On a General Class of Interior-Point Algorithms for

REPORTS ON COMPUTATIONAL MATHEMATICS, NO. 89/1996, DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF IOWA

On a General Class of Interior-Point Algorithms for Semide nite Programming with Polynomial Complexity and Superlinear Convergence Rongqin Sheng , Florian A. Potra  and Jun Ji y June, 1996

Abstract

We propose a uni ed analysis for a class of infeasible-start predictor-corrector algorithms for semide nite programming problems, using the Monteiro-Zhang uni ed direction. The algorithms are direct generalizations of the Mizuno-Todd-Ye predictorcorrector algorithm for linear programming. We show that the algorithms belonging to this class are globally convergent, provided the problem has a solution, and have optimal computational complexity. We also give simple sucient conditions for superlinear convergence. Our results generalize the results obtained by the rst two authors for the infeasible-interior-point algorithm proposed by Kojima, Shida and Shindoh and Potra and Sheng.

Key Words: semide nite programming, predictor-corrector, infeasible-interior-point algo-

rithm, polynomial complexity, superlinear convergence. Abbreviated Title: On a general class of algorithms for SDP.

 Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA. The work of these two authors was supported in part by NSF Grant DMS 9305760. y Department of Mathematics and Computer Science, Valdosta State University, Valdosta, GA 31698, USA.

1

1 Introduction In this paper we consider the semide nite programming (SDP) problem: (P ) minfC  X : Ai  X = bi ; i = 1; : : : ; m; X  0g; and its associated dual problem: (D) maxfbT y :

m X i=1

yiAi + S = C; S  0g;

(1.1) (1.2)

where C 2 IR nn; Ai 2 IR nn; i = 1; : : : ; m; b = (b1 ; : : : ; bm )T 2 IR m are given data, and X 2 S+n , (y; S ) 2 IR m  S+n are the primal and dual variables, respectively. Here S+n denotes the cone of all symmetric positive semide nite n  n - matrices and X  0 indicates that X 2 S+n . By G  H we denote the trace of GT H . Without loss of generality, we assume that the matrices C and Ai ; i = 1; : : : ; m, are symmetric (otherwise, replace C by (C + C T )=2 and Ai by (Ai + ATi )=2. Also, for simplicity we assume that Ai; i = 1; : : : ; m, are linearly independent. Throughout this paper we assume that both (1.1) and (1.2) have nite solutions and their optimal values are equal. Under this assumption, X  and (y; S ) are solutions of (1.1) and (1.2) if and only if they are solutions of the following nonlinear system: Ai  X = bi; i = 1; : : : ; m; (1.3a) m X yiAi + S = C; (1.3b) i=1

XS = 0; X  0; S  0: (1.3c) Over the last couple of years many interior-point methods for solving (1.3) have been investigated (cf. [1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]). In the present paper we consider a class of primal-dual interior-point algorithms for SDP that generalize the Mizuno-Todd-Ye predictor-corrector method [9]. The latter method was originally introduced for linear programming and was later generalized by Nesterov and Todd [12] for a more general class of optimization problems with self scaled cones that contains SDP. Several generalizations of the Mizuno-Todd-Ye predictor-corrector method for SDP have been recently analyzed by Lin and Saigal [7], Luo, Sturm and Zhang [8], Kojima, Shida and Shindoh [3, 4, 5], Potra and Sheng [14, 13, 15], and Zhang [19]. The algorithm proposed by Kojima, Shida and Shindoh [3] and Potra and Sheng [14] uses the Kojima-Shindoh-Hara search direction and has polynomial complexity. Also, Potra and Sheng [14] proposed a suf cient condition for the superlinear convergence of the algorithm while Kojima, Shida and Shindoh [3] established the superlinear convergence under the following three assumptions: 2

