A New Full-nt-step Infeasible Interior-point Algorithm for Sdp Based on a Specific Kernel Function

In this paper, we propose a new infeasible interior-point algorithm with full NesterovTodd (NT) steps for semidefinite programming (SDP). The main iteration consists of a feasibility step and several centrality steps. We used a specific kernel function to induce the feasibility step. The analysis is more simplified. The iteration bound coincides with the currently best known bound for infeasible interior-point methods.


Introduction
In this paper we deal with SDP problems, whose primal and dual forms are: , m  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 [1], Roos [2] and Wolkowicz et al. [3].Bai et al. [4] introduced a new class of so-called eligible kernel functions for Linear Optimization (LO) which are defined by some simple properties following the same way of Peng et al. who have designed a class of IPMs based on a so-called self-regular proximities [5].These methods use the new search directions which are different than the classic Newton directions.Some extensions were successfully made by Mansouri and Roos [6], Liu and Sun [7].In the current paper, we propose a new infeasible interior-point algorithm, whose feasibility step is induced by a specific kernel function.
In the sequel, we denotes e as the all one vector and   X  the vector of eigenvalues of n X   .Two dif-ferent forms of norm will be used : max , .

The Statement of the Algorithm
We start usually with assuming that the initial iterates 0 0 , X y and are as follows where I is the n n  identity matrix, 0  is the initial dual gap and 0 for some optimal solution  

The Feasible SDP Problem
The perturbed KKT condition for and  is   As in Mansouri and Roos [6], we use in this paper, the so-called NT-direction determined by the following system   . where We also define the square root matrix 1 2

D P
 .The matrix D can be used to rescale X and S to be the same matrix V, defined by It is clear that D and V are symmetric and positive definite.Let us further define a n d : where .Using the above notations, the third equation of the system (1) is then formulated as follows It is clear that the first two equations imply that X and S are orthogonal, i.e.
, which yields that X and S 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

The Perturbed Problem
For any  with 0 and its dual problem   , we denote this unique solution as


 is smaller than a threshold value  which is obviously true at the start of the first iteration.The following system is used to define where  .Inspired by [8], we used in the third equation for the above system, a linearization   means that we target the   -center of and After the feasibility step, the new iterates are given by . 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).

Infeasible IPMs Based on a Specific Kernel Function
Now we introduce the definition of a kernel function.We call a kernel function if differentiable and the following conditions are satisfied 1) We define By using the scaled search directions f X D and f S D as defined in (4), the system (7) can be reduced to According to (8), Equation ( 9) can be rewritten as It is clear that the right-hand side of the above equation is the negative gradient direction of the following barrier function whose kernel loga- Therefore, the aforementioned equation can be rewritten as Inspired by the work of [4,7,9], and by making a slight modification of the standard Newton direction, the new feasibility step used in this paper, is defined by the following different system: , where the kernel function of  is given by    , the third equation in the system (11) can be rewritten as In the sequel, the feasibility step will be based on the Equation (13).

Some Technical Results
We recall some interesting results from Klerk [1].In the sequel, we denote the iterates after a centrality step as Lemma 2 Let X, S satisfy the Slater's regularity condition and 0 , then the full-NT step is strictly feasible.
Corollary 3 Let X, S satisfy the Slater's regularity condition and 0 . Lemma 4 After a feasible full-NT step the proximity function satisfies The required number of centrality steps can easily be computed.After the  -update, one has .
From this, one deduces easily that   We give below a more formal description of the algorithm in Figure 1.
The following lemma stated without proof, will be useful for our analysis.
Lemma 6 (See [10], Lemma 2.5) For any , one has By applying Lemma (6), one can easily verify so that -update : : 1 ; for any , we have: and furthermore, according to (6), we obtain: Lemma 7 According to the result of Corollary (3), for any Proof.By applying Hölder inequality and using and the result follows.
The following Lemma gives an upper bound for the proximity-measure of the matrix X S be a primal-dual NT pair and , X S is a primal-dual pair and by applying Lemma (7), and the two inequalities (15) and ( 16), we can get: since the last term in the last equality is negative.This completes the proof of the Lemma.Lemma 9 (See [1], Lemma 6.1).If one has and   0 S   .Let Q be an n n  real symmetric matrix and M be an n n  real skew-symmetric matrix, we recall the following result.Lemma 10 (See [7], R denote a convex functions.Then, for any nonzero n R   , the following inequality V as in (3).We may write

Analysis of the Feasibility Step
According to (8), Equation (13) can be rewritten as and by multiplying both side from the left with V, we get To simplify the notation in the sequel, we denote Note that f XS D is symmetric and M is skew-symmetric.Now we may write, using (19), By subtracting and adding 1 2 Using ( 20) and ( 17), we get Note that due to (8), Proof.We begin by introducing a step length   .We may write to the right hand side of the above equality we obtain where the matrix . Lemma (11) implies that the determinant of X S   will be positive if the symmetric matrix . This means that X S   0 has positive determinant.By positiveness of X and and continuity of both 0 S X  and S  , we deduce that 1  X and are positive definite which completes the proof.

S
We continue this section by recalling the following Lemma.

 
V    be as given by (6) and   The proof of Lemma (15), together with makes clear that the elements of the vector   Furthermore, by using ( 8) and ( 22), we obtain the bounds of the elements of the vector   In the sequel we denote This implies Due to the fact that since M is skewsymmetric and Lemma (10), we may write where we apply for the third equality, Lemma (12) whose second condition is due to the requirement (24) given below.
For each 1, 2, , , we require By applying Lemma (15), the above inequality holds if By using Lemma (13), we may write  Furthermore, by using Lemma (8), we get The last equality is due to (24), which completes the proof.
Because we need to have   , it follows from this lemma that it suffices if Now we decide to choose Note that the left-hand side of (25) is monotonically increasing with respect to 2  .By some elementary calculations, for and

Upper Bound for    V
In this section we consider the linear space


It is clear that the affine space We can get from Mansouri and Roos [6], the following result.
Lemma 17 (See [6], Lemma 5.11) Let Q be the (unique) matrix in the intersection of the affine spaces and .Then


Note that (27) implies that we must have 0.35 . Due to the above lemma, this will certainly hold if Furthermore, according to Mansouri and Ross, we have     , we may write

       
By using 1 8n   , the above inequality becomes 2.27 0.28.8 Because we are looking for the value that we do not allow  to exceed and in order to guarantee that   holds.

Iteration Bound
In the previous sections, we have found that, if at the start of an iteration the iterates satisfies 

Concluding Remarks
In this paper we extended the full-Newton step infeasible interior-point algorithm to SDP.We used a specific kernel function to induce the feasibility step and we analyzed the algorithm based on this kernel function.The iteration bound coincides with the currently best known bound for IIPMs.Future research might focuses on studying new kernel functions.
different symmetrization schemes, several search directions have been proposed.

1 
both   P  and   D  are strictly feasible.Lemma 1 ([6], Lemma 4.1) Let the original problems,   P and   D , be feasible.Then for each  such that 0  the perturbed problems   P  and   D  are strictly feasible.We assume that   P and  are feasible.It fol- lows from Lemma (1) that the problems  strictly feasible.Hence their central path exists.The central path of   P  and   D  is defined by the solution sets , and hence after k centrality steps the iterates  
det and similar relations for y and S. It is clear that .We want to show that the determinant of the feasibility step, with  as defined in (26), the iterates satisfies   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