A CLASS OF NEW SEARCH DIRECTIONS FOR FULL-NT STEP FEASIBLE INTERIOR POINT METHOD IN SEMIDEFINITE OPTIMIZATION

. In this paper, based on Darvay et al. ’s strategy for linear optimization (LO) (Z. Darvay and P.R. Tak´acs, Optim. Lett. 12 (2018) 1099–1116.), we extend Kheirfam et al. ’s feasible primal-dual path-following interior point algorithm for LO (B. Kheirfam and A. Nasrollahi, Asian-Eur. J. Math. 1 (2020) 2050014.) to semidefinite optimization (SDO) problems in order to define a class of new search directions. The algorithm uses only full Nesterov-Todd (NT) step at each iteration to find an 𝜖 -approximated solution to SDO. Polynomial complexity of the proposed algorithm is established which is as good as the LO analogue. Finally, we present some numerical results to prove the efficiency of the proposed algorithm.


Introduction
Semidefinite optimization (SDO) problems are convex optimization problems, including linear optimization (LO), which minimize a linear function with the matrix variable over the intersection of an affine set and the cone of positive semidefinite matrices.SDO problems have a lot of significant applications in continuous and combinatorial optimization (see, e.g., [3,21]).
In the last decade, SDO has become a very active research area in mathematical programming because of the extension of the most algorithms for LO to the SDO case.Several primal-dual interior-point methods (IPMs) suggested for LO have been successfully extended to SDO [7,11,14,18], convex quadratic semidefinite optimization (CQSDO) [2,10] and other optimization problems [9,19,20,22] due to their polynomial complexity and practical efficiency.The first primal-dual feasible IPM with a full-Newton step for LO was proposed by Roos et al. [16].Later on, De Klerk [7], Achache and Guerra [2] extended Roos et al.'s algorithm for LO to SDO and CQSDO by using the full Nesterov-Todd (NT) direction as a search direction, respectively.Finding the search directions plays a crucial role in IPMs.In 2003, Darvay [4] introduced a new strategy for defining search directions for LO problems.The strategy is based on an algebraic equivalent transformation (AET) of the standard centering equations of the central path (   ) = () where () = √ .Achache [1], Wang and Bai [18,19], extended Darvay's algorithm for LO to convex quadratic optimization (CQO), SDO and second-order cone optimization (SOCO), respectively.In 2016, Darvay et al. [6] developed a new full-Newton step feasible IPM for LO based on a new reformulation of the standard centering equations of the central path with () =  − √ .Kheirfam [11], generalized this method for SDO and derived the currently best-known iteration bound for SDO problems.In 2018, Darvay and Takács [5] designed a feasible primal-dual interior point algorithm for LO.Their algorithm is based on a new reformulation of the nonlinear equations of the central path (   ) = ((   ) 2 ) where () =  2 , for  > 1 .Recently, Kheirfam et al. [12] extended this study to case () =   with  > 1  √ 2 and  ≥ 2 to determine a class of the search directions in LO and proved that the suggested approach has the same complexity bound obtained by Darvay et al. [5].
Motivated by the mentioned works, we propose a new feasible primal-dual path-following interior point algorithm for SDO based on a new transformation to define a class of new search directions.We adopt the basic analysis used in [12] for the SDO case.The iteration bound for the algorithm with the small-update method is as good as the bound for the LO case [5,12].Furthermore, our analysis is relatively simple and straightforward to the LO analogue.
The outline of the paper is as follows.In Section 2, the SDO problem and the central path are presented.In Section 3, we extend Darvay's new technique for LO to SDO and derive a class of new search directions for SDO problems based on the AET with () =   for  > 1  √ 2 and  ≥ 2. In Section 4, we present a new primal-dual interior point algorithm for SDO.In Section 5, the polynomial complexity result is established where we give the detailed proofs of it.Some numerical results are provided in Section 6.Finally, a conclusion is stated in Section 7.
The following notations are used throughout the paper.R  denotes the space of vectors with  components.S  denotes the space of real symmetric matrices of order  and S  + (S  ++ ) denotes the cone of  ×  symmetric positive definite (positive semidefinite) matrices.Furthermore,  ⪰ 0 ( ≻ 0) means that  ∈ S  + ( ∈ S  ++ ).For any matrix ,   () denote the  ℎ eigenvalues of  with  min () the smallest one and det  denotes its determinant whereas  () = ∑︀  =1   = ∑︀  =1   denotes its trace where   is the diagonal elements of , ‖.‖  denote the Frobenius norm and the symbol  •  denotes the trace inner-product in S  defined by  •  = Tr() = ∑︀  ,=1     .The symmetric positive definite square root of any symmetric positive definite matrix  is denoted by  1/2 .For  (), () : R  + → R  ++ ,  () = (()) if  () ≤ () for some positive constant .Finally, the notation  ∼  ⇔  =  −1 for some invertible matrix , means the similarity between the two matrices ,  ∈ R × , and the identity matrix is denoted by .