(A) SDP has a strictly complementary solution; (B) SDP is nondegenerate in the sense that the Jacobian matrix of its KKT system is nonsingular; (C) the iterates converge tangentially to the central path in the sense that the size of the neighborhood in which the iterates reside must approach zero, namely, lim k(X k )1=2 S k (X k )1=2 ? (X k  S k =n)I kF =(X k  S k =n) = 0:

k!1

These were the rst two papers investigating the local convergence properties of interior-point algorithms for semide nite programming. More recently, Kojima, Shida and Shindoh [5] proposed a predictor-corrector algorithm using the Alizadeh-Haeberly-Overton search direction, and proved the quadratic convergence of the algorithm under assumptions (A) and (B), but the algorithm does not seem to be polynomial. Using the Nesterov-Todd search direction, Luo, Sturm and Zhang [8] investigated a symmetric primal-dual path following algorithm, which was proposed originally by Nesterov and Todd[12] and derived dierently in [16]. They proved the superlinear convergence under assumptions (A) and (C), and then dropped (C) by enforcing it in later iterations. In a recent paper, Potra and Sheng [15] proved the superlinear convergence of the infeasibleinterior-point algorithm of Kojima, Shida and Shindoh [3] and Potra and Sheng [14], under assumption (A) and a weaker condition than (C), namely, (D)

p

lim X k S k = X k  S k = 0: k!1

In a very recent paper, Monteiro and Zhang [11] proposed a uni ed analysis for a class of long-step interior-point algorithms for SDP. In what follows, we will call the uni ed direction the Monteiro-Zhang direction. Using this direction, we propose a uni ed analysis for a class of infeasible-start predictor-corrector algorithms which generalize the Mizuno-Todd-Ye predictor-corrector algorithm for linear programming. By extending the analysis of Potra and Sheng [14, 15], we show that this class of predictor-corrector algorithms shares similar global and local convergence properties with one of its members { the infeasible-interiorpoint algorithm proposed earlier by Kojima, Shida and Shindoh and Potra and Sheng, as long as the condition number of each matrix Jxk , which commutes with (X k )1=2 S k (X k )1=2 and de nes the scaling matrix P k by (P k )T P k = (X k )?1=2 Jxk (X k )?1=2 , is bounded. In particular we prove polynomial complexity for general problems and superlinear convergence for problems satisfying assumptions (A) and (D). 3

The following notation and terminology are used throughout the paper:

IR p : the p-dimensional Euclidean space; IR p+ : the nonnegative orthant of IR p ; IR p++ : the positive orthant of IR p; IR pq : the set of all p  q matrices with real entries; S pp: the set of all p  p symmetric matrices; S+p : the set of all p  p symmetric positive semide nite matrices; S++: the set of all p  p symmetric positive matrices; [M ]ij : the (i; j )-th entry of a matrix M; Tr(M ): the trace of a p  p matrix, Tr(M ) = Ppi=1[M ]ii ;

M  0: M is positive semide nite; M  0: M is positive de nite; i(M ); i = 1; : : : ; n: the eigenvalues of M 2 S n ; max(M ); min(M ): the largest, smallest, eigenvalue of M 2 S n ; G  H  Tr(GT H ); k
k: Euclidean norm of a vector and the corresponding norm of a matrix, i.e., qP p 2 kyk  qi=1 yi ; kM k  maxfkMyk : kyk = 1g ; kM kF  Ppi=1 Pqj=1[M ]2ij ; M 2 IR pq : Frobenius norm of a matrix; M k = o(1): kM k k ! 0 as k ! 1; M k = O(1): kM k k is bounded; M k = (1): 1=?  kM k k  ? for some constant ? > 0; M k = o(k ): M k =k = o(1); M k = O(k ): M k =k = O(1); M k = (k ): M k =k = (1):

2 The uni ed direction We denote the feasible set of the problem (1.3) by F = f(X; y; S ) 2 S+n  IR m  S+n : (X; y; S ) satis es (1:3a) and (1:3b)g and its solution set by F , i.e., F  = f(X; y; S ) 2 F : X  S = 0g: The residues of (1.3a) and (1.3b) are denoted by: Ri = bi ? Ai  X; i = 1; : : : ; m; 4

(2.1a)

Rd = C ?

m X i=1

yiAi ? S:

(2.1b)

