GLOBAL COMPETITIVE RANKING FOR CONSTRAINTS

HANDLING WITH MODIFIED DIFFERENTIAL EVOLUTION

Abul Kalam Azad and Edite M. G. P. Fernandes

Algoritmi R&D Center, School of Engineering, University of Minho, 4710-057 Braga, Portugal

Keywords:

Constrained nonlinear programming, Constraints handling, Ranking, Differential evolution.

Abstract:

Constrained nonlinear programming problems involving a nonlinear objective function with inequality and/or

equality constraints introduce the possibility of multiple local optima. The task of global optimization is to

ﬁnd a solution where the objective function obtains its most extreme value while satisfying the constraints.

Depending on the nature of the involved functions many solution methods have been proposed. Most of the

existing population-based stochastic methods try to make the solution feasible by using a penalty function

method. However, to ﬁnd the appropriate penalty parameter is not an easy task. Population-based differential

evolution is shown to be very efﬁcient to solve global optimization problems with simple bounds. To handle

the constraints effectively, in this paper, we propose a modiﬁed constrained differential evolution that uses

self-adaptive control parameters, a mixed modiﬁed mutation, the inversion operation, a modiﬁed selection and

the elitism in order to progress efﬁciently towards a global solution. In the modiﬁed selection, we propose a

ﬁtness function based on the global competitive ranking technique for handling the constraints. We test 13

benchmark problems. We also compare the results with the results found in literature. It is shown that our

method is rather effective when solving constrained problems.

1 INTRODUCTION

Problems involving global optimization over contin-

uous spaces are ubiquitous throughout the scientiﬁc

community. Many real world problems are formu-

lated as mathematical programming problems involv-

ing continuous variables with linear/nonlinear objec-

tive function and constraints. The constraints can be

of inequality and equality type. Generally, the con-

strained nonlinear programming problems are formu-

lated as follows:

minimize f(x)

subject to g

(x) ≤ 0 k = 1, 2,. . ., m

(x) = 0 l = 1, 2, . . . , m

≤ x

≤ u

j = 1, 2, . . . , n,

(1)

where, f, g

, h

: R

−→ R with feasible set F = {x ∈

: g(x) ≤ 0, h(x) = 0 and l ≤ x ≤ u}. f, g

, h

may

be differentiable and the information about deriva-

tives may or may not be provided.

Problem (1) involving global optimization (here

we consider a minimization problem) of a multivari-

ate function with constraints is widespread in the

mathematical modeling of real world systems. Many

problems can be described only by nonlinear relation-

ships, which introduce the possibility of multiple lo-

cal minima. The task of the global optimization is to

ﬁnd a solution where the objectivefunction obtains its

most extreme value, the global minimum, while satis-

fying the constraints.

In the last decades, manystochastic solution meth-

ods with different constraints handling techniques

have been proposed to solve (1). Stochastic meth-

ods involve a random sample of solutions and the

subsequent manipulation of the sample to ﬁnd good

local (and hopefully global) minima. The stochas-

tic methods can be based on a point-to-point search

or on a population-based search. Most of the exist-

ing population-based stochastic methods try to make

the solution feasible by repairing the infeasible one

or penalizing an infeasible solution with a penalty

function method. However, to ﬁnd the appropriate

penalty parameter is not an easy task. (Deb, 2000)

proposed an efﬁcient constraints handling technique

for genetic algorithms based on the feasibility and

dominance rules. The author used a penalty function

that does not require any penalty parameter. (Bar-

bosa and Lemonge, 2003) proposed a parameter-less

adaptive penalty scheme for genetic algorithms ap-

plied to constrained optimization problems. In the

Kalam Azad A. and M. G. P. Fernandes E..

GLOBAL COMPETITIVE RANKING FOR CONSTRAINTS HANDLING WITH MODIFIED DIFFERENTIAL EVOLUTION.

DOI: 10.5220/0003672200420051

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 42-51

ISBN: 978-989-8425-83-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

very recent paper the authors proposed this adap-

tive penalty scheme for differential evolution (Silva

et al., 2011). (Hedar and Fukushima, 2006) proposed

a ﬁlter simulated annealing method for constrained

continuous global optimization problems. The au-

thors used the ﬁlter method (Fletcher and Leyffer,

2002) rather than the penalty method to handle the

constraints. Runarsson and Yao proposed a stochas-

tic ranking (Runarsson and Yao, 2000) and a global

competitive ranking (Runarsson and Yao, 2003) tech-

nique for constrained evolutionary optimization based

on evolution strategy. The authors presented a new

view on the usual penalty function methods in terms

of the dominance of penalty and objective func-

tions. (Dong et al., 2005) proposed a swarm op-

timization based on constraint ﬁtness priority-based

ranking technique. (Zahara and Hu, 2008) proposed

constrained optimization with a hybrid Nelder-Mead

simplex method and a particle swarm optimization.

