In this paper we deal with SDP problems, whose primal and dual forms are:
and
where
,
. The matrices
, 
are assumed to be linearly independent.
The use of Interior-Point Methods (IPMs) based on the kernel functions becomes more desirable because of the efficiency from a computational point of view. Many researchers have been attracted by the Primal-Dual IPMs for SDP. For a comprehensive study, the reader is referred to Klerk [
In the sequel, we denotes e as the all one vector and 
the vector of eigenvalues of
. Two different forms of norm will be used
and 
are as follows
identity matrix, 
is the initial dual gap and 
is such that
of 
and
.
and 
is
.
. Using the above notations, the third equation of the system (1) is then formulated as follows
and 
are orthogonal, i.e.
, which yields that 
and 
are both zero if and only if
. In this case, X and S satisfy
, implying that X and S are the 
-centers. Hence, we can use the norm 
as a quantity to measure closeness to the 
-centers. Let us define
with
, we consider the perturbed problem
, defined by
given by
then 
and 
, yielding that both 
and 
are strictly feasible.
and
, be feasible. Then for each 
such that 
the perturbed problems 
and 
are strictly feasible.
and 
are feasible. It follows from Lemma (1) that the problems 
and 
are strictly feasible. Hence their central path exists. The central path of 
and 
is defined by the solution sets 
of the following system
and
, we denote this unique solution as
. 
is the 
-center of
, and 
the 
-center of
. By taking
, one has 
. Initially, one has 
and
, whence 
and
. In what follows, we assume that at the start of each iteration, 
is smaller than a threshold value 
which is obviously true at the start of the first iteration. The following system is used to define the step 
and
.
, which means that we target the 
-center of 
and
.
, 
and
. The algorithm begins with an infeasible interior point 
such that 
is feasible for the perturbed problems, 
and
. First we find a new point 
which is feasible for the perturbed problems with
. Then 
is decreased to
. A few centering steps are applied to produce new points 
such that
. This process is repeated until the algorithm terminates. Starting at the iterates 
and targeting the 
-center, the centering steps are obtained by solving the system (1).
a kernel function if 
is twice differentiable and the following conditions are satisfied 
for all 
.
and 
as defined in (4), the system (7) can be reduced to
whose kernel logarithmic barrier function is
.
is given by
, the third equation in the system (11) can be rewritten as
, 
,
.
. If
, then the fullNT step is strictly feasible.
. If
. One has
.
, then
-update, one has 
, and hence after k centrality steps the iterates 
satisfy
holds, after at most
, one has
, we have:
one has
, we obtain
.
be a primal-dual NT pair and 
such that
. Moreover let 
and
. Then
is a primal-dual pair and by applying Lemma (7), and the two inequalities (15) and (16), we can get:
, 
, then
, and
.
real symmetric matrix and M be an 
real skew-symmetric matrix, we recall the following result.
.
.
, 
and
. Then
, let 
denote a convex functions. Then, for any nonzero
, the following inequality
, 
and 
that satisfy the Feasibility conditions for 
and 
(i.e., primal feasible and dual feasible), except possibly the positive semidefinite conditions. A crucial element in the analysis is to show that after the feasibility step, the inequality 
holds, i.e., that the new iterates are within the region where the Newton process targeting at the 
-centers of 
and 
is quadratically convergent. Let X, y and S denote the iterates at the start of an iteration and assume that
. Recall that at the start of the first iteration this is true since
. Defining 
and 
as in (4) and 
as in (3). We may write
which implies that
is symmetric and M is skew-symmetric. Now we may write, using (19),
, to the last expression we obtain
is positive definite.
and
. Then the iterates 
are strictly feasible if
.
, and we define
, 
and similar relations for y and S. It is clear that
. We want to show that the determinant of 
remains positive for all
. We may write
and
to the right hand side of the above equality we obtain
. Lemma (11) implies that the determinant of 
will be positive if the symmetric matrix
. This means that 
has positive determinant. By positiveness of 
and 
and continuity of both 
and
, we deduce that 
and 
are positive definite which completes the proof.
be as given by (6) and
. Then
, makes clear that the elements of the vector 
satisfy
, one has
since M is skewsymmetric and Lemma (10), we may write
, we define
is convex in 
if
. Taking
, we require
, it follows from this lemma that it suffices if
. By some elementary calculations, for 
and
, we obtain
.
and
. We can get from Mansouri and Roos [
and
. Then
to guarantee
. Due to the above lemma, this will certainly hold if 
satisfies
, we may write
, the above inequality becomes
to exceed and in order to guarantee that 
, (28) holds if 
satisfies
, since
. This will be certainly satisfied if
. Hence, combining this with (29), we deduce that 
holds.
, with
, then after the feasibility step, with 
as defined in (26), the iterates satisfies
. According to (14), at most 
centering steps suffice to get iterates that satisfy
. So each main iteration consists of at most 3 so-called inner iterations. In each main iteration both the duality gap and the norms of the residual vectors are reduced by the factor
. Hence, using
, the total number of main iterations is bounded above by
, the total number of inner iterations is so bounded above by