For any given  > 0 we de ne the set of -approximate solutions of (1.3) as F = fZ = (X; y; S ) 2 S+n  IR m  S+n : X  S  ; jRij  ; i = 1; : : : ; m; kRd k  g: We consider the symmetrization operator i h HP (M ) = 21 PMP ?1 + (PMP ?1)T ; 8M 2 IR nn: introduced by Zhang [19]. Since, as observed by Zhang, HP (M ) = I i M = I; for any nonsingular matrix P , any matrix M with real spectrum and any  2 IR it follows that for any given nonsingular matrix P , (1.3) is equivalent to Ai  X = bi ; i = 1; : : : ; m; (2.2a) m X yiAi + S = C; (2.2b) i=1

HP (XS ) = 0; X  0; S  0: (2.2c) n S n we de ne the set of permissible matrices associated Following [11], for any (X; S ) 2 S++ ++ with (X; S ) IP (X; S ) = fP : P 2 IR nn is nonsingular and PXSP ?1 2 S n g: n  IR m  S n . For every P 2 IP (X; S ), a search direction (U; w; V ) 2 Let (X; y; S ) 2 S++ ++ m S n  IR  S n is de ned by the following linear system HP (US + XV ) = I ? HP (XS ); (2.3a) Ai  U = (1 ?  )Ri; i = 1; : : : ; m; (2.3b) m X wiAi + V = (1 ?  )Rd; (2.3c) i=1

where  > 0 is a parameter, and  2 [0; 1]. The choice of  = 0 de nes the predictor direction while  = 1 corresponds to a corrector or centering direction. Using Proposition 3.1 and Theorem 3.1 of [17] it follows that for every P 2 IP (X; S ) the system (2.3) has a unique symmetric solution (U; w; V ) 2 S n  IR m  S n . The following characterization of IP (X; S ) will be frequently used in our analysis. 5

n , Then, Lemma 2.1 Let X; S 2 S++

IP (X; S )

n g (2.4) = fP : P T P = X ?1=2 JxX ?1=2; Jx(X 1=2 SX 1=2 ) = (X 1=2 SX 1=2)Jx; Jx 2 S++ n g: = fP : P T P = S 1=2 JsS 1=2; Js(S 1=2 XS 1=2 ) = (S 1=2 XS 1=2)Js; Js 2 S++ (2.5)

Proof. Observe that

is equivalent to

PXSP ?1 = (PXSP ?1)T

(2.6)

[X 1=2P T PX 1=2][X 1=2 SX 1=2 ] = [X 1=2 SX 1=2 ][X 1=2 P T PX 1=2]: Then (2.4) is proved by letting Jx = X 1=2 P T PX 1=2 . Equality (2.5) can be proved similarly by observing that (2.6) is also equivalent to [S ?1=2 P T PS ?1=2][S 1=2 XS 1=2 ] = [S 1=2 XS 1=2 ][S ?1=2P T PS ?1=2]: We mention that the above lemma can also be derived from Proposition 3.4 and Theorem 3.1 of Monteiro and Zhang [11]. n  S n ! IR nn such that In the sequel, we assume that we are given a mapping P : S++ ++ n  Sn : P (X; S ) 2 IP (X; S ) , for any (X; S ) 2 S++ ++ n such that If P = P (X; S ) then from Lemma 2.1 it follows that there are Jx; Js 2 S++ P T P = X ?1=2JxX ?1=2 = S 1=2 JsS 1=2 . Moreover, Jx commutes with X 1=2 SX 1=2 and Js commutes with S 1=2 XS 1=2 .

Lemma 2.2 For any P 2 IP (X; S ), there exist orthogonal matrices Qx and Qs such that P = QxJx1=2 X ?1=2 = QsJs1=2 S 1=2: Proof. Since

we have and

P T P = X ?1=2JxX ?1=2 = S 1=2 JsS 1=2; [Jx1=2 X ?1=2 P ?1]T [Jx1=2 X ?1=2 P ?1] = I; [Js1=2 S 1=2 P ?1]T [Js1=2 S 1=2 P ?1] = I: 6

The lemma is proved by taking QTx = Jx1=2 X ?1=2 P ?1 and QTs = Js1=2 S 1=2 P ?1. Note that Jx = Js = I and P = X ?1=2 or S 1=2 de ne the directions formulated by Monteiro [10] which are particular cases of the direction originally proposed by Kojima, Shindoh and Hara [6]. The direction de ned by Jx = Js = I and P = S 1=2 was derived independently by Helmberg, Rendl, Vanderbei and Wolkowicz [2]. Finally, the case Jx = [X 1=2 SX 1=2]1=2 ; or Js = [S 1=2 XS 1=2 ]?1=2 corresponds to the Nesterov-Todd direction [12] (see [17] and [16] ).