The authors also used constraint ﬁtness priority-based

ranking technique for constraints handling. Rocha

and Fernandes proposed the feasibility and domi-

nance rules (Rocha and Fernandes, 2008) and the self-

adaptive penalties (Rocha and Fernandes, 2009) in

the electromagnetism-like algorithm for constrained

global optimization problems. (Coello, 2000) pro-

posed constraints handling using an evolutionary mul-

tiobjective optimization technique. This author intro-

duced the concept of nondominance (commonly used

in multiobjective optimization) as a way to incorpo-

rate constraints into the ﬁtness function of a genetic

algorithm. (Coello and Cort´es, 2004) proposed hy-

bridizing a genetic algorithm with an artiﬁcial im-

mune system for global optimization. The authors

used genotypic-based distances to move from infeasi-

ble solution to feasible one. Another constraints han-

dling technique is the multilevel Pareto ranking based

on the constraints matrix (Ray and Tai, 2001; Ray and

Liew, 2003). This technique is based on the concepts

of Pareto nondominance in multiobjective optimiza-

tion. (Ray and Tai, 2001) proposed an evolutionary

algorithm with a multilevel pairing strategy and (Ray

and Liew, 2003) proposed a society and civilization

algorithm based on the simulation of social behaviour.

Differential evolution (DE) proposed by (Storn

and Price, 1997) is a population-based heuristic ap-

proach that is very efﬁcient to solve global opti-

mization problems with simple bounds. DE’s per-

formance depends on the ampliﬁcation factor of dif-

ferential variation and crossover control parameter.

Hence self-adaptive control parameters have been im-

plemented in DE in order to obtain a competitive al-

gorithm. Further, to improve solution accuracy, tech-

niques that are able to exploit locally certain regions,

detected in the search space of the problem as promis-

ing, are also required. A local search starts from

a candidate solution and then iteratively moves to a

neighbour solution. Typically, every candidate solu-

tion has more than one neighbour solutions and the

choice of movement depends only on the information

about the solutions in the neighbourhood of the cur-

rent one. When the solutions ought to be restricted

to a set of equality and inequality constraints, an ef-

ﬁcient constraints handling technique is also required

in the solution method. In this paper, we propose a

modiﬁed constrained differential evolution algorithm

(herein denoted as m-CDE) that uses self-adaptive

control parameters (Brest et al., 2006), a mixture of

modiﬁed mutations (Kaelo and Ali, 2006), and also

includes the inversion operation, a modiﬁed selection

and the elitism to be able to progress efﬁciently to-

wards a global solution of problems (1). To handle

the constraints effectively, the modiﬁed selection in-

corporates the global competitive ranking technique

to assess the ﬁtness of all individual points in the pop-

ulation.

The organization of this paper is as follows. We

describe the constraints handling techniques in Sec-

tion 2. In Section 3 the modiﬁed constrained differ-

ential evolution is outlined. Section 4 describes the

experimental results and ﬁnally we draw the conclu-

sions of this study in Section 5.

2 CONSTRAINTS HANDLING

TECHNIQUES

Stochastic methods are mostly developed for the

global optimization of unconstrained problems. Fi-

nally, they are extended to the constrained problems

with the modiﬁcation of solution procedures or by

applying penalty function methods. In population-

based techniques, the widely used approach to deal

with constrained optimization problems is based on

penalty functions. In penalty functions, a penalty term

is added to the objective function in order to penalize

the constraint violation. This enable us to transform

a constrained optimization problem into a sequence

of unconstrained subproblems, whose objective func-

tion is

ψ(x) = f(x) + µH[ζ(x)], (2)

where ζ(x) is a real-valued function and is greater

than or equal to 0 aiming at measuring the constraint

violation, H is a function of the constraint violation,

and µ deﬁnes a positive penalty parameter aiming at

balancing objective and constraint violation. An indi-

vidual point is feasible if ζ(x) = 0.

GLOBAL COMPETITIVE RANKING FOR CONSTRAINTS HANDLING WITH MODIFIED DIFFERENTIAL

EVOLUTION

In constrained optimization, it is very important

to ﬁnd the right balance between the objective func-

tion and the constraint violation. The penalty func-

tion method can be applied to any type of constraints,

but the performance of penalty-type methods is not

always satisfactory. Usually, µ is updated throughout

the iterative process, so that the sequence of the solu-

tions of the unconstrained subproblems converges to

the solution of the constrained problem. Small values

of µ can produce almost optimal but infeasible solu-

tions. On the other hand, large values of µ can give

feasible solutions although an optimal solution in the

boundary may not be found. Hence, the most difﬁ-

cult aspect of a penalty function method is to deter-

mine the appropriate value to initialize the parameter

µ, as well as the rule for its updating. For this rea-

son alternative constraints handling techniques have

been proposed in the last decades. Here, three differ-

ent constraints handling techniques, usually used in

population-based methods - stochastic ranking, global

competitive ranking, and the feasibility and domi-

nance rules - have been implemented, and extensively

tested, in our m-CDE algorithm. They are brieﬂy de-

scribed below.

2.1 Stochastic Ranking

(Runarsson and Yao, 2000) ﬁrst proposed stochastic

ranking for the constrained optimization. This is a

bubble-sort-like algorithm to give ranks to individ-

uals in a population stochastically. In this ranking

method, two adjacent individual points are compared

and given ranks and swapped. The algorithm is halt

if there is no swap. Individuals are ranked primar-

ily based on their constraint violations. The objective

function values are then considered if: i) individu-

als are feasible, or ii) a uniform random number be-

tween 0 and 1 is less than or equal to P

. The proba-

bility P

is used only for comparisons of the objective

function in the infeasible region of the search space.

