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Algorithm for Solving a QGP Problem (EA-QGP)
1. Set k := 0,
best
:= −
inf
, ξ
opt
:= −1 and x
opt
:=
−1 or other infeasible values.
2. Using an EA, generate a population P(k) =
n
ξ
(k)
i
o
i=I
i=1
, such that ξ 6 ξ
(k)
i
6
¯
ξ, for all individ-
uals i; I is the size of the population.
3. For each individual ξ
(k)
i
solve the standard GP
problem (10) w.r.t λ and x. This gives, w.r.t ξ
(k)
i
,
the optimal solution denoted λ
(k)
i
and x
(k)
i
. This
step represents the evaluation of the population
P(k). If problem (10) is not feasible, then set
J(ξ
(k)
) := −
inf
else set J(ξ
(k)
) := λ
(k)
.
4. If J(ξ
(k)
b
) >
best
, then set:
best
:= J(ξ
(k)
b
),
ξ
opt
:= ξ
(k)
b
and x
opt
:= x
(k)
b
, where ξ
(k)
b
represents
the best individual of the population P(k) and x
(k)
b
the solution to the corresponding GP problem.
5. If the termination condition is satisfied, go to step
7 (the termination condition can be, for instance,
a defined number of iterations).
6. From the results obtained step 3, generate a new
population P(k) where k := k + 1 (this is done by
using the usual operators of EA, i.e. selection,
crossover and mutation operators), go to step 3.
7. The optimal solution is given by (x
opt
,ξ
opt
), stop.
In this algorithm,
inf
represents the IEEE arith-
metic representation for positive infinity, and
best
is
a variable containing the current best objective func-
tion. Note that the use of “ global optimization meth-
ods ” like, for instance GA, increases the probability
of finding a global optimum but this is not guaran-
teed, except perhaps if the search space of problem
(9) is explored very finely, but this cannot be done in
a reasonable time.
4 ROBUSTNESS ISSUE
Until now it was implicitly assumed that the parame-
ters (i.e. the problem data) which enter in the formu-
lation of a QGP problem are precisely known. How-
ever, in many practical applications some of these pa-
rameters are subject to uncertainties. It is then im-
portant to be able to calculate solutions that are in-
sensitive to parameters uncertainties; this leads to the
notion of optimal robust design. We say that the de-
sign is robust, if the various specifications (i.e. the
constraints) are satisfied for a set of values of the pa-
rameters uncertainties. In this section we show how
to use the methods presented above to developdesigns
that are robust with respect to some parameters uncer-
tainties.
Let θ = [θ
1
θ
2
··· θ
q
]
T
be the vector of uncertain
parameters. It is assumed that θ lie in a bounded set
Θ defined as follows:
Θ =
θ ∈ R
q
: θ θ
¯
θ
, (11)
where the notation denotes the componentwise in-
equality between two vectors: v w means v
i
6 w
i
for all i. The vectors θ = [θ
1
···θ
q
]
T
,
¯
θ = [
¯
θ
1
···
¯
θ
q
]
T
are the bounds of uncertainty of the parameters vec-
tor θ. Thus, the uncertain vector belong to the q-
dimensional hyperrectangle Θ also called the param-
eter box. In these conditions, the QGP problem (7),
or equivalently (8), must be expressed in term of func-
tions of (x,ξ), the design variables, and θ the vector of
uncertain parameters. The robust version of the quasi
geometric problem (8) is then written as follows:
maximize λ
subject to λ+ ϕ
0
(x,ξ,θ) 6 ϕ
′
0
(ξ,θ)
ϕ
i
(x,ξ,θ) 6 Q
i
(ξ,θ), i = 1,··· ,m
h
j
(x,ξ,θ) = Q
′
j
(ξ,θ), j = 1,··· , p
(12)
for all θ ∈ Θ. The functions ϕ
i
, i = 0,··· , m, are
posynomial functions of (x,ξ), for each value of θ,
and the functions h
j
, j = 1, · · · , p, are monomial func-
tions of (x,ξ), for each value of θ. The functions ϕ
′
0
,
Q
i
and Q
′
j
are only assumed to be positive for each θ.
We consider the resolution of the robust QGP
problem in the case of a finite set. Let Θ
N
=
{θ
(1)
, θ
(2)
,··· ,θ
(N)
} be a finite set of possible vec-
tor parameter values. This finite set can be imposed
by the problem itself or can be obtained by sampling
the continuous set Θ defined in (11). For instance, we
might sample each interval [θ
i
,
¯
θ
i
] with three values:
θ
i
,
θ
i
+
¯
θ
i
2
and
¯
θ
i
, and form every possible combination
of parameter values, this lead to N = 3
q
different vec-
tor parameters.
Whatever how the finite set is obtained, we have to
determine a solution (x, ξ) that satisfy the QGP prob-
lem for all possible vector parameters. To do so, we
have only to replicate the constraints for all possible
vector parameters. Thus, in the case of a finite set Θ
N
,
the robust QGP problem is formulated as follows:
maximize λ
subject to λ+ ϕ
0
(x,ξ,θ
(k)
) 6 ϕ
′
0
(ξ,θ
(k)
)
ϕ
i
(x,ξ,θ
(k)
) 6 Q
i
(ξ,θ
(k)
)
h
j
(x,ξ,θ
(k)
) = Q
′
j
(ξ,θ
(k)
)
(13)
where i = 1,··· ,m and j = 1,··· , p and k = 1,··· ,N.
As we can see, problem (13) can be solved as a stan-
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