3 A class of predictor-corrector algorithms In this section, we propose an infeasible-interior-point predictor-corrector algorithm for solving (1.3), which generalizes the interior{point method for linear programming proposed by Mizuno, Todd and Ye [9]. The algorithm performs in a neighborhood of the infeasible central path: n  IR m  S n : C ( ) = fZ = (X; y; S ) 2 S++ ++ 0 XS = I; Ri = (=0 )Ri ; i = 1; : : : ; m; Rd = (=0 )Rd0 g: The positive parameter  is driven to zero and therefore the residues are also driven to zero at the same rate as  . The iterates reside in the following neighborhood of the above central path: n  IR m  S n : kH (XS ) ? I k   g N ( ;  ) = f(X; y; S ) 2 S++ P ++ F m n n ? 1 = f(X; y; S ) 2 S++  IR  S++ : kPXSP ? I kF   g =

n  IR m  S n f(X; y; S ) 2 S++ ++

:

m X

(i(XS ) ?  )2

!1=2

i=1 m n n f(X; y; S ) 2 S++  IR  S++ : kX 1=2 SX 1=2 ? I kF

  g   g;

= where is a constant such that 0 < < 1. It is interesting to note that the neighborhood N ( ;  ) is independent on the scaling matrix P . Throughout the paper we also use the notation:  = (X  S )=n: (3.1) In order to de ne the neighborhood size in our algorithm, we need to assume that the spectral condition numbers of Jx or Js are bounded, that is,   sup minfx; sg < 1; (3.2) 7

where x = kJxkkJx?1k; s = kJskkJs?1 k and the supremum is taken over all matrices P that are used in our algorithm. Our algorithm depends on two positive parameters ; satisfying the inequalities

p 2 p < 1: p   < < (3.3a) 2 2(1 ?  ) p 1 ?  (3.3b) ?   (1= ): p p For example,  = 0:25= ; = 0:41=  verify (3.3). At a typical step of our algorithm we are given (X; y; S ) 2 N (;  ) and obtain a predictor direction (U; w; V ) 2 S n  IR m  S n by solving the linear system HP (US + XV ) = ?HP (XS ); (3.4a) Ai  U = Ri ; i = 1; : : : ; m; (3.4b) m X i=1

wi A i + V = R d :

(3.4c)

As we mentioned before the above linear system has a unique symmetric solution, which we call the ane scaling direction. If we take a steplength  along this direction we obtain the points X () = X + U; y() = y + w; S () = S + V: Theoretically we would like to compute the step length 8 !1=2 n < X 2  = max :~ 2 [0; 1] : (i(X ()S ()) ? (1 ? ) ) i=1

 (1 ? );

9 = 8  2 [0; ~]; :

(3.5) However this involves computing the root of a complicated nonlinear equation. In Lemma 3.5 we will show that   b (3.6) where 2 ; (3.7) b  q 1 + 4=( ? ) + 1 and   1 kPUV P ?1kF : (3.8) 8

Actually, b is the positive root of 2 + ( ? ) ? ( ? ) = 0. In what follows we assume that a steplength  satisfying     b (3.9) is computed, and we consider the predicted points X = X + U; y = y + w; S = S + V: (3.10) In case  = 1 (which is very unlikely), it is easily seen that (X; y; S ) 2 F  and therefore the algorithm terminates with an exact solution. Now suppose that  < 1. Then X and S are symmetric positive de nite matrices since i(X ()S ())  (1 ? )(1 ? ) > 0; i = 1; : : : ; n; 8 2 [0; ]. Therefore we can de ne the corrector direction (U; w; V ) as the unique symmetric solution of the following linear system (3.11a) HP (US + XV ) = (1 ? )I ? HP (XS ); Ai  U = 0; i = 1; : : : ; m; (3.11b) m X wi Ai + V = 0; (3.11c) i=1

where P = P (X; S ): By taking a unit steplength along this direction we obtain a new point X + = X + U; y+ = y + w; S + = S + V : (3.12) Clearly (3.13) Ri+ = (1 ? )Ri; i = 1; : : : ; m; Rd+ = (1 ? )Rd : Correspondingly, we de ne  + = (1 ? ): (3.14) Summarizing, we can formally de ne our algorithm as follows: Algorithm 3.1 Choose (X 0; y0; S 0) 2 N (; 0) with 0 = 0 = (X 0  S 0)=n and set 0 = 1. For k = 0; 1;


