Differential Evolution with DEoptim
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C ONTRIBUTED R ESEARCH A RTICLES 27
Differential Evolution with DEoptim
An Application to Non-Convex Portfolio Optimiza- tween upper and lower bounds (defined by the user)
tion or using values given by the user. Each generation
involves creation of a new population from the cur-
by David Ardia, Kris Boudt, Peter Carl, Katharine M. rent population members { xi | i = 1, . . . , NP}, where
Mullen and Brian G. Peterson i indexes the vectors that make up the population.
This is accomplished using differential mutation of
Abstract The R package DEoptim implements the population members. An initial mutant parame-
the Differential Evolution algorithm. This al- ter vector vi is created by choosing three members of
gorithm is an evolutionary technique similar to the population, xi1 , xi2 and xi3 , at random. Then vi is
classic genetic algorithms that is useful for the generated as
solution of global optimization problems. In this
note we provide an introduction to the package .
vi = xi1 + F · ( xi2 − xi3 ) ,
and demonstrate its utility for financial appli-
cations by solving a non-convex portfolio opti- where F is a positive scale factor, effective values for
mization problem. which are typically less than one. After the first mu-
tation operation, mutation is continued until d mu-
tations have been made, with a crossover probability
Introduction CR ∈ [0, 1]. The crossover probability CR controls the
fraction of the parameter values that are copied from
Differential Evolution (DE) is a search heuristic intro- the mutant. Mutation is applied in this way to each
duced by Storn and Price (1997). Its remarkable per- member of the population. If an element of the trial
formance as a global optimization algorithm on con- parameter vector is found to violate the bounds after
tinuous numerical minimization problems has been mutation and crossover, it is reset in such a way that
extensively explored (Price et al., 2006). DE has the bounds are respected (with the specific protocol
also become a powerful tool for solving optimiza- depending on the implementation). Then, the ob-
tion problems that arise in financial applications: for jective function values associated with the children
fitting sophisticated models (Gilli and Schumann, are determined. If a trial vector has equal or lower
2009), for performing model selection (Maringer and objective function value than the previous vector it
Meyer, 2008), and for optimizing portfolios under replaces the previous vector in the population; oth-
non-convex settings (Krink and Paterlini, 2011). DE erwise the previous vector remains. Variations of
is available in R with the package DEoptim. this scheme have also been proposed; see Price et al.
In what follows, we briefly sketch the DE algo- (2006).
rithm and discuss the content of the package DEop- Intuitively, the effect of the scheme is that the
tim. The utility of the package for financial applica- shape of the distribution of the population in the
tions is then explored by solving a non-convex port- search space is converging with respect to size and
folio optimization problem. direction towards areas with high fitness. The closer
the population gets to the global optimum, the more
the distribution will shrink and therefore reinforce
Differential Evolution the generation of smaller difference vectors.
For more details on the DE strategy, we refer
DE belongs to the class of genetic algorithms (GAs) the reader to Price et al. (2006) and Storn and Price
which use biology-inspired operations of crossover, (1997).
mutation, and selection on a population in order
to minimize an objective function over the course The package DEoptim
of successive generations (Holland, 1975). As with
other evolutionary algorithms, DE solves optimiza- DEoptim (Ardia et al., 2011) was first published on
tion problems by evolving a population of candi- CRAN in 2005. Since becoming publicly available, it
date solutions using alteration and selection opera- has been used by several authors to solve optimiza-
tors. DE uses floating-point instead of bit-string en- tion problems arising in diverse domains. We refer
coding of population members, and arithmetic op- the reader to Mullen et al. (2011) for a detailed de-
erations instead of logical operations in mutation, in scription of the package.
contrast to classic GAs. DEoptim consists of the core function DEoptim
Let NP denote the number of parameter vectors whose arguments are:
(members) x ∈ Rd in the population, where d de- • fn: the function to be optimized (minimized).
notes dimension. In order to create the initial genera-
tion, NP guesses for the optimal value of the param- • lower, upper: two vectors specifying scalar real
eter vector are made, either using random values be- lower and upper bounds on each parameter to
The R Journal Vol. 3/1, June 2011 ISSN 2073-485928 C ONTRIBUTED R ESEARCH A RTICLES
be optimized. The implementation searches be- > set.seed(1234)
tween lower and upper for the global optimum > DEoptim(fn = Rastrigin,
of fn. + lower = c(-5, -5),
+ upper = c(5, 5),
• control: a list of tuning parameters, among + control = list(storepopfrom = 1))
which: NP (default: 10 · d), the number of pop-
ulation members and itermax (default: 200), The above call specifies the objective function to
the maximum number of iterations (i.e., pop- minimize, Rastrigin, the lower and upper bounds
ulation generations) allowed. For details on on the parameters, and, via the control argument,
the other control parameters, the reader is re- that we want to store intermediate populations from
ferred to the documentation manual (by typ- the first generation onwards (storepopfrom = 1).