The central path
The standard primal form of semidefinite optimization (SDO) problems is as follows and its Lagrange dual problem () max where ,   ∈ S  and  ∈ R  .Throughout the paper, we make the following assumptions on () and ().
• Interior point condition (IPC).There exists a triple ( 0 ,  0 ,  0 ) such that: If the IPC holds, it is well known that finding an optimal solutions of () and () is equivalent to solving the following system: The basic idea of primal-dual IPMs is to replace the third equation  = 0 in the system (1), the so-called complementarity condition for () and (), by the parameterized equation  =  ( > 0).Thus we consider Since the IPC holds and the   are linearly independent, the parameterized system (2), has a unique solution ((), (), ()) for any  > 0 [13,15].The set of all such solutions defines the central-path of () and ().
If  → 0, then the limit of the central-path exists and since the limit satisfies the complementarity condition, the limit yields a primal-dual optimal solutions for () and () [8].
Inspired by [5], we replace the standard centering equation  =  by  (︁ )︂ , then the system (2) can be written as: Applying Newton method's on system (3), we obtain the following system for the search directions ∆, ∆ and Applying Lemma 2.5 in [18], the third equation of the last system can be written as Then we consider the following system ( to obtain search directions (∆, ∆, ∆).It is obvious that ∆ is symmetric due to the second equation in (5) but ∆ may be not symmetric.Many researchers have proposed several methods for symmetrizing the third equation in (5) such that the resulting new system has a unique symmetric solution.
In this paper, we use the Nesterov-Todd symmetrization scheme [2,7,10,11,[17][18][19]21], which defines the so-called NT-direction.Let us define the matrix We replace the term ∆ −1 in the third equation of ( 5) by  ∆  .Then the system (5) becomes Furthermore, we define  =  , where  1 2 denotes the symmetric square roote of  .The matrix  can be used to scale  and  to the same matrix  as follows Note that both matrices  and  are symmetric and positive definite.It is easy to verify from (7) that In addition, the scaling directions   and   are: Then it follows from ( 6) that the scaled NT search directions (  , ∆,   ) are defined by the following system Since the   are linearly independent so the Ā , then the system (9) has a unique solution   , ∆, and   with   and   are symmetric matrices, and Darvay et al. [5] developed a full Newton step primal-dual path-following IPM for LO.They also established that their approach solves the LO problem in polynomial time and has ( √ )-iteration complexity bound.
Their analysis is based on the function () =  2 case, and Kheirfam [12] extended this study to the case 2 and  ≥ 2 to derive a class of new search directions for LO problems.In this paper, we extend the proposed strategy in [5] with () =   for  > 1  √ 2 and  ≥ 2 to the SDO case, this yields We begin by recalling the following technical lemma.
Hence,  is a distance which measure the closeness of primal-dual points (, , ) to the central path.

The algorithm
A primal-dual path following interior point algorithm based on new search directions for SDO is given in Figure 1.