, do A1 through A5: A1 Set X = XPk , y = y k , S = S k and de ne Rd = C ? mi=1 yiAi ? S; Ri = bi ? Ai  X; i = 1; : : : ; m: A2 If maxfX  S; kRd k; jRi j; i = 1; : : : ; m:g  ; then report (X; y; S ) 2 F and terminate. A3 Find the unique symmetric solution U; w; V of the linear system (3.4), de ne X; y; S as in (3.10), and set + = (1 ? ) , for a  satisfying (3.9). If  = 1, then report (X; y; S ) 2 F  and terminate. 9

A4 Find the unique symmetric solution U; w; V of the linear system (3.11) and de ne X +; y+; S +;  + as in (3.12) and (3.14). A5 Set X k+1 = X +; S k+1 = s+; k+1 = + ; k = ; k+1 = + ; Rdk = Rd ; Rik = Ri; i = 1; : : : ; m:

In analysing our algorithm we need the following technical results of Monteiro [10, Lemma 2.6]. Lemma 3.2 Suppose that M 2 IR pp is a nonsingular matrix and E 2 IR pp has at least one real eigenvalue. Then, max(E )  12 max(MEM ?1 + (MEM ?1 )T ); (3.15) (3.16) min(E )  12 min(MEM ?1 + (MEM ?1 )T ); (3.17) kGkF  21 kMGM ?1 + (MGM ?1 )T kF ; 8G 2 S p : Lemma 3.3 Let n(X;n y; Sm) 2 Nn( ;n  ) for some 2 [0; 1=p) and  > 0. Suppose that (Dx;
y; Ds) 2 IR  IR  IR is a solution of the linear system: HP (DxS + XDs) = H; (3.18a) Ai  Dx = 0; i = 1; : : : ; m; (3.18b) m X
yi Ai + Ds = 0; (3.18c) i=1

for some H 2 IR nn . Then, the following three statements hold: (a) if x  s, then

kX 1=2 D

2 sX 1=2 kF

+  2 kX ?1=2DxX ?1=2 k2F

(b) if s  x, then

kS 1=2 DxS 1=2 k2F

+  2 kS ?1=2D

pkH k2 F : p (c) kPDx DsP ?1 kF  2(1 ?  )2

10

2 k H k F ;  (1 ? p )2

2 k H k F p F  (1 ?  )2 ;

2 s S ?1=2 k

Proof. Let us prove (a). Suppose that x  s , then x  . Writing

2H = 2HP (XDs + DxS ) = PXDsP ?1 + PDxSP ?1 + [PXDsP ?1 + PDxSP ?1]T = PXDsP ?1 + PDxX ?1P ?1 + PDxX ?1P ?1(PXSP ?1 ? I ) +[PXDsP ?1 + PDxX ?1P ?1 + PDxX ?1P ?1(PXSP ?1 ? I )]T = B + B T + PDxX ?1P ?1(PXSP ?1 ? I ) +[PDxX ?1P ?1(PXSP ?1 ? I )]T ; where

B = PXDsP ?1 + PDxX ?1P ?1;

we have

2kH kF  kB + B T kF ? 2kPDxX ?1P ?1(PXSP ?1 ? I )kF  kB + B T kF ? 2kPDxX ?1P ?1kF kPXSP ?1 ? I kF  kB + B T kF ? 2  kPDxX ?1P ?1kF According to Lemma 2.2, P = QxJx1=2 X ?1=2 , so that we have

kPDxX ?1P ?1kF = kQxJx1=2 X ?1=2DxX ?1X 1=2 Jx?1=2 QTx kF = kJx1=2 X ?1=2DxX ?1X 1=2 Jx?1=2 kF  kpJx1=2 kkJx?1=2kkX ?1=2 DxXp?1=2kF =  kX ?1=2D X ?1=2 k  kX ?1=2 D X ?1=2k : x