That is, given any pair of two adjacent individual

points, the probability of comparing them (in order to

determine which one is ﬁtter) according to the objec-

tive function is 1 if both individuals are feasible; oth-

erwise it is P

. Such ranking ensures that good feasi-

ble solutions as well as promising infeasible ones are

ranked in the top of the population.

In our implementation of the stochastic ranking

(SR) method (Runarsson and Yao, 2000) in the modi-

ﬁed constrained differential evolution, each individual

point x

is evaluated according to the ﬁtness function

) =

− 1

N − 1

, (3)

where I

represents the rank of the point x

of the pop-

ulation and N is the number of individuals in a pop-

ulation. From (3), the ﬁtness of an individual point

having the highest rank will be 0 and that with the

lowest rank will be 1. The best individual point in a

population has the lowest ﬁtness value.

2.2 Global Competitive Ranking

(Runarsson and Yao, 2003) proposed another con-

straints handling technique for constrained problems

in order to strike the right balance between the ob-

jective function and the constraint violation. This

method is called global competitive ranking. In this

method, an individual point is ranked by comparing it

against all other members of the population.

In this ranking process, after calculating f and ζ

for all the individuals, f and ζ are sorted separately

in ascending order (since we consider the minimiza-

tion problem) and given ranks. Special considera-

tion is given to the tied individuals. In the case of

tied individuals the same higher rank will be given.

For example, in these eight individuals, already in as-

cending order, h6, (5, 8), 1, (2, 4, 7), 3i (individuals in

parentheses havesame value) the correspondingranks

are I(6) = 1, I(5) = I(8) = 2, I(1) = 4, I(2) = I(4) =

I(7) = 5, I(3) = 8. After the ranking of all the in-

dividuals based on the objective function f and the

constraint violation ζ, separately, the ﬁtness function

of each individual point x

is given by

) = P

i, f

− 1

N − 1

+ (1− P

)

i,ζ

− 1

N − 1

, (4)

where Φ

means ﬁtness based on the global com-

petitive ranking (GR), and I

i, f

and I

i,ζ

are the ranks of

point x

based on the objective function and the con-

straint violation, respectively. P

indicates the prob-

ability that the ﬁtness is calculated based on the rank

of objective function. It is clear from the above that

can be used easily to bias the calculation of ﬁtness

accordingto the objectivefunction or the average con-

straint violation. The probability should take a value

0.0 < P

< 0.5 in order to guarantee that a feasible

solution may be found. From (4), the ﬁtness of an in-

dividual point is a value between 0 and 1, and the best

individual point in a population has the lowest ﬁtness

value.

2.3 Feasibility and Dominance Rules

(Deb, 2000) proposed another constraints handling

technique for population-based solution methods. It

is based on a set of rules that use feasibility and domi-

nance (FD) principles, as follows. First, the constraint

violation ζ is calculated for all the individuals in a

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

population. Then the objective function f is evalu-

ated only for feasible individuals. Two individuals are

compared at a time, and the following criteria are al-

ways enforced:

1. any feasible point is preferred to any infeasible

point;

2. between two feasible points, one having better ob-

jective function is preferred;

3. between two infeasible points, one having smaller

constraint violation is preferred.

In this case, the ﬁtness of each individual point x

calculated as follows

) =



f(x

) if x

is feasible

max, f

+ ζ(x

) otherwise,

(5)

where f

max, f

is the objective function of the worst fea-

sible solution in the population. When all individuals

are infeasible then its value is set to zero. This ﬁtness

function is used to choose the best individual point in

a population.

3 MODIFIED CONSTRAINED

DIFFERENTIAL EVOLUTION

The population-based differential evolution algorithm

has become popular and has been used in many prac-

tical cases, mainly because it has demonstrated good

convergence properties and is easy to understand. DE

is a ﬂoating point encoding that creates a new candi-

date point by adding the weighted difference between

two individuals to a third one in the population. This

operation is called mutation. The mutant point’s com-

ponents are then mixed with the components of target

point to yield the trial point. This mixing of com-

ponents is referred to as crossover. In selection, a

trial point replaces a target point for the next genera-

tion only if it is considered an equal or better point.

In unconstrained optimization, the selection opera-

tion relies on the objective function. DE has three

parameters: ampliﬁcation factor of differential varia-

tion F, crossover control parameter Cr, and popula-

tion size N.

It is not an easy task to set the appropriate param-

eters since these depend on the nature and size of the

optimization problems. Hence, self-adaptive control

parameters ought to be implemented. (Brest et al.,

2006) proposed self-adaptive control parameters for

DE when solving global optimization problems with

simple bounds. In most original DE, three points are

chosen randomly for mutation and the base point is

then chosen at random within the three. This has an

exploratory effect but it slows down the convergence

of DE. (Kaelo and Ali, 2006) proposed a modiﬁed

mutation for differential evolution.

The herein presented modiﬁed constrained differ-

ential evolution algorithm - m-CDE - for constrained

nonlinear programming problems (1) includes:

1. the self-adaptive control parameters F and Cr, as

proposed by (Brest et al., 2006);

2. a modiﬁed mutation that mixes the modiﬁcation

proposed by (Kaelo and Ali, 2006) with the cycli-

cal use of the overall best point as base point;

3. the inversion operation;

4. a modiﬁed selection that is based on the ﬁtness of

individuals;

5. the elitism.

