A New Full-NT-Step Infeasible Interior-Point Algorithm for SDP Based on a Specific Kernel Function

Abstract

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.

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Bouali, S. and Kabbaj, S. (2012) A New Full-NT-Step Infeasible Interior-Point Algorithm for SDP Based on a Specific Kernel Function. Applied Mathematics, 3, 1014-1022. doi: 10.4236/am.2012.39150.

1. Introduction

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 [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 the vector of eigenvalues of. Two different forms of norm will be used

2. The Statement of the Algorithm

We start usually with assuming that the initial iterates and are as follows

where I is the identity matrix, is the initial dual gap and is such that

for some optimal solution of and.

2.1. The Feasible SDP Problem

The perturbed KKT condition for and is

Based on different symmetrization schemes, several search directions have been proposed.

As in Mansouri and Roos [6], we use in this paper, the so-called NT-direction determined by the following system

(1)

where

(2)

We also define the square root matrix.

The matrix D can be used to rescale X and S to be the same matrix V, defined by

(3)

It is clear that D and V are symmetric and positive definite. Let us further define

(4)

where. Using the above notations, the third equation of the system (1) is then formulated as follows

(5)

It is clear that the first two equations imply that 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

(6)

2.2. The Perturbed Problem

For any with, we consider the perturbed problem, defined by

and its dual problem given by

where

Note that if then and , yielding that both and are strictly feasible.

Lemma 1 ([6], Lemma 4.1) Let the original problems, and, be feasible. Then for each such that the perturbed problems and are strictly feasible.

We assume that 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

If 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

(7)

where and.

Inspired by [8], we used in the third equation for the above system, a linearization, which means that we target the -center of and.

After the feasibility step, the new iterates are given by, 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).

2.3. Infeasible IPMs Based on a Specific Kernel Function

Now we introduce the definition of a kernel function. We call a kernel function if is twice differentiable and the following conditions are satisfied

1)

2) for all

3).

We define

(8)

By using the scaled search directions and as defined in (4), the system (7) can be reduced to

(9)

According to (8), Equation (9) can be rewritten as

(10)

It is clear that the right-hand side of the above equation is the negative gradient direction of the following barrier function whose kernel logarithmic barrier function is.

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:

(11)

where the kernel function of is given by

(12)

Since, the third equation in the system (11) can be rewritten as

(13)

In the sequel, the feasibility step will be based on the Equation (13).

3. 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. If, then the fullNT step is strictly feasible.

Corollary 3 Let X, S satisfy the Slater’s regularity condition and. If. One has.

Lemma 4 After a feasible full-NT step the proximity function satisfies

Lemma 5 If, then

The required number of centrality steps can easily be computed. After the -update, one has , and hence after k centrality steps the iterates satisfy

From this, one deduces easily that holds, after at most

(14)

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

Figure 1. Primal-dual infeasible IPMs.

for any, we have:

(15)

and furthermore, according to (6), we obtain:

(16)

Lemma 7 According to the result of Corollary (3), for any one has

Proof. By applying Hölder inequality and using, we obtain

and the result follows.

The following Lemma gives an upper bound for the proximity-measure of the matrix.

Lemma 8 Let be a primal-dual NT pair and such that. Moreover let and. Then

Proof. Since the 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 , , then, and.

Let Q be an real symmetric matrix and M be an real skew-symmetric matrix, we recall the following result.

Lemma 10 (See [7], Lemma 3.8). If Q is positive definite, then.

Lemma 11 (See [1], Lemma 6.3). If Q is positive definite, then.

Lemma 12 (See [7], Lemma 3.10). Let A, , and. Then

Lemma 13 (See [11], Lemma A.1) For, let denote a convex functions. Then, for any nonzero, the following inequality

holds.

4. Analysis of the Feasibility Step

4.1. The Feasibility Step

As established in Section 2, the feasibility step generates new iterates, 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

Therefore which implies that

(17)

According to (8), Equation (13) can be rewritten as

(18)

and by multiplying both side from the left with V, we get

(19)

To simplify the notation in the sequel, we denote

(20)

Note that is symmetric and M is skew-symmetric. Now we may write, using (19),

By subtracting and adding, to the last expression we obtain

Using (20) and (17), we get

(21)

Note that due to (8), is positive definite.

Lemma 14 Let and. Then the iterates are strictly feasible if

.

Proof. We begin by introducing a step length, and we define

We then have, 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

to the right hand side of the above equality we obtain

where the matrix

is skew-symmetric for all. Lemma (11) implies that the determinant of will be positive if the symmetric matrix

is positive definite which is true for all. 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.

We continue this section by recalling the following Lemma.

Lemma 15 (See [12], Lemma II. 60). Let be as given by (6) and. Then

The proof of Lemma (15), together with, makes clear that the elements of the vector satisfy

(22)

Furthermore, by using (8) and (22), we obtain the bounds of the elements of the vector

(23)

In the sequel we denote

where

This implies

and

Lemma 16 Assuming, one has

Proof. Using (6), we get

where

From (21), one has

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, we define

It is clear that is convex in if

. Taking, we require

By applying Lemma (15), the above inequality holds if

(24)

By using Lemma (13), we may write

where

Furthermore, by using Lemma (8), we get

We deduce

where

Hence

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

(25)

Now we decide to choose

(26)

Note that the left-hand side of (25) is monotonically increasing with respect to. By some elementary calculations, for and, we obtain

(27)

4.2. Upper Bound for

In this section we consider the linear space

.

It is clear that the affine space

equals and. 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 to guarantee. Due to the above lemma, this will certainly hold if satisfies

(28)

Furthermore, according to Mansouri and Ross, we have

Since, we may write

By using, the above inequality becomes

(29)

Because we are looking for the value that we do not allow 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.

4.3. Iteration Bound

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

Since, the total number of inner iterations is so bounded above by

5. 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.

6. Acknowledgements

The authors gratefully acknowledge the help of the guest editor and anonymous referees in improving the readability of the paper.

Conflicts of Interest

The authors declare no conflicts of interest.

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