x

Using Lemma 3.2 with M = PX 1=2 and

x

F

F G = X 1=2 DsX 1=2 + X ?1=2 DxX ?1=2 ,

we obtain

kB + B T kF  2kX 1=2 DsX 1=2 + X ?1=2 DxX ?1=2kF = 2(kX 1=2 DsX 1=2k2F +  2 kX ?1=2DxX ?1=2 k2F )1=2 (since Dx  Ds = 0): Therefore,

kH kF  21 kB + B T kF ?  kPDxX ?1P ?1kF  (kX 1=2 DsX 1=2 k2F +  2 kX ?1=2DxX ?1=2 k2F )1=2 ? p  kX ?1=2 DxX ?1=2 kF  (kX 1=2 DsX 1=2 k2F +  2 kX ?1=2DxX ?1=2 k2F )1=2 (1 ? p); 11

which proves (a). We can prove (b) similarly. To prove (c), we may assume x  s without loss of generality, and deduce kPDxDsP ?1kF = kQxJx1=2 X ?1=2 DxDsX 1=2 Jx?1=2 QTx kF = kJx1=2 X ?1=2 DxDsX 1=2Jx?1=2 kF  [pkJx1=2 kkJx?1=2 k]kX ?1=2DxDsX 1=2 kF  [ kX ?1=2 DxX ?1=2 kF ][kX 1=2DsX 1=2 kF ]=  p(1=2)[ 2 kX ?1=2DxX ?1=2 k2F + kX 1=2 DsX 1=2 k2F ]= pkH k2  2(1 ? pF)2 ( from (a)): The next corollary will be essentially used in the proof of global and local convergence properties of Algorithm 3.1. Corollary 3.4 Under the hypothesis of Lemma 3.2, )kH kF ; kX 1=2 DsX 1=2 kF  (11+? p  H kF  kX ?1=2 DxX ?1=2 kF  (1 ? pk : )(1 ? ) Proof. If x  s, then the results follow immediately from (a) of Lemma 3.2. Suppose x  s. Then from (b) of Lemma 3.2, we obtain kS 1=2DxS 1=2 kF  1 k?HpkF ;  kS ?1=2 DsS ?1=2kF  1 k?HpkF : Hence, kX 1=2 DsX 1=2 kF = k[X 1=2 S 1=2 ]S ?1=2DsS ?1=2 [S 1=2 X 1=2 ]kF  kX 1=2 S 1=2k2 kS ?1=2DsS ?1=2 kF = max(X 1=2 SX 1=2 )kS ?1=2DsS ?1=2 kF )kH kF :  (11+? p  12

 kX ?1=2 DxX ?1=2 kF =  k[X ?1=2 S ?1=2 ]S 1=2 DxS 1=2[S ?1=2 X ?1=2 ]kF   kX ?1=2 S ?1=2 k2kS 1=2 DxS 1=2 kF =  kS 1=2 DxS 1=2 kF =min(X 1=2SX 1=2 ) H kF  (1 ? pk : )(1 ? ) The next lemma justi es our de nition of steplength  in the algorithm. Lemma 3.5 If (X; y; S ) 2 N (;  ), then  2 [0; 1] de ned by (3.5) satis es   ^, where b is given by (3.7) and (3.8). Proof. By de nition, we have

X ()S () ? (1 ? )I = (X + U )(S + V ) ? (1 ? )I = (1 ? )(XS ? I ) + XS + (XV + US ) + 2UV:

(3.19)

If we set

R()  P [X ()S () ? (1 ? )I ]P ?1 ; then, in view of (3.19) and (3.4a), we obtain R() + R()T = 2(1 ? )(HP (XS ) ? I ) +[2HP (XV + US )PX 1=2 + 2HP (XS )] + 2 [PUV P ?1 + (PUV P ?1)T ] = 2(1 ? )(X 1=2 SX 1=2 ? I ) + 2 [PUV P ?1 + (PUV P ?1)T ]: Therefore, 1 kR() + R()T k F 2  (1 ? )kX 1=2 SX 1=2 ? I kF + 2kPUV P ?1kF   (1 ? ) + 2 :

