Applied Mathematics and Computation 217 (2011) 6437–6440

Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Finding a strict feasible solution of a linear semidefinite program

Djamel Benterki ⇑, Abdelkrim KeraghelLaboratory of Numerical and fundamental Mathematics, Department of Mathematics, Faculty of Sciences, University Ferhat Abbas, Setif, Algeria

a r t i c l e i n f o a b s t r a c t

Keywords:Linear programmingSemidefinite programmingProjective interior point method

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.12.083

⇑ Corresponding author.E-mail address: dj_benterki@yahoo.fr (D. Benterk

This study deals with the performance of projective interior point methods for linear semi-definite program. We propose a modification in the initialization phases of the method inorder to reduce the computation time.

This purpose is confirmed by numerical experiments showing the efficiency which arepresented in the last section of the paper.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

The linear semidefinite programming (SDP) is a model which traduces many real applications. It can be found in controltheory, combinatory optimization and nonlinear programming. In term of research, it is one of subject treated with fervour,in particular the problem of initialization in (SDP) [1,3,4,5,8,7]. We are interested in all aspects: theoretical, algorithmical andnumerical. But the last aspect, is the principal object of our study.

A linear semidefinite program in its standard form is of type:

z� ¼minX½trðCXÞ : X 2 K; trðAiXÞ ¼ bi for i ¼ 1; . . . ;m�: ðSDPÞ

Here b 2 Rm, K denotes the cone of positive semidefinite matrices in the linear space of n � n symmetric matrices E. Thematrices C and Ai, i = 1, . . . ,m, are given and belong to E. The trace of the product on E of two matrices A and B is definedby trðABÞ ¼

Pni¼1

Pnj¼1aijbij and norm A is:

kAk ¼Xn

i¼1

Xn

j¼1

a2ij

!12

:

Finally, we denote by int(K), the cone of positive definite matrices of E.In interior point methods, the successive iterates should be strictly feasible. In consequence, a major concern is to find an

initial feasible solution. This is the object of this paper.

2. Strict feasibility problem in SDP

The strict feasibility problem associated to the problem (SDP) is to find a matrix X such that:

½X 2 intðKÞ; trðAiXÞ ¼ bi for i ¼ 1; . . . ;m�: ðSFÞ

One way to solve a strictly feasible problem consists in introducing an additional variable k as follows:

. All rights reserved.

i).

6438 D. Benterki, A. Keraghel / Applied Mathematics and Computation 217 (2011) 6437–6440

minðX;kÞ

k :trðAiXÞ þ kðbi � trðAiX0ÞÞ ¼ bi for i ¼ 1; . . . ;m

X 2 intðKÞ; k P 0

� �; ðPaÞ

where X0 2 int(K) is arbitrary.This problem can be equivalently formulated as a following linear semidefinite program:

minX0

trðC 0X 0Þ : X0 2 eK ; trðA0iX0Þ ¼ bi for i ¼ 1; . . . ;m

h i; ðSDPaÞ

where C0 is the (n + 1) � (n + 1) symmetric matrix defined by

C 0½i; j� ¼1; if i ¼ j ¼ nþ 1;0; otherwise

�

for i ¼ 1; . . . ;m; A0i is the (n + 1) � (n + 1) symmetric matrix defined by

A0i ¼Ai 00 bi � trðAiX0Þ

� �

and X0 is the (n + 1) � (n + 1) symmetric matrix such that

X 0 ¼X 00 k

� �:

Finally, eK denotes the cone of positive semidefinite matrices in the linear space of (n + 1) � (n + 1) symmetric matrices.For solving the problem (SDPa), we use the variant of projective interior point method described in [2].

Lemma 1 [6]. X⁄ is a solution of problem (SF) if and only if X� 00 e

� �is an optimal solution of problem (SDPa) with X⁄ 2 int(K)

and e sufficiently small.

To compute the optimal solution of problem (SDPa), we use only the second phase of the projective interior point methoddescribed in a former paper [2]. The corresponding algorithm is:

3. Algorithm for solving (SDPa) [2]

Description of the algorithm

� (a) Initialization: e > 0 is fixed, X0 = I; k0 = 1, q 2 (0,1) and k = 0

If maxijbi � tr(AiXk)j 6 e, Stop: Xk is an e-approximate solution of (SF).If not go to (b).

� (b) If kk 6 e, Stop: Xk is an e-approximate solution of (SF).

If not: go to (c)

� (c) Step k1. Set zk ¼ trðC0X 0kÞ.2. Determine Lk such that X 0k ¼ LkLt

k (Cholesky decomposition). Next, compute

– Ck ¼ LtkC0Lk,

– AðkÞi ¼ LtkA0iLk; i ¼ 1; . . . ;m:

3. Compute the matrix M and the vector d by� �

– Mij ¼ tr AðkÞi AðkÞj þ bibj; i; j ¼ 1; . . . ;m,

– di ¼ �bizk � tr CkAðkÞi

� �i ¼ 1; . . . ;m.

4. Solve the linear system

Mu ¼ d:

5. Compute P

– Vk ¼ Ck þ m

i¼1uiAðkÞi ,

– vk ¼ �Pm

i¼1biui � zk,

– s ¼ ðkVkk2 þ v2kÞ

12,

D. Benterki, A. Keraghel / Applied Mathematics and Computation 217 (2011) 6437–6440 6439

– �k ¼ 1ðnþ1Þs trðVkÞ,

– r ¼ 1ðnþ1Þs2 kVkk2 � �k2,

– bk ¼ q max vks ;

�kþ rffiffiffinp �� �1.

