A Scalarized Augmented Lagrangian Algorithm (SCAL) for
Multi-objective Optimization Constrained Problems
Lino Costa
1,2
, Isabel Esp
´
ırito Santo
1,2
and Pedro Oliveira
3,4
1
Algoritmi R&D Center, University of Minho, Portugal
2
Department of Production and Systems Engineering, University of Minho, Portugal
3
Instituto de Ci
ˆ
encias Biom
´
edicas Abel Salazar, Universidade do Porto, Portugal
4
ISPUP-EPIUnit, Universidade do Porto, Rua das Taipas, n
◦
135, 4050-600 Porto, Portugal
Keywords:
Multi-objective Constrained Optimization, Augmented Weighted Tchebycheff, Pattern Search, Augmented
Lagrangian.
Abstract:
In this paper, a methodology to solve constrained multi-objective problems is presented, using an Augmented
Lagrangian technique to deal with the constraints and the Augmented Weighted Tchebycheff method to tackle
the multi-objective problem and find the Pareto Frontier. We present the algorithm, as well as some preliminary
results that seem very promising when compared to previous state-of-the- art work. As far as we know, the
idea of incorporating an Augmented Lagrangian in multi-objective optimization is rarely used so, the obtained
results are very encouraging to pursuit further in this line of investigation, namely with the tuning of the
Augmented Lagrangian parameters as well as testing other algorithms to solve the subproblems or to handle
the multi-objective problems. It is also our intention to investigate the resolution of problems with three or
more objectives.
1 INTRODUCTION
A multi-objective optimization (MO) problem with m
objectives and n decision variables, without loss of
generality, can be mathematically formulated as fol-
lows:
minimize: f (x) = ( f
1
(x), f
2
(x), . . . , f
m
(x))
T
subject to:
c
i
(x) = 0, i = 1, . . . , q
g
j
(x) ≥ 0, j = 1, . . . , p
x ∈ Ω
(1)
where x is the decision vector, Ω ⊆ R
n
is the feasible
decision space, f (x) is the objective vector defined
in the objective space R
m
, c(x) = 0 are the equality
constraints and g(x) ≥ 0 are the inequality constraints.
When several objectives are optimized at the same
time, the search space becomes partially ordered. In
such scenario, solutions are compared on the basis of
the Pareto dominance. For two solutions a and b from
Ω, a solution a is said to dominate a solution b (de-
noted by a ≺ b) if:
∀i ∈ {1, . . . , m} : f
i
(a) ≤ f
i
(b)∧
∃ j ∈ {1, . . . , m} : f
j
(a) < f
j
(b).
(2)
Since solutions are compared against different ob-
jectives, there is no longer a single optimal solution
but a set of optimal solutions, generally known as
the Pareto optimal set. This set contains equally im-
portant solutions representing different trade-offs be-
tween the given objectives and can be defined as:
P S = {x ∈ Ω | @y ∈ Ω : y ≺ x}. (3)
Approximating the Pareto optimal set is the main
goal in multi-objective optimization.
Constraint handling in multi-objective optimiza-
tion is still a very challenging research endeavor.
Constraints are present in every real world problem
and, so far, there is no superior approach to the han-
dling of constraints. The inclusion of constraints, in
particular, equality constraints, further complexifies
multi-objective optimization since its inclusion can
greatly transform the Pareto Frontier and, thus, make
its approximation much more difficult.
In this paper, a Scalarized Augmented Lagrangian
Algorithm (SCAL) for constrained multi-objective
optimization problems is presented and tested in sev-
eral constrained problems.
Costa, L., Santo, I. and Oliveira, P.
A Scalarized Augmented Lagrangian Algorithm (SCAL) for Multi-objective Optimization Constrained Problems.
DOI: 10.5220/0006720603350340
In Proceedings of the 7th International Conference on Operations Research and Enterprise Systems (ICORES 2018), pages 335-340
ISBN: 978-989-758-285-1
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
335

