﬁeld

is always accepted. A (temporary) degradation in

quantum potential level is accepted with a probabil-

ity P

(eq. 7) determined by the level of degradation.

As long as there exists at least one pair tx

, x

u Ă

S with x

‰ x

, the potential ﬁeld has a minimum at

with x

‰x

. Thus, the sampling will ﬁnd

new candidates. The sample variance σ

is decreased

in each iteration by a rate ω. Finally, for the next iter-

ation, the solution set X is updated by selecting the µ

best from offspring S.

The process is repeated until any stopping crite-

rion is met; apart from having come near enough the

minimum we regularly used an upper bound for the

maximum number of iterations (or rather: number of

ﬁtness evaluations respectively).

5 RESULTS

We evaluated our evolution strategy with a set of

well known test functions developed for benchmark-

ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications

0 100 200 300 400

500

´10

´7

´4

´1

iterations

log error

plain

quantum

Figure 3: Comparing the convergence of using the quantum

ﬁeld as surrogate with the plain approach working on the

objective function directly. For testing, the 2-dimensional

Alpine function has been used.

ing optimization heuristics. We used the follow-

ing functions: Alpine, Goldstein-Price, Himmelblau,

Bohachevsky 1, generalized Rosenbrock, Griewank,

Sphere, Booth, Chichinadze and Zakharov (Ulmer

et al., 2003; Ahrari and Shariat-Panahi, 2013; Him-

melblau, 1972; Yao et al., 1999); see also appendix

A. These functions represent a mix of multi-modal, 2-

dimensional and multi-dimensional functions, partly

with a huge number of local minima and steep as

well as shoal surroundings of the global optimum and

broad variations in characteristics and domain sizes.

Figure 1 shows some of the used functions (left

column) together with the respective evolution of the

quantum potential ﬁeld that guides the search towards

the minimum at p0, 0q for the Alpine function and

p0, ´1q for Goldstein-Price; the Himmelblau function

has four global minima which are all found. The ﬁg-

ure also shows the evolution of the solution popula-

tion. In the next step, we tested the performance of

our approach against competitive approaches.

First, we tested the effect of using the quantum

ﬁeld as adaptive surrogate compared with the same

update strategy working directly on the ﬁtness land-

scape of the objective function. Figure 3 shows the

convergence of the error on the Alpine test function

for both cases. Although the approach with surrogate

converges slightly slower in the beginning it clearly

outperforms the plain approach without quantum sur-

rogate. Figure 4 shows the same effect for the 20-

dimensional case. Both results show the convergence

of the mean error for 100 runs each.

In a next step, we compared our approach

with two well-known heuristics: particle swarm

optimization (PSO) from (Kennedy and Eber-

hart, 1995) and the covariance matrix adaption

500

1,000

1,500

2,000

2,500

3,000

´6

´4

´2

iterations

log error

plain

quantum

Figure 4: Convergence of using the quantum ﬁeld as sur-

rogate compared to the plain approach. Depicted are the

means of 100 runs on the 20-dimensional Alpine function.

0 20 40

80 100 120 140

´22

´15

´8

´1

iterations

log error

quantum

CMA-ES

Figure 5: Comparing the convergence (means of 100

runs) of CMA-ES and the quantum approach on the 2-

dimensional Booth function.

500

1,000

1,500

´2

´1

iterations

log error

quantum

CMA-ES

Figure 6: Comparing the convergence of CMA-ES and the

quantum approach on the 20-dimensional Griewank func-

tion on the domain r´2048, 2048s

. Both algorithms have

been stopped after 1500 iterations.

A Quantum Field Evolution Strategy - An Adaptive Surrogate Approach

evolution strategy (CMA-ES) by (Hansen, 2011).

Both strategies are well-known, established, and

have been applied to wide range of optimiza-

tion problems. We used readily available and

evaluated implementations from Jswarm-PSO

(http://jswarm-pso.sourceforge.net) and commons

math (http://commons.apache.org/proper/commons-

math).

All algorithms have an individual, strategy spe-

ciﬁc set of parameters that usually can be tweaked to

some degree for a problem speciﬁc adaption. Never-

theless, default values that are applicable for a wide

range of functions are usually available. For our ex-

periments, we used the following default settings. For

the CMA-ES, the (external) strategy parameters are

λ, µ, w

i“1...µ

, controlling selection and recombination;

and d

for step size control and c

and µ

cov

control-

ling the covariance matrix adaption. We have chosen

to set these values after (Hansen, 2011).