The modiﬁcation in mutation allows m-CDE to en-

hance the local search around the overall best point.

In modiﬁed selection of m-CDE, we introduce and

test the three different techniques described so far for

calculating the ﬁtness function of individuals that are

capable to handle the constrained problems (1). The

modiﬁed constrained differential evolution is outlined

below.

The target point of m-CDE, at iteration/generation

z, is deﬁned by x

i,z

= (x

i1,z

, x

i2,z

, . . . , x

in,z

), where n

is the number of variables of the optimization prob-

lem and i = 1, 2, . . . , N. N does not change during the

optimization process. The initial population is cho-

sen randomly and should cover the entire component

spaces.

Self-adaptive Control Parameters. In m-CDE, we

use self-adaptive control parameters for F andCr pro-

posed by (Brest et al., 2006) by generating a different

set (F

,Cr

) for each point x

in the population. The

new control parameters for next generation F

i,z+1

and

i,z+1

are calculated by

i,z+1



+ λ

× F

if λ

< τ

i,z

otherwise

i,z+1



if λ

< τ

i,z

otherwise,

(6)

where λ

∼ U[0, 1], k = 1, . . . , 4 and τ

= τ

= 0.1 rep-

resent the probabilities to adjust parameters F

and

, respectively. F

= 0.1 and F

= 0.9, so the new

i,z+1

takes a value from [0.1, 1.0] in a random man-

ner. The new Cr

i,z+1

takes a value from [0, 1]. F

i,z+1

and Cr

i,z+1

are obtained before the mutation is per-

formed. So, they inﬂuence the mutation, crossover

and selection operations of the new point x

i,z+1

Modiﬁed Mutation. In m-CDE, this is a mixture of

two different types of mutation operations. We use

the mutation proposed in (Kaelo and Ali, 2006). After

choosing three points randomly, the best point among

GLOBAL COMPETITIVE RANKING FOR CONSTRAINTS HANDLING WITH MODIFIED DIFFERENTIAL

EVOLUTION

three based on the ﬁtness function is selected for the

base point and the remaining two points are used as

differential variation, i.e., for each target point x

i,z

, a

mutant point is created according to

i,z+1

= x

+ F

i,z+1

− x

), (7)

where r

, r

are randomly chosen from the set

{1, 2, . . . , N}, mutually different and different from

the running index i and r

is the index with the best ﬁt-

ness (among the three points). This modiﬁcation has

a local effect when the points of the population form

a cluster around the global minimizer.

Furthermore, at every B generations, the best point

found so far is used as the base point and two ran-

domly chosen points are used as differential variation,

i.e.,

i,z+1

= x

best

+ F

i,z+1

− x

). (8)

These mixed modiﬁcations allow m-CDE to maintain

its exploratory feature and at the same time to exploit

the region around the best individual of the population

expediting the convergence.

Crossover. In order to increase the diversity of the

mutant points’ components, crossover is introduced.

To this end, the crossover point u

i,z+1

is formed,

where

ij,z+1



ij,z+1

if (r

≤ Cr

i,z+1

) or j = s

ij,z

if (r

> Cr

i,z+1

) and j 6= s

(9)

In (9), r

∼ U[0, 1] performs the mixing of jth com-

ponent of points, s

is randomly chosen from the set

{1, 2, . . . , n} and ensures that u

i,z+1

gets at least one

component from v

i,z+1

Inversion. Since in m-CDE, a point has n-

dimensional real components, inversion (Holland,

1975) can easily be applicable. With the inversion

probability (p

inv

∈ [0, 1]), two positions are chosen on

the point u

, the point is cut at those positions, and

the cut segment is reversed and reinserted back into

the point to create the trial point u

′

. In practice, m-

CDE with the inversion has been shown to give better

results than those obtained without the inversion. An

illustrative example of inversion is shown in Figure 1.

Bounds Check. When generating the mutant point

and when the inversion operation is performed, some

components can be generated outside the bound con-

straints. So, in m-CDE after inversion the bounds of

each component should be checked with the follow-

ing projection of bounds:

′

ij,z+1







if u

′

ij,z+1

< l

if u

′

ij,z+1

> u

′

ij,z+1

otherwise.

(10)

Modiﬁed Selection. In original DE, the target and the

trial points are compared based on their correspond-

ing objective function to decide which point becomes

a member of the next generation, that is if the trial

point’s objective function is less than or equal to the

that of target point, then the trial point will be the tar-

get point for the next generation.

In this paper, for constrained nonlinear program-

ming problems, we propose a modiﬁed selection

based on one of the ﬁtness functions of individuals

discussed so far. When using the global competitive

ranking technique, all the target points at generation

z and trial points at generation z + 1 are ranked to-

gether and their corresponding ﬁtness Φ

are calcu-

lated. Then the modiﬁed selection is performed, i.e.,

the trial and the target points are compared to decide

which will be the new target points for next genera-

tion based on their calculated ﬁtness by the following

way

i,z+1



′

i,z+1

if Φ

′

i,z+1

) ≤ Φ

i,z

)

i,z

otherwise.