2 AUGMENTED WEIGHTED
TCHEBYCHEFF METHODS
The main goal of the MO optimization is to obtain the
set of Pareto-optimal solutions that correspond to dif-
ferent trade-offs between objectives. In this study, a
scalarization method is used to obtain an approxima-
tion to the Pareto-optimal set (Miettinen, 1999).
The Augmented Weighted Tchebycheff method,
proposed by Steuer and Choo (Steuer and Choo,
1983), has the advantage that it can converge to non-
extreme final solutions and may also be applicable
to nonlinear and nonconvex multi-objective optimiza-
tion problems (Steuer and Choo, 1983). The Aug-
mented Weighted Tchebycheff method can be formu-
lated as:
minmax[w
i
| f
i
(x) − z
?
i
|] +ρ
m
∑
i=1
| f
i
(x) − z
i
| (4)
where w
i
are the weighting coefficients for objective
i, z
?
i
are the components of a reference point, and ρ is
a small positive value (Dchert et al., 2012). The solu-
tions of the optimization problem defined by Eq. 4 are
Pareto-optimal solutions. Different combinations al-
low computing approximations to the Pareto-optimal
set using a single objective optimization method. An
approximation to the ideal vector can be used as refer-
ence point z
?
= (z
?
1
, . . . , z
?
m
)
T
= (min f
1
, . . . , min f
m
)
T
.
3 AUGMENTED LAGRANGIAN
TECHNIQUE USING THE
HOOKE AND JEEVES
PATTERN SEARCH METHOD
The Augmented Lagrangian technique herein pre-
sented solves a sequence of simple subproblems
where the objective function penalizes all or some of
the constraint violation. This objective function is an
Augmented Lagrangian that depends on a penalty pa-
rameter and on the multiplier vectors and is based in
the ideas presented in (Bertsekas, 1999; Conn et al.,
1991; Lewis and Torczon, 2002):
Φ(x;λ, δ, µ) = f (x) + λ
T
c(x) +
1
2µ
kc(x)k
2
(5)
+
µ
2





δ +
g(x)
µ

+




2
− kδk
2
!
where µ is a positive penalty parameter, λ =
(λ
1
, . . . , λ
m
)
T
, δ = (δ
1
, . . . , δ
p
)
T
are the Lagrange
multiplier vectors associated with the equality and in-
equality constraints, respectively. Function Φ aims
to penalize solutions that violate the equality and in-
equality constraints, not including the simple bounds
l ≤ x ≤ u. To force feasibility, the inner iterative pro-
cess must return an approximate solution that satis-
fies the bound constraints. While using the Hooke
and Jeeves version of the pattern search (Hooke and
Jeeves, 1961), any computed solution x that does not
satisfy the bounds is projected onto the set Ω compo-
nent by component (for all i, . . . , n) as follows:
x
i
=



l
i
if x
i
< l
i
x
i
if l
i
≤ x
i
≤ u
i
u
i
if x
i
> u
i
. (6)
The corresponding subproblem is then formulated as:
minimize
x∈Ω
Φ(x;λ
j
, δ
j
, µ
j
) (7)
where, for each set of fixed λ
j
, δ
j
and µ
j
, the solu-
tion of subproblem (7) provides an approximation x
j
to the problem formulated in Eq. (4), where the index
j is the iteration counter of the outer iterative process.
We refer to (Bertsekas, 1999) for details. In practice,
common safeguarded schemes maintain the sequence
of penalty parameters far away from zero so that solv-
ing subproblem (7) is an easy task.
To evaluate the equality and inequality constraint
violation, and the complementarity, the following er-
ror function is used:
E(x, δ) = max

kc(x)k
∞
1 +kxk
,
k[g(x)]
+
k
∞
1 +kδk
,
max
i
δ
i
|g
i
(x)|
1 +kδk

.
(8)
The Lagrange multipliers λ
j
and δ
j
are estimated in
this iterative process using the first-order updating
formulae
¯
λ
j+1
i
= λ
j
i
+
c
i
(x
j
)
µ
j
, i = 1, . . . , m (9)
and
¯
δ
j+1
i
= max

0, δ
j
i
+
g
i
(x
j
)
µ
j

, i = 1, . . . , p (10)
where: for all j ∈ N, and for i = 1, . . . , m and l =
1, . . . , p, λ
j+1
i
is the projection of
¯
λ
j+1
i
on the inter-
val [λ
min
, λ
max
] and δ
j+1
i
is the projection of
¯
δ
j+1
i
on
the interval [0, δ
max
], where −∞ < λ
min
≤ λ
max
< ∞
and 0 ≤ δ
max
< ∞. After the new approximation x
j
has been computed, the Lagrange multiplier vector δ
associated with the inequality constraints is updated,
in all iterations, since δ
j+1
is required in the error
function (8) to measure constraint violation and com-
plementarity. We note that the Lagrange multipliers
λ
i
, i = 1, . . . , m are updated only when feasibility and
complementarity are at a satisfactory level, herein de-
fined by the condition
E(x
j
, δ
j+1
) ≤ η
j
(11)
ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems
336