λ “ 4 `t3 lnnu, µ “

„



, (8)

“

lnp

`0.5q´lni

j“1

`0.5q´lni

, i “ 1, . . . , µ (9)

“

n `4

, µ

cov

“ µ

f f , (10)

cov

“

cov

pn `

1 ´

cov

min

2µ

cov

´1

pn `2q

`µ

cov

(11)

These settings are speciﬁc to the dimension N of the

objective function. An in-depth discussion of these

parameters is also given in (Hansen and Ostermeier,

2001).

For the PSO, we used values of 0.9 for the weights

and 1 for the inertia parameter as default setting (Shi

and Eberhart, 1998).

For the quantum ﬁeld strategy, we empirically

found the following values as a useful setting for a

range of objective functions. The initial mutation

variance has been set to σ “ d{10 for an initial di-

ameter d of the search space (domain of the objec-

tive function). The shrinking rate of the variance has

been set to ω “ 0.98 and the bandwidth in the poten-

tial equation (4) has been set to σ

pot

“ 0.4. For the

population size we chose µ “ 10 and λ “ 50 if not

otherwise stated.

First, we compared the convergence of the quan-

tum strategy with the CMA-ES. Figure 5 shows a

ﬁrst result for the 2-dimensinal Booth function (with

a minimum of zero). The quantum approach has been

stopped at errors below 1 ˆ10

´21

to avoid numeri-

cal instabilities. The used CMA-ES implementation

has a similar condition integrated into its code. Com-

paring iterations, the quantum approach converges

faster than CMA-ES. Figure 5 shows the result for

the 20-dimensional Griewank function with a search

domain of r´2049, 2048s

. Here, the quantum ap-

proach achieves about the same result as the CMA-ES

within less iterations. Comparing iterations does not

yet shed light on performance.

As the performance is determined by the number

of operations that have to be conducted in each itera-

tion, the following experiments consider the number

of function evaluation calls rather than iterations. Ta-

ble 1 shows the results for a bunch of 2-dimensional

test functions. For each test function and each al-

gorithm the achieved mean (averaged over 100 runs)

solution quality and the needed number of function

evaluations is displayed. The solution quality is ex-

pressed as the error in terms of remaining difference

to the known global optimum. As stopping criterion

this time each algorithm has been equipped with two

conditions: error below 5 ˆ10

´17

and a given budget

of at most 5 ˆ10

evaluations. For the quantum ap-

proach two counts of evaluation functions are given,

because due to the nature of surrogate approaches a

share of function evaluations is substituted by surro-

gate evaluations. Thus, total evaluations refers to the

sum of function and surrogate evaluations.

The results show that CMA-ES is in general un-

beatable in terms of function evaluations whereas the

quantum approach in half of the cases gains the more

accurate result. The PSO succeeds for the Griewank

and the Chichinadze function. For the 20-dimensional

cases in Table 2 the quantum approach gains the most

accurate result in most of the cases. The CMA-ES

winning margin of a low number of evaluations de-

creases compared with the quantum approach, but is

still prominent. Nevertheless, the number of neces-

sary function evaluations for the quantum approach

can still be reduced when using a lower population

size. But, such tuning is subject to the problem at

hand. On the other hand, notwithstanding the low

number of objective evaluations, the CMA-ES needs

higher processing time for high-dimensional prob-

lems due to the fact that CMA-ES needs – among oth-

ers – to conduct eigenvalue decompositions of its co-

variance matrix (Opn

q) with number of dimensions n

(Knight and Lunacek, 2007). Table 3 gives an expres-

sion for necessary computation (CPU-) times (Java 8,

2.7 GHz Quadcore) for the 100-dimensional Sphere

function for CMA-ES and a quantum approach with

reduced populations size (µ “ 4, λ “ 12).

All in all, the quantum approach is competitive to

the established algorithms and in some cases even su-

perior.

ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications

Table 1: Results for comparing CMA-ES, PSO and the quantum approach with a set of 2-dimensional test functions. The

error denotes the remaining difference to the known optimum; obj. evaluations and total evaluations refer to the number of

conducted function evaluations and the sum of function and quantum surrogate evaluations respectively. The latter is only

applicable to the quantum approach.

problem algorithm error obj. evaluations total evaluations

Alpine

CMA-ES 3.954 ˆ10

´12

˘ 4.797 ˆ10

´12

746.38 ˘ 90.81 n/a

PSO 5.543 ˆ10

´9

˘ 3.971 ˆ10

´8

500000.00 ˘ 0.00 n/a

quantum 8.356 ˆ10

´16

˘ 4.135 ˆ10

´16

99363.00 ˘ 66100.02 198996.53 ˘ 132200.50

Griewank

CMA-ES 2.821 ˆ10

˘ 2.254 ˆ10

174.22 ˘ 137.63 n/a

PSO 4.192 ˆ10

´4

˘ 1.683 ˆ10

´3

500000.00 ˘ 0.00 n/a

quantum 6.577 ˆ10

´3

˘ 5.175 ˆ10

´3

201361.50 ˘ 47207.02 472348.75 ˘ 94420.25

GoldsteinPrice

CMA-ES 0.459 ˆ10

˘ 1.194 ˆ10

613.96 ˘ 203.03 n/a

PSO 1.698 ˆ10

˘ 1.196 ˆ10

500000.00 ˘ 0.00 n/a

quantum 1.130 ˆ10

˘ 1.255 ˆ10

250000.00 ˘ 0.00 500012.75 ˘ 3.48

Bohachevsky1

CMA-ES 3.301 ˆ10

´2

˘ 1.094 ˆ10

´1

672.70 ˘ 59.91 n/a

PSO 1.626 ˆ10

´10

˘ 1.568 ˆ10

´9

500000.00 ˘ 0.00 n/a

quantum 4.707 ˆ10

´16

˘ 3.246 ˆ10

´16

39641.50 ˘ 1672.63 87548.03 ˘ 3377.00

Booth

CMA-ES 4.826 ˆ10

´17

˘ 1.124 ˆ10

´16

605.68 ˘ 49.55 n/a

PSO 5.985 ˆ10

´14

˘ 3.743 ˆ10

´13

500000.00 ˘ 0.00 n/a

quantum 5.695 ˆ10

´16

˘ 2.957 ˆ10

´16

33696.00 ˘ 1357.07 68283.18 ˘ 2711.51

Chichinadze

CMA-ES 0.953 ˆ10

˘ 4.125 ˆ10

698.44 ˘ 200.31 n/a

PSO 0.226 ˆ10

˘ 2.247 ˆ10

´1

500000.00 ˘ 0.00 n/a

quantum 1.327 ˆ10

˘ 8.376 ˆ10

50.00 ˘ 0.00 180.68 ˘ 12.63

Table 2: Results for comparing CMA-ES, PSO and the quantum approach with a set of 20-dimensional test functions with the

same setting as in Table 1.