ing ?DEoptim). For convenience, the function Storing intermediate populations allows us to exam-
DEoptim.control() returns a list with default ine the progress of the optimization in detail. Upon
elements of control. initialization, the population is comprised of 50 ran-
dom values drawn uniformly within the lower and
• ...: allows the user to pass additional argu-
upper bounds. The members of the population gen-
ments to the function fn.
erated by the above call are plotted at the end of
The output of the function DEoptim is a member different generations in Figure 2. DEoptim consis-
of the S3 class DEoptim. Members of this class have a tently finds the minimum of the function (market
plot and a summary method that allow to analyze the with an open white circle) within 200 generations us-
optimizer’s output. ing the default settings. We have observed that DE-
Let us quickly illustrate the package’s usage with optim solves the Rastrigin problem more efficiently
the minimization of the Rastrigin function in R2 , than the simulated annealing method available in the
which is a common test for global optimization: R function optim (for all annealing schedules tried).
Note that users interested in exact reproduction of
> RastriginC ONTRIBUTED R ESEARCH A RTICLES 29
Generation 1 Generation 20
4 4
2 2
x2
x2
0 0
−2 −2
−4 −4
−4 −2 0 2 4 −4 −2 0 2 4
x1 x1
Generation 40 Generation 80
4 4
2 2
x2
x2
0 0
−2 −2
−4 −4
−4 −2 0 2 4 −4 −2 0 2 4
x1 x1
Figure 2: The population (market with black dots) associated with various generations of a call to DEoptim as
it searches for the minimum of the Rastrigin function at point x = (0, 0)0 (market with an open white circle).
The minimum is consistently determined within 200 generations using the default settings of DEoptim.
The R Journal Vol. 3/1, June 2011 ISSN 2073-485930 C ONTRIBUTED R ESEARCH A RTICLES
Modern portfolio selection considers additional age estimators, and their implementation in portfolio
criteria such as to maximize upper-tail skewness and allocation, we refer the reader to Würtz et al. (2009,
liquidity or minimize the number of securities in the Chapter 4). These estimators might yield better per-
portfolio. In the recent portfolio literature, it has formance in the case of small samples, outliers or de-
been advocated by various authors to incorporate partures from normality.
risk contributions in the portfolio allocation prob-
lem. Qian’s (2005) Risk Parity Portfolio allocates port- > library("quantmod")
folio variance equally across the portfolio compo- > tickers getSymbols(tickers,
nents. Maillard et al. (2010) call this the Equally-
+ from = "2000-12-01",
Weighted Risk Contribution (ERC) Portfolio. They de-
+ to = "2010-12-31")
rive the theoretical properties of the ERC portfolio > P for(ticker in tickers) {
of the minimum variance and equal-weight portfo- + tmp pContribCVaR rbind(tickers, round(100 * pContribCVaR, 2))
show here, the package DEoptim is well suited to [,1] [,2] [,3] [,4] [,5]
solve these problems. Note that it is also the evo- tickers "GE" "IBM" "JPM" "MSFT" "WMT"
lutionary optimization strategy used in the package "21.61" "18.6" "25.1" "25.39" "9.3"
PortfolioAnalytics (Boudt et al., 2010b).
To illustrate this, consider as a stylized example a We see that in the equal-weight portfolio, 25.39%
five-asset portfolio invested in the stocks with tickers of the portfolio CVaR risk is caused by the 20%
GE, IBM, JPM, MSFT and WMT. We keep the dimen- investment in MSFT, while the 20% investment in
sion of the problem low in order to allow interested WMT only causes 9.3% of total portfolio CVaR. The
users to reproduce the results using a personal com- high risk contribution of MSFT is due to its high stan-
puter. More realistic portfolio applications can be dard deviation and low average return (reported in
obtained in a straightforward manner from the code percent):
below, by expanding the number of parameters opti-
> round(100 * mu , 2)
mized. Interested readers can also see the DEoptim
GE IBM JPM MSFT WMT
package vignette for a 100-parameter portfolio opti-
-0.80 0.46 -0.06 -0.37 0.0
mization problem that is typical of those encountered > round(100 * diag(sigma)^(1/2), 2)
in practice. GE IBM JPM MSFT WMT
We first download ten years of monthly data us- 8.90 7.95 9.65 10.47 5.33
ing the function getSymbols of the package quant-
mod (Ryan, 2010). Then we compute the log-return We now use the function DEoptim of the package
series and the mean and covariance matrix estima- DEoptim to find the portfolio weights for which the
tors. For on overview of alternative estimators of the portfolio has the lowest CVaR and each investment
covariance matrix, such as outlier robust or shrink- can contribute at most 22.5% to total portfolio CVaR
The R Journal Vol. 3/1, June 2011 ISSN 2073-4859C ONTRIBUTED R ESEARCH A RTICLES 31
risk. For this, we first define our objective function from the last two lines, this minimum risk portfolio
to minimize. The current implementation of DEop- has a higher expected return than the equal weight
tim allows for constraints on the domain space. To portfolio. The following code illustrates that DEop-
include the risk budget constraints, we add them to tim yields superior results than the gradient-based
the objective function through a penalty function. As optimization routines available in R.