CF1
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(a) CF1
CF2
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
(b) CF2
CF3
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(c) CF3
CF4
0
0.5
1
1.5
2
2.5
3
(d) CF4
CF5
0.1
0.2
0.3
0.4
0.5
0.6
(e) CF5
CF6
0.03
0.035
0.04
0.045
0.05
0.055
(f) CF6
CF7
0.1
0.2
0.3
0.4
0.5
0.6
(g) CF7
BNH
1.35
1.4
1.45
1.5
1.55
1.6
1.65
(h) BHN
CONSTR
-1
0
1
2
3
4
5
6
(i) CONSTR
OSY
66
68
70
72
74
76
(j) OSY
SRN
5
10
15
20
(k) SRN
TNK
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(l) TNK
Figure 1: Boxplots of IGD on test problems.
for a positive tolerance η
j
. It is required that {η
j
}
defines a decreasing sequence of positive values con-
verging to zero, as j → ∞. This is easily achieved by
η
j+1
= πη
j
for 0 < π < 1.
We consider that an iteration j failed to provide
an approximation x
j
with an appropriate level of fea-
sibility and complementarity if condition (11) does
not hold. In this case, the penalty parameter is de-
creased using µ
j+1
= γµ
j
where 0 < γ < 1. When
condition (11) holds, then the iteration is considered
satisfactory. This condition says that the iterate x
j
is
feasible and the complementarity condition is satis-
fied within some tolerance η
j
and, consequently, the
algorithm maintains the penalty parameter value. We
remark that when (11) fails to hold infinitely many
times, the sequence of penalty parameters tends to
zero. To be able to define an algorithm where the se-
quence {µ
j
} does not reach zero, the following update
is used instead:
µ
j+1
= max{µ
min
, γµ
j
}, (12)
A Scalarized Augmented Lagrangian Algorithm (SCAL) for Multi-objective Optimization Constrained Problems
337

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
f
1
(x)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f
2
(x)
(a) CF1
0 0.5 1 1.5 2 2.5
f
1
(x)
0
0.2
0.4
0.6
0.8
1
1.2
f
2
(x)
CF2
(b) CF2
0 0.2 0.4 0.6 0.8 1 1.2
f
1
(x)
0
0.5
1
1.5
2
2.5
f
2
(x)
CF3
(c) CF3
0 1 2 3 4 5 6 7 8
f
1
(x)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
f
2
(x)
CF4
(d) CF4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
f
1
(x)
0
0.5
1
1.5
2
2.5
f
2
(x)
CF5
(e) CF5
0 0.2 0.4 0.6 0.8 1 1.2
f
1
(x)
0
0.5
1
1.5
2
2.5
f
2
(x)
CF6
(f) CF6
0 0.2 0.4 0.6 0.8 1 1.2
f
1
(x)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
f
2
(x)
CF7
(g) CF7
0 20 40 60 80 100 120 140
f
1
(x)
0
5
10
15
20
25
30
35
40
45
50
f
2
(x)
BNH
(h) BNH
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
f
1
(x)
1
2
3
4
5
6
7
8
9
f
2
(x)
CONSTR
(i) CONSTR
-300 -250 -200 -150 -100 -50 0
f
1
(x)
0
10
20
30
40
50
60
70
80
f
2
(x)
OSY
(j) OSY
0 50 100 150 200 250
f
1
(x)
-250
-200
-150
-100
-50
0
50
f
2
(x)
SRN
(k) SRN
0 0.2 0.4 0.6 0.8 1 1.2
f
1
(x)
0
0.2
0.4
0.6
0.8
1
1.2
f
2
(x)
TNK
(l) TNK
Figure 2: Illustrations of the Pareto fronts and an example run on test problems.
where µ
min
is a sufficiently small positive real value.
In our algorithm, ∆
k
s
k
is computed by the Hooke
and Jeeves (HJ) search method (Hooke and Jeeves,
1961). This algorithm differs from the traditional co-
ordinate search since it performs two types of moves:
the exploratory move and the pattern move. An ex-
ploratory move is a coordinate search – a search along
the coordinate axes – around a selected approxima-
tion, using a step length ∆
k
. A pattern move is a
promising direction that is defined by z
k
− z
k−1
when
the previous iteration was successful and z
k
was ac-
cepted as the new approximation. A new trial ap-
proximation is then defined as z
k
+ (z
k
− z
k−1
) and an
exploratory move is then carried out around this trial
point. If this search is successful, the new approxi-
mation is accepted as z
k+1
. We refer to (Hooke and
Jeeves, 1961; Lewis and Torczon, 1999) for details.
This HJ iterative procedure terminates, providing a
new approximation x
j
to the problems (4), x
j
← z
k+1
,
when the following stopping condition is satisfied,
ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems
338