problem algorithm error obj. evaluations total evaluations

Rosenbrock

CMA-ES 1.594 ˆ10

˘ 6.524 ˆ10

10678.60 ˘ 6514.47 n/a

PSO 1.629 ˆ10

˘ 1.759 ˆ10

50000000 ˘ 0.00 n/a

quantum 3.884 ˆ10

˘ 1.470 ˆ10

23291850.00 ˘ 3166.67 50001008.27 ˘ 557.53

Griewank

CMA-ES 5.878 ˆ10

˘ 1.082 ˆ10

8734.48 ˘ 7556.50 n/a

PSO 1.429 ˆ10

˘ 3.539 ˆ10

50000000 ˘ 0.00 n/a

quantum 2.267 ˆ10

´3

˘ 3.973 ˆ10

´3

6929090.0 ˘ 974963.9 17269617.9 ˘ 1949920.8

Zakharov

CMA-ES 9.184 ˆ10

´16

˘ 1.068 ˆ10

´15

8902.24 ˘ 845.05 n/a

PSO 7.711 ˆ10

˘ 5.961 ˆ10

50000000 ˘ 0.00 n/a

quantum 8.978 ˆ10

´17

˘ 9.645 ˆ10

´18

1021370.00 ˘ 4802.47 2962737.19 ˘ 10126.34

Spherical

CMA-ES 1.200 ˆ10

´15

˘ 1.258 ˆ10

´15

8678.80 ˘ 912.13 n/a

PSO 2.637 ˆ10

˘ 7.517 ˆ10

50000000 ˘ 0.00 n/a

quantum 8.943 ˆ10

´17

˘ 1.036 ˆ10

´17

973750.00 ˘ 4243.53 2674716.70 ˘ 8844.04

Alpine

CMA-ES 9.490 ˆ10

´12

˘ 3.331 ˆ10

´11

15196.48 ˘ 570.76 n/a

PSO 4.021 ˆ10

˘ 2.442 ˆ10

50000000 ˘ 0.0 n/a

quantum 9.272 ˆ10

´17

˘ 5.857 ˆ10

´18

1867830.0 ˘ 3795.41 4771580.5 ˘ 8390.4

A Quantum Field Evolution Strategy - An Adaptive Surrogate Approach

Table 3: Comparing computational performace of CMA-ES and quantum approach with the 100-dimensional Sphere function.

algorithm error total evaluations CPU time / nsec.

CMA-ES 1.593 ˆ10

´16

˘ 1.379 ˆ10

´16

129204.4 ˘ 18336.1 5.189 ˆ10

˘ 6.918 ˆ10

quantum 9.861 ˆ10

´17

˘ 1.503 ˆ10

´18

161172.8 ˘ 626.8 4.029 ˆ10

˘ 1.040 ˆ10

6 CONCLUSION

We introduced a novel evolution strategy for global

optimization that uses the quantum potential ﬁeld de-

ﬁned by elitist solutions for generating the offspring

solution set.

By using the quantum potential, information about

the ﬁtness landscape of scattered points is condensed

into a surrogate for guiding further sampling instead

of looking at different single solutions; one at a time.

In this way, the quantum surrogate tries to ﬁt the

search distribution to the shape of the objective func-

tion like CMA-ES (Hansen, 2006). The quantum sur-

rogate adapts continuously as the optimization pro-

cess zooms into areas of interest.

Compared with a population based solver and

CMA-ES as established evolution strategy, we

achieved a competitive and sometimes faster conver-

gence with less objective function calls. We tested

our method on ill-conditioned problems as well as on

simple problems ﬁnding it performing equally good

on both.

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APPENDIX

Used test functions (Ulmer et al., 2003; Ahrari and

Shariat-Panahi, 2013; Himmelblau, 1972; Yao et al.,

1999; Mishra, 2006).

Alpine:

xq “

i“1

sinpx

q`0.1x

|, (12)

´10 ď x

ď 10 with x

“ p0, . . . , 0q and f

q “ 0.

Goldstein-Price:

xq “p1 `px

`1q

p19 ´14x

`2x

´14x

`6x

`3x

qq¨

p30 `p2x

´3x

p18 ´32x

`12x

`48x

´36x1x2 `27x2

qq,

(13)

´2 ď x

, x

ď 2 with x

“ p0, ´1q and f

q “ 3.

Himmelblau:

xq “ px

´11q

`px

´7q

(14)

´10 ď x

, x

ď 10 with f

q “ 0 at four identical

local minima.

Bohachevsky 1:

xq“x

`2x

´0.3 cosp2πx

q´0.4cosp4πx

q`0.7,

(15)

´100 ď x

, x

ď100 with x

“p0, 0qand f

q“ 0.

Generalized Rosenbrock:

xq “

n´1

i“1

pp1 ´x

`100px

i`1

´x

q, (16)

´2048 ď x

ď 2048 with x

“ p1, . . . , 1q f

q “ 0.

Griewank:

xq “ 1 `

200

i“1

i“0

cosp

q, (17)

´100 ď x

ď 100 with x

“ p0, . . . , 0q f

q “ 0.

Zakharov:

xq “

i“1

0.5ix

i“1

0.5ix

, (18)

´5 ď x

ď 10 with x

“ p0, . . . , 0q f

q “ 0.

Sphere:

xq “

i“1

, (19)

´5 ď x

ď 5 with x

“ p0, . . . , 0q f

q “ 0.

Chichinadze:

xq “ x

´12x

`11 `10cospπx

{2q

`8sinp5πx

q´p1{5q

0.5

´0.5px

´0.5q

(20)

´30 ďx

, x

ď30 with x

“p5.90133, 0.5q f

q“

´43.3159.

Booth:

0px

xq “ px

`2x

´7q

p2x

´5q

, (21)

´20 ď x

, x

ď 20 with x

“ p1, 3q f

q “ 0.

A Quantum Field Evolution Strategy - An Adaptive Surrogate Approach