![](https://csdnimg.cn/release/download_crawler_static/12188639/bg4.jpg)
Combining the first constraint condition in (2.1) with (2.3), we can define
e
A(x) :=
A(x) 0
0 −ρI
2n
!
and
D
e
A(x)d :=
n
X
i=1
e
E
i
d
i
, where
e
E
i
:=
∂A(x)
∂x
i
0
0 E
i
!
.
It follows that the subproblem (2.1) can be reformulated as
min
d∈R
n
q(d)
s.t.
e
A(x) + D
e
A(x)d 0,
g(x) + Dg(x)d = 0,
which is exactly the subproblem in [3].
The QSD subproblem (2.1) also requires the specification of a matrix B,
although this plays a relatively minor part in the analysis of global convergence.
For this reason we do not make the dependence on B explicitly in notations.
We let d denote the global solution (if it e xists) of QSD subproblem (2.1).
Then we denote
∆q
k
= q(0) − q(d) = −Df (x
k
)
>
d −
1
2
d
>
B
k
d (2.4)
as the predicted reduction in f(x) at k-th iteration, and
∆f
k
= f(x
k
) − f (x
k
+ d) (2.5)
as the actual reduction in f(x) at k-th iteration. The measure of constraint
infeasibility is
h
k
:= h(x
k
) =k g(x
k
) k
1
+λ
1
(A(x
k
))
+
, (2.6)
and we use f
k
to denote f(x
k
).
We now turn to the definition of filter for nonlinear programming as intro-
duced in [5, 6, 7]. The two aims in a constrained optimization problem are
to minimize f(x) and to satisfy the constraints, that is, to minimize h(x). In
a filter we consider pairs of values (h
k
, f
k
) obtained by evaluating h(x
k
) and
f(x
k
) for each iterate x
k
. A pair (h
i
, f
i
) obtained in the iteration i is said
to dominate another pair (h
j
, f
j
) if and only if both h
i
≤ h
j
and f
i
≤ f
j
,
indicating that the point x
i
is at least as good as x
j
with respect to both
measures. The filter is defined to be a list of pairs (h
i
, f
i
) such that no pair
dominates any other. We use F
k
to denote the set of iteration indices j
(j < k) such that (h
j
, f
j
) is an entry in the current filter. A point x is said to
4
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