Composite Materials Research Progress

80
Michaël Bruyneel
Solving the primal problem (2.2) requires the manipulation of one design function, m
structural restrictions and 2 × n side constraints (for mono-objective problems). When the
dual formulation is used, the resulting quasi-unconstrained problem (7.10) includes one
design function and m side constraints, if the side constraints in the primal problem are treated
separately. In relation (7.10), L( x,
λ)
is the Lagrangian function of the optimization problem,
which can be written
pij
qij
L ( x,
λ)
= ∑λj( c j + ∑ + ∑ )
(7.11)
k
U − x x − L
j i i i i
according to the general definition of the involved approximations g
~
j ( X ) of the functions.
The parameter λj is the dual variable associated to each approximated function g
~
j ( X ) . Given
that the approximations are separable, the Lagrangian function is separable too. It turns that:
and the Lagrangian problem of (7.10)
can be split in n one dimensional problems
= ∑ x ) , ( ) ( λ
λ x, L
L
i
i
k
minL ( x, λ)
x
i
i
k
minL ( x , λ)
(7.12)
x
i
The primal-dual relations are obtained by solving (7.12) for each primal variable xi:
∂Li
( xi
, λ)
= 0
∂xi
i
i
⇒
xi
= xi
( λ)
k i
(7.13)
Relation (7.13) asserts the stationnarity conditions of the Lagrangian function over the
primal variables xi. Once the primal-dual relations (7.13) are known, (7.10) can be replaced
by
maxl ( λ)
⇔ maxL(
x(
λ)
, λ)
(7.14)
λ
λ
λ j ≥ 0 j = 1,...,
m
Solving problem (2.2) is then equivalent to maximize the dual function l (λ)
with non
negativity constraints on the dual variables (7.14). As it is explained by Fleury (1993), the