Table 1: Performance of MOEA/D-IEpsilon (SBX), MOEA/D-IEpsilon (DE), and SCAL on CF1-CF7, BNH, CONSTR,
OSY, SRN and TNK problems in terms of the mean and standard deviation values of IGD.
Problem MOEA/D-IEpsilon (SBX) MOEA/D-IEpsilon (DE) SCAL
CF1 mean 6.06E-03 5.13E-03 3.54E-01
std 1.36E-03 9.43E-04 1.13E-16
CF2 mean 1.04E-01 1.41E-02 9.49E-02
std 4.62E-02 2.15E-02 9.46E-02
CF3 mean 3.15E-01 2.33E-01 3.30E-01
std 1.25E-01 1.47E-01 1.79E-01
CF4 mean 1.18E-01 2.73E-02 3.97E-01
std 3.12E-02 9.17E-03 6.65E-01
CF5 mean 2.91E-01 2.49E-01 2.94E-01
std 1.33E-01 1.09E-01 2.00E-01
CF6 mean 1.34E-01 6.46E-02 3.73E-02
std 7.21E-02 3.59E-02 8.14E-03
CF7 mean 2.48E-01 2.08E-01 1.96E-01
std 8.81E-02 8.50E-02 1.77E-01
BNH mean 3.91E-01 4.59E-01 1.41E+00
std 1.17E-01 1.11E-01 7.81E-02
CONSTR mean 8.33E-03 1.28E-02 2.10E+00
std 9.96E-04 1.29E-03 2.75E+00
OSY mean 3.86E+00 4.31E+00 6.51E+00
std 5.60E-01 8.67E-01 2.19E+00
SRN mean 3.54E-01 3.56E-01 6.62E+00
std 5.89E-03 8.26E-03 2.63E+00
TNK mean 2.21E-03 1.82E-03 3.00E-01
std 9.08E-05 4.47E-05 3.36E-01
∆
k
≤ ε
j
. However, if this condition can not be sat-
isfied in k
max
iterations, then the procedure is stopped
with the last available approximation.
More detailed explanation on this can be found in
(Costa et al., 2012).
4 RESULTS AND DISCUSSION
In order to test the Augmented Lagrangian in con-
straint handling, several multi-objective test problems
are used: CF1-CF7 (Zhang et al., 2009), BNH (Binh
and Korn, 1997), CONSTR (Deb, 2001), OSY (Osy-
czka and Kundu, 1995), SRN (Srinivas and Deb,
1994), and TNK (Tanaka et al., 1995). Furthermore,
the results with this new approach are compared with
the results of (Fan et al., 2017), in particular, the al-
gorithm MOEA/D-IEpsilon, since this was the best
algorithm of all those tested, using simulated binary
crossover (SBX) and differential evolution crossover
(DE) operators (Fan et al., 2017). The compari-
son is based on the Inverted Generational Distance
(IGD) measure; this measure evaluates algorithm per-
formance in terms of convergence to the Pareto front
as well as the diversity of the approximation along the
frontier. For this purpose it is mandatory to know the
exact definition of the Pareto Frontier. These are pre-
liminary results and no thorough study has been made
of the parameters of the Augmented Lagrangian. Fig-
ures 1 presents the box plots for 30 executions for
each problem and Figure 2 shows the Pareto Fron-
tier (line) for the different problems with the approx-
imation (dots in the graphs) resulting from one of
the executions. It can be observed that the algorithm
closely approximates the Pareto Frontier, exhibiting
some extreme points which have a large impact on the
IGD measure. Table 1 presents the results of the new
approach and the published results with algorithm
MOEA/D-IEpsilon (Fan et al., 2017). Although, the
new algorithm only wins in two of the problems, with-
out much investigation of the Augmented Lagrangian
parameters, these results seem promising.
5 CONCLUSIONS
In this work a new approach for dealing with con-
straints in multi-objective optimization problems is
proposed based on the Augmented Lagrangian cou-
pled with weighted Tchebycheff method. Although
A Scalarized Augmented Lagrangian Algorithm (SCAL) for Multi-objective Optimization Constrained Problems
339

the reported results are a preliminary work on this ap-
proach, the results are very encouraging. Therefore,
future work will address the study of the parameters
of the Augmented Lagrangian, the use of an achieve-
ment scalarizing function, and the testing of the algo-
rithm in problems with three objectives.
ACKNOWLEDGEMENTS
This work has been supported by the Portuguese
Foundation for Science and Technology (FCT) in the
scope of the project UID/CEC/00319/2013 (ALGO-
RITMI R&D Center).
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