(11)

After performing selection in m-CDE, the best point

is chosen in the current generation based on the lowest

ﬁtness function of the target points.

A similar procedure is performed when the

stochastic ranking technique is implemented.

On the other hand, when using the feasibility and

dominance principles, the trial and the target points

are compared based on the three feasibility and dom-

inance rules to decide which will be the new target

points for next generation. After performing selec-

tion, the ﬁtness function Φ

for all the target points

are calculated, and the best point based on the lowest

ﬁtness function in the current generation is chosen.

We remark that this point is the overall best point in

the entire generations so far.

Elitism. The elitism is also performed to keep the

best point found so far in the entire generations. The

elitism aims at preserving in the entire generations the

individual point that, with the constraint violation 0

or smaller than others, has the smaller objective func-

tion. This is only required when either the stochas-

tic ranking or the global competitive ranking is used

to calculate ﬁtness of individuals. We remark that in

these two techniques, ﬁtness values of individuals are

calculated at every generation based on their corre-

sponding ranks. Thus, the ﬁtness of best individual

point (based on the objective function and the con-

straint violation) may not be the lowest one.

Termination Criterion. Let G

max

be the maximum

number of generations. If f

max,z

and f

min,z

are the

objective function values of the points that have the

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

| |

i,z+1

i1,z+1

i2,z+1

i3,z+1

i4,z+1

i5,z+1

i6,z+1

i7,z+1

i8,z+1

⇓

| |

′

i,z+1

i1,z+1

i2,z+1

i6,z+1

i5,z+1

i4,z+1

i3,z+1

i7,z+1

i8,z+1

Figure 1: Inversion used in m-CDE.

highest and the lowest ﬁtness function values respec-

tively, attained at generation z, then our m-CDE algo-

rithm terminates if (z > G

max

or ( f

max,z

− f

min,z

) ≤ η),

for a small positive number η.

3.1 The m-CDE Algorithm

The algorithm of the herein proposed modiﬁed con-

strained differential evolution for constrained global

optimization is described in the following:

Step 1. Set the values of parameters N, G

max

, B, P

, F

, τ

, p

inv

, and η.

Step 2. Set z = 1. Initialize the population x

, F

and

Step 3. Calculate the ﬁtness function Φ(x

Step 4. Choose f

max,z

and f

min,z

from target points.

For stochastic and global competitive ranking,

perform the elitism to choose f

best

and x

best

Otherwise for feasibility and dominance rules

technique, set f

best

= f

min,z

and x

best

= x

min,z

Step 5. If the termination criterion is met stop. Oth-

erwise set z = z+ 1.

Step 6. Compute the control parameters F

and Cr

Step 7. Compute the mutant point v

If MOD(z, B) = 0; use (8). Otherwise use (7).

Step 8. Perform the crossover to make point u

Step 9. If a random number γ ∼ U[0, 1] ≤ p

inv

, per-

form inversion to make trial point u

′

Step 10. Check the bounds of the trial points.

Step 11. Calculate the ﬁtness function Φ(x

) for all

the target and trial points.

Step 12. Perform the modiﬁed selection discussed

above.

Step 13. Go to step 4.

4 EXPERIMENTAL RESULTS

We code m-CDE in C with AMPL (Fourer et al.,

1993) interfacing and compile with Microsoft Visual

Studio 9.0 compiler in a PC having 2.5 GHz In-

tel Core 2 Duo processor and 4 GB RAM. We set

the value of parameters N = min(100, 10n), B = 10,

= 0.45, p

inv

= 0.05 and η = 10

−6

. We con-

sider 13 benchmark constrained nonlinear program-

ming problems (Runarsson and Yao, 2000). The char-

acteristics of these test problems are outlined in Ta-

ble 1. We model these problems in AMPL mod-

eling systems. To access these models please visit

http://www.norg.uminho.pt/emgpf/problems.htm.

Table 1: Characteristics of the test problems.

Prob.

Type of

opt

Var. Constraints

f n m

g01 quadratic -15.0000 13 9 0 9

g02 general -0.8036 20 2 0 2

g03 polynomial -1.0005 10 0 1 1

g04 quadratic -30665.5387 5 6 0 6

g05 cubic 5126.4967 4 2 3 5

g06 cubic -6961.8139 2 2 0 2

g07 quadratic 24.3062 10 8 0 8

g08 general -0.0958 2 2 0 2

g09 general 680.6301 7 4 0 4

g10 linear 7049.2480 8 6 0 6

g11 quadratic 0.7499 2 0 1 1

g12 quadratic -1.0000 3 1 0 1

g13 general 0.0539 5 0 3 3

The following average measure of constraint vio-

lation of an individual point x is used:

ζ(x) =

∑

k=1

max{0, g

(x)} +

∑

l=1

(x)|

where m = m

+ m

is the total number of constraints.

In this paper, we consider an individual point as a fea-

sible one if ζ(x) ≤ δ, where δ is a very small positive

number. Here we set δ = 10

−8

We test m-CDE with the previously described

constraints handling techniques and compare their

performance using performance proﬁles (Dolan and

Mor´e, 2002). A comparison with other solution meth-

ods found in literature is also included.