such, we allow the search algorithm to consider in-
> out out$value
pression of these violations is α × |violation| p . Crama [1] 0.1255093
and Schyns (2003) describe several ways to calibrate > out out$objective
[1] 0.1158250
term becomes negligible with respect to the penalty;
thus, small variations of w can lead to large varia- Even on this relatively simple stylized example,
tions of the penalty term, which mask the effect on the optim and nlminb routines converged to local
the portfolio CVaR. We set α = 103 and p = 1, but minima.
recognize that better choices may be possible and de- Suppose now the investor is interested in the
pend on the problem at hand. most risk diversified portfolio whose expected re-
turn is higher than the equal-weight portfolio. This
> obj32 C ONTRIBUTED R ESEARCH A RTICLES
DEoptim is displayed in Figure 3. Gray elements for the solution of global optimization problems in
depict the results for all tested portfolios (using a wide variety of fields. We have referred interested
hexagon binning which is a form of bivariate his- users to Price et al. (2006) and Mullen et al. (2011) for
togram, see Carr et al. (2011); higher density regions a more extensive introduction, and further pointers
are darker). The yellow-red line shows the path of to the literature on DE. The utility of using DEoptim
the best member of the population over time, with was further demonstrated with a simple example of a
the darkest solution at the end being the optimal port- stylized non-convex portfolio risk contribution allo-
folio. We can notice how DEoptim does not spend cation, with users referred to PortfolioAnalytics for
much time computing solutions in the scatter space portfolio optimization using DE with real portfolios
that are suboptimal, but concentrates the bulk of the under non-convex constraints and objectives.
calculation time in the vicinity of the final best port- The latest version of the package includes the
folio. Note that we are not looking for the tangency adaptive Differential Evolution framework by Zhang
portfolio; simple mean/CVaR optimization can be and Sanderson (2009). Future work will also be di-
achieved with standard optimizers. We are looking rected at parallelization of the implementation. The
here for the best balance between return and risk con- DEoptim project is hosted on R-forge at https://
centration. r-forge.r-project.org/projects/deoptim/.
We have utilized the mean return/CVaR concen-
tration examples here as more realistic, but still styl-
ized, examples of non-convex objectives and con- Acknowledgements
straints in portfolio optimization; other non-convex
objectives, such as drawdown minimization, are also The authors would like to thank Dirk Eddelbuettel,
common in real portfolios, and are likewise suitable Juan Ospina, and Enrico Schumann for useful com-
to application of Differential Evolution. One of the ments and Rainer Storn for his advocacy of DE and
key issues in practice with real portfolios is that a making his code publicly available. The authors are
portfolio manager rarely has only a single objective also grateful to two anonymous reviewers for help-
or only a few simple objectives combined. For many ful comments that have led to improvements of this
combinations of objectives, there is no unique global note. Kris Boudt gratefully acknowledges financial
optimum, and the constraints and objectives formed support from the National Bank of Belgium.
lead to a non-convex search space. It may take sev-
eral hours on very fast machines to get the best an-
swers, and the best answers may not be a true global Disclaimer
optimum, they are just as close as feasible given poten-
tially competing and contradictory objectives. The views expressed in this note are the sole respon-
When the constraints and objectives are relatively sibility of the authors and do not necessarily reflect
simple, and may be reduced to quadratic, linear, or those of aeris CAPITAL AG, Guidance Capital Man-
conical forms, a simpler optimization solver will pro- agement, Cheiron Trading, or any of their affiliates.
duce answers more quickly. When the objectives Any remaining errors or shortcomings are the au-
are more layered, complex, and potentially contra- thors’ responsibility.
dictory, as those in real portfolios tend to be, DE-
optim or other global optimization algorithms such
as those integrated into PortfolioAnalytics provide Bibliography
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The R Journal Vol. 3/1, June 2011 ISSN 2073-4859C ONTRIBUTED R ESEARCH A RTICLES 33
0.4
monthly mean (in %)
0.2
0
−0.2
−0.4
14 16 18 20 22 24
monthly CVaR (in %)
Figure 3: Risk/return scatter chart showing the results for portfolios tested by DEoptim.
The R Journal Vol. 3/1, June 2011 ISSN 2073-485934 C ONTRIBUTED R ESEARCH A RTICLES
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