6. Compute � �

– X 0kþ1 ¼ X 0k �

bks�bkvk

LkðVk � vkIÞLtk, with X 0kþ1 ¼

Xkþ1 00 kkþ1

:

� (d) Take k = k + 1, and go back to (b).

4. Modification in algorithm [2]

At iteration k, the algorithm [2] gives a strictly feasible solution Xk 00 kk

� �for (SDPa). To find the next iteration

X 0kþ1 ¼Xkþ1 0

0 kkþ1

� �, we search a matrix Wk such that the matrix Xk + Wk is a solution of problem (SF), i.e.:

trðAiðXk þWkÞÞ ¼ bi for i ¼ 1; . . . ;m ð1Þ

and

Xk þWk 2 intðKÞ: ð2Þ

Eq. (1) is equivalent to:

trðAiWkÞ ¼ kkqi for i ¼ 1; . . . ;m: ð3Þ

Where

qi ¼ bi � trðAiX0Þ for i ¼ 1; . . . ;m:

Eq. (3) is equivalent to the following convex optimization problem:

min kWkk2; trðAiWkÞ ¼ kkqi for i ¼ 1; . . . ;m

h i: ðCPÞ

Lemma 2. By optimality condition, the optimal solution Wk of the problem (CP) can be done as:

Wk ¼ �Xm

i¼1

hki Ai:

Where hk ¼ ðhk1; h

k2; . . . ; hk

mÞt is a solution of positive definite symmetric linear system:

Xm

i¼1

trðAiAjÞhki ¼ �kkqj; j ¼ 1; . . . ;m: ðlsÞ

4.1. The modified algorithm

Description of the algorithm

� (a’) Initialization: e > 0 is fixed, X0 = I; k0 = 1, q 2 (0,1) and k = 0.– If maxijbi � tr(AiXk)j 6 e, Stop: Xk is an e-approximate solution of (SF).– If not compute the solution h0 of the linear system (ls) and go to (b’).� (b’)

– If kk 6 e, Stop: Xk is an e-approximate solution of (SF).– If not Take hk = kkh

0 and Wk ¼ �Pm

i¼1hki Ai:

– If Xk + Wk is positive definite, Stop: Xk + Wk is an e-approximate solution of (SF).– If not go to (c’).� (c’) is identical to (c) of the algorithm [2].� (d’) Take k = k + 1, and go back to (b’).

6440 D. Benterki, A. Keraghel / Applied Mathematics and Computation 217 (2011) 6437–6440

5. Numerical tests

The algorithm has been tested on some benchmark problems issued from the library of test problems SDPLIB [9]. We havetaken q = 0.90 and the stopping criterion � = 10�8. The following table contain the number of iteration of the first phase(phase 1) corresponding to the search of an initial strictly feasible solution of the problem (SDP).

Example

Size (m,n) Nbr. of iterations classic algorithm Nbr. of iterations modified algorithm

control1

(21,15) 10 7 hinf1 (13,14) 6 5 hinf2 (13,16) 7 6 hinf3 (13,16) 6 4 hinf4 (13,16) 7 4 hinf5 (13,16) 7 5 hinf7 (13,16) 7 5 hinf9 (13,16) 5 5 hinf10 (21,18) 7 5 truss1 (6,13) 15 12 truss4 (12,19) 25 21

We have also tested the infeasible problems infd1 and infd2 of SDPLIB. Both algorithms concludes to their infeasibility.

6. Conclusion

Based on our modification, we have improved the numerical behavior of the algorithm, reducing the number of iterationscorresponding to the search of an initial strictly feasible solution of the problem (SDP). It is also possible to applied this ideato the second phase corresponding to the search of an optimal solution of the problem (SDP).

Acknowledgements

The authors thank the referees for their careful reading and their precious comments. Their help is much appreciated.

References

[1] D. Benterki, Résolution des problèmes de programmation semi-définie par la méthode de réduction du potentiel, Thèse de Doctorat, Université FerhatAbbas, Sétif, 2004.

[2] D. Benterki, J.P. Crouzeix, B. Merikhi, A numerical feasible interior point method for linear semidefinite programs, RAIRO Oper. Res. 41 (2007) 49–59.[3] D. Benterki, J.P. Crouzeix, B. Merikhi, A numerical implementation of an interior point method for semidefinite programming, Pesquisa Operacional 23

(1) (2003) 49–59.[4] D. Benterki, B. Merikhi, A modified algorithm for the strict feasibility, RAIRO Oper. Res. 35 (2001) 395–399.[5] I.J. Lustig, Feasibility issues in a primal-dual interior point method for linear programming, Math. Program. 49 (1991) 145–162.[6] B. Merikhi, Extension de quelques méthodes de points inté rieurs pour la programmation semi-définie, Université Ferhat Abbas de Sétif, Algérie, 2006.[7] M.R. Oskoorouchi, J.L. Goffin, The analytic center cutting plane method with semidefinite cuts, SIAM J. Optim. 13 (2003) 1029–1053.[8] J. Sun, K.C. Toh, J.Y. Zhao, An analytic center cutting plane method for semidefinite feasibility problems, Math. Oper. Res. 27 (2002) 332–346.[9] <http://infohost.nmt.edu/�sdplib/>.