4.1 Performance Proﬁles

For a fair comparison we run all the variants of m-

CDE in comparison for 30 times and reported the re-

sults. We used different G

max

for the 13 problems,

GLOBAL COMPETITIVE RANKING FOR CONSTRAINTS HANDLING WITH MODIFIED DIFFERENTIAL

EVOLUTION

but used the same value for all the variants. We

used the performance proﬁles proposed by (Dolan

and Mor´e, 2002) to evaluate and compare the per-

formance of the variants. Performance proﬁles are

the graphical representation of performance of differ-

ent solvers/variants/codes for a set of test problems

on the basis of their performance ratio. The perfor-

mance proﬁle plot represents the cumulative distribu-

tion function of the performance ratio based on an ap-

propriate performance metric. The authors proposed

the computing time required to solve a problem as a

performance metric for different solvers (Dolan and

Mor´e, 2002), but other performance metrics could be

used.

In our comparative study, let P be the set of all

problems and S be the set of all variants of m-CDE.

Also let m

(p,s)

be the performance metric found by

variant s ∈ S on problem p ∈ P that measures the rel-

ative improvement of the objective function values, a

scaled distance to the optimal objectivefunction value

opt

(Ali et al., 2005), deﬁned by

(p,s)

− f

opt

− f

opt

. (12)

In (12), f

(p,s)

is the average/best of objective func-

tion values found by variant s on problem p after 30

runs and f

is the worst objective function value of

problem p after 30 runs among all variants. Since

a zero value of min{m

(p,s)

: s ∈ S } may appear, the

performance ratios used in our comparative study are

deﬁned (Vaz and Vicente, 2007) by

(p,s)







1+ m

(p,s)

− q if q ≤ 10

−5

(p,s)

otherwise,

where q = min{m

(p,s)

: s ∈ S }. Then ρ

(τ), the frac-

tion of problems for which the variant s has a perfor-

mance ratio r

(p,s)

within a factor τ, is given by

(τ) =

where n

is the number of problems in P with r

(p,s)

≤

τ and n

is the total number of problems in P . ρ

(τ)

is the probability (for variant s ∈ S ) that the perfor-

mance ratio r

(p,s)

is within a factor τ ∈ R of the best

possible ratio.

At ﬁrst, we tested our program with the stochastic

ranking and the global competitive rankings for ob-

taining solutions of the above 13 constrained prob-

lems to check which technique is better than the oth-

ers in comparison. The variants were stochastic rank-

ing, global competitive ranking 1, with a ﬁxed value

of P

, and global competitive ranking 2, with a ran-

dom value of P

in the interval (0, 0.45). Figure 2

shows the proﬁles of the performance metric f

avg

, the

average of the best solutions obtained over the 30

runs. If we are only interested in knowing which vari-

1.0 1.1 1.2 1.3 1.4 1.5 1.6

0.0

0.2

0.4

0.6

0.8

1.0

ρ(τ)

Performance profile based on f_avg after 30 runs

Stochastic Ranking

Global Competitive Ranking 1

Global Competitive Ranking 2

Figure 2: Performance proﬁle of average objective function

value for different variants.

ant is the most efﬁcient, in the sense that it reaches the

best solutions mostly, we compare the values of ρ for

τ = 1, and ﬁnd the highest value which is the prob-

ability that the variant will win over the remaining

ones. However, to know the robustness of variants,

we compare the values of ρ(τ) for large values of τ. It

means that variants with the largest probabilities ρ(τ)

for large values of τ are the most robust ones. In this

ﬁgure it is shown that variant global competitive rank-

ing 1 (for ﬁxed value of P

) wins over the other two

variants of m-CDE.

1.0 1.1 1.2 1.3 1.4 1.5 1.6

0.0

0.2

0.4

0.6

0.8

1.0

Performance profile based on f_avg after 30 runs

ρ(τ)

Feasibility and Dominance Rules

Stochastic Ranking

Global Competitive Ranking 1

Global Competitive Ranking 2

Figure 3: Performance proﬁle of average objective function

value using ﬁtness (5).

In the second experiment, we tested the follow-

ing four variants of m-CDE: feasibility and domi-

nance rules, stochastic ranking, global competitive

ranking 1 and global competitive ranking 2 (as pre-

viously deﬁned). To be able to fairly compare with

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

Table 2: Experimental results of the 13 test problems by m-CDE1.