Analysis of the algorithm
In this section, we will show that the proposed algorithm can solve the SDO in polynomial time.
For simplicity, we may write  + = (1) and  + = (1), then The following lemma gives a lower bound for the smallest eigenvalue of  Proof.From (15), in Lemma 5.3, letting  = 1, we get It follows that Using Lemmas 3.1, 5.2 and the skew-symmetric of  (1), it implies that: Where the last equality is follows from (14).This completes the proof.
The next result shows the quadratic convergence of the proximity.
By using the Frobenius norm, we obtain Where the last equality is due to ( 14) and the third inequality becomes from the following inequality Here, the inequality is due to 2  4 is skew-symmetric and  (1)  (1) is positive semidefinite.Substituting ( 16) into (17) yields the result.This completes the proof.
The next lemma, shows the influence of a full NT-step on the duality gap.
Lemma 5.6.After a full NT-step, we have Proof.From (15) in the proof of Lemma 5.3, we have by skew-symmetry of  (1) and Lemma 3.1, it implies From the orthogonality of the matrices   and   , we get This completes the proof.
In the next lemma, we investigate the effect of a full NT-step on the proximity after an update of the parameter .
From Lemma 5.7, we deduce that Algorithm 4 is well defined.The next lemma gives an upper bound for the number of iterations produced by Algorithm 4.
Proof.It is a straightforward from Lemma 5.8.This completes the proof.

Numerical results
In order to compare the efficiency of the algorithm with the existing methods and to show the influence of the parameter  on the number of iterations produced by the algorithm, we present some numerical results under Matlab 8.1 where the implementation is done on a computer with an Intel core 2.3 GHz processor and 4 GB RAM, for solving some semidefinite optimization problems.
Note that the value of the parameter  may be very large, which leads to a very small value of the parameter , see Theorem 5.9.This motivated us to make some changes in the implementation of the proposed algorithm.The initial primal dual point ( 0 ,  0 ,  0 ) with  0 =  0 • 0  is chosen such that the pair is strictly feasible, the proximity ( 0 ,  0 ;  0 ) ≤  and the smallest eigenvalue of the matrix  is greater than a positive constant.At each iteration, the value of the parameter  was calculated as  + =  min{(++) : 1≤≤}  , where 0 <  < 1 and  is a given lower bound, which in this case is 0.5 2  ,  ≥ 2 and  = 0.2.The technique for determining the value of the parameter  + ensures that  min ( ) ≥ 1  √ 2 with  ≥ 2, which is signifiant in our case for the used search direction.Moreover, to guarantee that the iterates remain interior, we use the following strategy: we compute at each iteration a maximum step size  max such that  +  max ∆ ≻ 0 and  +  max ∆ ≻ 0 with  max = min(  ,   ) and  ∈ (0, 1), where   and   are the primal and the dual feasible step size given by and )︁ ≥ 0.
To ensure the strict feasibility of the new iterates we used a factor  = 0.95.In our computational study, we compared our algorithm where () =   ,  ≥ 2 with the variant of interior point algorithms that use the following AET for solving SDO problems: () =  , () = √  and () =  − √  (see [7,11,18], respectively) where the value of  is 1  4 .In all cases, the accuracy parameter had a value  = 10 −5 .Here, we use the following notations: "iter" means the number of iterations performed by the algorithm in order to get an approximate optimal solution."CPU" denotes the time (in seconds) necessary to get an approximate optimal solution for SDO.Problem 6.1.We consider the SDO problem in [10], where We take  0 =  0 =  and  0 = [1, 1, 1] ⊤ as a feasible starting point.An exact optimal solution for Problem 6.1 is given by ]︀ ⊤ , the optimal value of both problems is equal to −1.0957.We summarize the obtained numerical results in Table 1 where the parameter  used in the implementation is as follows:  ∈ {2, 3, 5, 7, 15, 20, 30}.   . ., .The optimal value of both problems is equal to −.We summarize the obtained numerical results in Table 2 where the parameter  used in the implementation is as follows: ∈ {2, 3 , 5, 7, 10}.
Comment.Across the numerical results obtained by the algorithm the minimal number of iterations is achieved by the AET () =   with  = 2 for different size (, ).

Conclusion
In this work we have extended a primal-dual path-following interior-point method for LO to SDO problems with full NT-step.Based on the new Darvay's technique [5], we used the function () =   with  ≥ 2 in order to determine a class of new search directions.The associated short-step algorithm deserves the best well-known polynomial complexity, which is the same iteration bound as in the LO case.Moreover, the resulting analysis is relatively simple and straightforward to the LO analogue in [12].We also presented some numerical results to show the efficiency of the proposed method.

√ 2 .
They established that the iteration bound of it is  (︀√  log   )︀

Table 1 .
Number of iterations for Problem 6.1.