Prob. best worst avg. median std. dev. avg. vio. G

avg

g01 -15.0000 -15.0000 -15.0000 -15.0000 1.16E-06 0.00E+00 196

g02 -0.8036 -0.7926 -0.8007 -0.8036 4.95E-03 0.00E+00 1408

g03 -1.0000 -0.9998 -1.0000 -1.0000 3.90E-05 5.88E-09 1750

g04 -30665.5387 -30665.5387 -30665.5387 -30665.5387 2.38E-05 8.66E-09 1091

g05 5126.4978 5126.4986 5126.4979 5126.4978 1.83E-04 9.23E-09 1750

g06 -6961.8161 -6624.1610 -6950.5609 -6961.8161 6.16E+01 9.67E-09 1750

g07 24.2316 24.2319 24.2317 24.2316 7.44E-05 1.60E-09 1750

g08 -0.0958 -0.0958 -0.0958 -0.0958 2.71E-06 0.00E+00 42

g09 680.6301 680.6301 680.6301 680.6301 1.38E-06 0.00E+00 727

g10 7049.2533 7076.5860 7053.3441 7050.6330 6.99E+00 1.13E-09 1750

g11 0.7500 0.7670 0.7506 0.7500 3.11E-03 6.36E-09 158

g12 -1.0000 -1.0000 -1.0000 -1.0000 2.33E-06 0.00E+00 30

g13 0.0539 0.0539 0.0539 0.0539 3.53E-17 9.96E-09 1750

the variant feasibility and dominance rules, the other

variants were fairly modiﬁed. After the modiﬁed se-

lection step of the algorithm, the ﬁtness function is re-

calculated now using (5) so that the best and the worst

target points of the populationare identiﬁed according

to the objective function and constraint violation val-

ues. Figure 3 shows the proﬁles of the performance

metric f

avg

for the four variants. In this ﬁgure it is

shown that variant global competitive ranking 1 here

also wins over the other three variants of m-CDE in

comparison.

From the above discussion it is clear that in both

cases the variant of m-CDE based on global competi-

tive ranking with ﬁxed value of P

gave better perfor-

mance over the other ones. We also tested our m-CDE

with global dense ranking technique but the obtained

results were signiﬁcantly worse than those obtained

by global competitive ranking technique.

4.2 Comparing with other Methods

We also compare our m-CDE (based on the global

competitive ranking with ﬁxed value of P

) with the

stochastic ranking presented in (Runarsson and Yao,

2000) and the global competitive ranking presented

in (Runarsson and Yao, 2003). The authors proposed

these techniques based on a (30, 200) evolution strat-

egy. Here, the stochastic ranking technique is de-

noted by SRES and the global competitive ranking

technique is GRES. An adaptive penalty scheme for

constraint handling with dynamic use of variants of

differential evolution (DUVDE) (Silva et al., 2011)

is also used in the comparison. We set the same

value of maximum number of generations according

to (Runarsson and Yao, 2000; Runarsson and Yao,

2003). We set G

max

= 1750 for all the problems ex-

cept problem g12 for all the runs. G

max

= 175 was set

for g12. Here, we aim to get a solution within 0.001%

of the optimal solution f

opt

Firstly, we run all the 13 problems 30 times with

the global competitive ranking technique using modi-

ﬁed constrained differential evolution. Here we iden-

tiﬁed f

min

and f

max

of the objective function based on

the current ﬁtness function Φ

at generation z. Then

elitism was performed to determine the f

best

. Here-

after we will denote this version by m-CDE1. We re-

port ‘best’, ‘worst’, ‘avg.’ (average), ‘median’ and

‘std. dev.’ (standard deviation) of the best objective

functions and ‘avg. vio.’ (average constraint viola-

tion) among all the 30 runs. The average number of

generations ‘G

avg

’ attained among the 30 runs is also

reported. These results are shown in Table 2.

Secondly, we run again all the 13 problems 30

times with the global competitive ranking technique

using modiﬁed constrained differential evolution, but

this time, both f

min

and f

max

are identiﬁed after recal-

culating the ﬁtness of all individuals in the population

using function Φ

, at each generation z. Then we

set f

best

= f

min

. Hereafter we will denote this version

by m-CDE2. We remark that m-CDE2 is the vari-

ant global competitive ranking 1 of the second expe-

rience reported in the previous subsection. The re-

sults are shown in Table 3. It is shown from Tables 2

and 3 that in almost all problems with respect to the

different objective function measures the results ob-

tained by m-CDE1 are relatively better than those of

m-CDE2.

Finally, to compare the results obtained by m-

CDE1, shown in Table 2, with DUVDE, SRES and

GRES, results of ‘best’, ‘avg.’ and ‘std. dev.’ of the

best objective functions among 30 runs are shown in

Table 4. These results are taken from the cited papers

(Silva et al., 2011), (Runarsson and Yao, 2000) and

(Runarsson and Yao, 2003) respectively. In m-CDE1,

we use the population size N dependent on the dimen-

sion of test problem and use the maximum number of

generations according to (Runarsson and Yao, 2000;

Runarsson and Yao, 2003), although in DUVDE the

GLOBAL COMPETITIVE RANKING FOR CONSTRAINTS HANDLING WITH MODIFIED DIFFERENTIAL

EVOLUTION

Table 3: Experimental results of the 13 test problems by m-CDE2.

Prob. best worst avg. median std. dev. avg. vio. G

avg

g01 -15.0000 -15.0000 -15.0000 -15.0000 1.22E-06 0.00E+00 196

g02 -0.8036 -0.7926 -0.8004 -0.8036 5.03E-03 2.07E-10 1438

g03 -1.0000 -0.9966 -0.9997 -0.9999 6.53E-04 4.54E-09 1750

g04 -30665.5387 -30665.5387 -30665.5387 -30665.5387 2.47E-05 8.24E-09 1039

g05 5126.4978 5126.4978 5126.4978 5126.4978 1.85E-12 1.00E-08 1750

g06 -6961.8161 -6615.7067 -6950.2790 -6961.8161 6.32E+01 9.94E-09 1695

g07 24.2316 24.2325 24.2318 24.2317 1.98E-04 1.75E-09 1750

g08 -0.0958 -0.0958 -0.0958 -0.0958 2.86E-06 0.00E+00 30

g09 680.6301 680.6301 680.6301 680.6301 1.10E-06 0.00E+00 730

g10 7049.2604 7250.9916 7100.8072 7051.2523 7.57E+01 1.38E-09 1750

g11 0.7500 1.0000 0.9353 1.0000 1.09E-01 1.74E-09 1515

g12 -1.0000 -1.0000 -1.0000 -1.0000 2.64E-06 0.00E+00 20

g13 0.0539 0.0539 0.0539 0.0539 1.14E-05 9.70E-09 1750

Table 4: Results from DUVDE, SRES and GRES.

Prob.

DUVDE SRES GRES

best avg. std. dev. best avg std. dev. best avg. std. dev.

g01 -15.0000 -12.5000 2.37E+00 -15.0000 -15.0000 0.00E+00 -15.0000 – 0.00E+00

g02 -0.8036 -0.7688 3.57E-02 -0.8035 -0.7820 2.00E-02 -0.8035 – 1.70E-02

g03 -1.0000 -0.2015 3.45E-01 -1.0000 -1.0000 1.90E-04 -1.0000 – 2.60E-05

g04 -30665.5000 -30665.5000 0.00E+00 -30665.5390 -30665.5390 2.00E-05 -30665.5390 – 5.40E-01

g05 5126.4965 5126.4965 0.00E+00 5126.4970 5128.8810 3.50E+00 5126.4970 – 1.10E+00

g06 -6961.8000 -6961.8000 0.00E+00 -6961.8140 -6875.9400 1.60E+02 -6943.5600 – 2.90E+02

g07 24.3060 30.4040 2.16E+01 24.3070 24.3740 6.60E-02 24.3080 – 1.10E-01

g08 -0.0958 -0.0958 0.00E+00 -0.0958 -0.0958 2.60E-17 -0.0958 – 2.60E-17

g09 680.6300 680.6300 3.00E-05 680.6300 680.6560 3.40E-02 680.6310 – 5.80E-02

g10 7049.2500 7351.1700 5.26E+02 7054.3160 7559.1920 5.30E+02 * – *

g11 0.7500 0.9875 5.59E-02 0.7500 0.7500 8.00E-05 0.7500 – 7.20E-05

g12 † † † -1.0000 -1.0000 0.00E+00 -1.0000 – 0.00E+00

g13 † † † 0.0539 0.0675 3.10E-02 0.0539 – 1.30E-04

(†) not considered; (–) not available; (*) not solved

authors used the population size 50 and the maximum

number of generations 3684 for all the tested prob-

lems. Problems g12 and g13 were not tested with

DUVDE (Silva et al., 2011). From Tables 2 and 4

we may conclude that for most of the problems, and

with respect to all measures of comparison, m-CDE1

performs rather well when compared with DUVDE,

SRES and GRES.

From the above discussion it is clear that the

proposed modiﬁed constrained differential evolution

based on the global competitive ranking technique to

handle constraints, is rather effective when converg-

ing to constrained global solutions.

5 CONCLUSIONS

In this paper, to make the DE methodology more ef-

ﬁcient to handle the constraints in constrained global

optimization problems, a modiﬁed constrained differ-

ential evolution algorithm has been proposed. The

modiﬁcations focus on self-adaptive control param-

eters, modiﬁed mutation, modiﬁed selection and

elitism. Inversion operation has also been imple-

mented in the proposed m-CDE.

To conclude, we emphasize the modiﬁcations that

mostly inﬂuence the efﬁciency of the algorithm. The

mixed modiﬁed mutation, in the m-CDE algorithm,

aims at exploring both the entire search space (when

using the mutation proposed in (Kaelo and Ali, 2006))

and the neighbourhood of the best point found so far

(when using the best point as the base point cycli-

cally). The modiﬁed selection, to handle the con-

straints effectively, uses a ﬁtness function based on

the global competitiveranking technique. In this tech-

nique, ﬁtness of all target and trial points are cal-

culated all together after ranking them based on the

objective function and the constraint violation sepa-

rately, for competing in modiﬁed selection to decide

which points win for next generation population. This

technique seems to have stricken the right balance be-

tween the objective function and the constraint vio-

lation for obtaining a global solution while satisfying

the constraints.

To test the effectiveness of the new m-CDE, 13

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

benchmark constrained nonlinear programming prob-

lems have been considered. These problems have also

been solved with the stochastic ranking and the feasi-

bility and dominance rules techniques and compari-

son has been made based on their performance pro-

ﬁles. It is shown that m-CDE with the global com-

petitive ranking, based on a ﬁxed value of P

, is rel-

atively better than the other two techniques. A com-

parison has also been made with other results from the

literature: the adaptive penalty-based differential evo-

lution, the stochastic ranking based on an evolution

strategy, and the global competitive ranking based on

an evolution strategy. It is shown that m-CDE is rather

competitive when compared with the other solution

methods. Future developments will focus on the ex-

tension of the m-CDE to problems with mixed integer

variables.

ACKNOWLEDGEMENTS

This work is supported by FCT (Fundac¸˜ao para a

Ciˆencia e a Tecnologia) and Ciˆencia 2007, Portugal.

We thank two anonymous referees for their valuable

comments to improve this paper.

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