(3.20)

Hence, for any given parameter  2 [0; 1), we must have X ()  0; S ()  0 for all  2 [0; min(;b  )). Otherwise, there must exist 0  0  min(;b  )   < 1 such that X (0)S (0) is singular, which means

min(X (0)S (0) ? (1 ? 0) )  ?(1 ? 0): 13

(3.21)

However, using (3.16) with M = P and E = X (0)S (0) ? (1 ? 0) , we have

min(X (0)S (0) ? (1 ? 0 ) )  21 min(R(0) + R(0 )T )  ? 21 kR(0 ) + R(0)T kF  ?[ (1 ? 0 ) + (0)2  ] (from (3:20))  ? (1 ? 0); which contradicts (3.21). Since X ()  0, its square root X ()1=2 exists and is uniquely de ned. Applying (3.17) of Lemma 3.2 with G = X ()1=2 S ()X ()1=2 ? (1 ? )I , M = PX ()1=2, and noting that R() = MGM ?1 , we obtain n X i=1

(i(X ()S ()) ? (1 ? ) )2

!1=2

= kX ()1=2 S ()X ()1=2 ? (1 ? )I kF  21 kR() + R()T kF   (1 ? ) + 2  (from (3:20))  (1 ? ); for all   :b Therefore, (X (); y(); S ()) 2 N ( ; (1 ? ) ), for all  2 [0; min(;b  )]. If b < 1 we can choose  = b, which gives   b. Finally, if b = 1, then (X (); y(); S ()) 2 N ( ; (1 ? ) ), for all  2 [0; 1), which implies X (1)  0; S (1)  0 and X (1)S (1) = 0, and therefore  = 1 = b. Before stating our main result let us note that the standard choice of starting points

X 0 = pI; y0 = 0; S 0 = d I is perfectly centered and satis es (X 0; y0; S 0) 2 N (; 0), as required in the algorithm. We will see that if the problem has a solution, then for any  > 0 Algorithm 3.1 terminates in a nite number (say K) of iterations. If  = 0 then the algorithm is likely to generate an in nite sequence. However it may happen that at a certain iteration (let us say at iteration K0) we have  = 1, which implies that an exact solution is obtained, and therefore the algorithm terminates at iteration K0. If this (unlikely) phenomenon does not happen we set K0 = 1. 14

Theorem 3.6 For any integer 0  k < K0 , Algorithm 3.1 de nes a triple (X k ; yk ; S k ) 2 N (; k) and the corresponding residuals satisfy Rdk = k Rd0 ; Rik = k Ri0 ; i = 1; : : : ; m; k = k 0 ; (1 ? )k  k = (X k  S k )=n  (1 + )k where kY ?1 (1 ? j ); 0 = 1; k= j =0

(3.22) (3.23) (3.24) (3.25)

and j is de ned by (3.9). Proof. The proof is by induction. For k = 0, (3.22){(3.25) are clearly satis ed. Suppose they are satis ed for some k  0. As in Algorithm 3.1 we will omit the index k. Therefore we can write (X; y; S ) 2 N (;  ); Rd = Rd0 ; Ri = Ri0; i = 1; : : : ; m;  = 0; (1 ? )    (1 + ): The fact that (3.23) and (3.24) hold for k + 1 follows immediately from (3.13) and (3.14). From (3.12) and (3.11a) we have X +S + ? + I = (X + U )(S + V ) ? +I = XS ? (1 ? )I + XU + US + UV : (3.26) Let P = P (X; S ); and de ne B = P (X +S + ? +I )P ?1: Then, recalling (3.26) and (3.11a), we obtain B + B T = 2[HP (XS ) ? (1 ? )I ] +[2HP (US + XV )] + [PUV P ?1 + (PUV P ?1 )T ] = PUV P ?1 + (PUV P ?1)T : (3.27)

15

Since k < K0, we see that  < 1 and that X  0; S  0. Applying (c) of Lemma 3.3, we deduce

kPUV P ?1kF pkH (XS ) ? (1 ? ) k2  2(1 P? p )2(1 ? ) F p 2(1 ? )  2(1 ? p )2 (3.28)  (1 ? ) = + (from (3:3)): Without loss of generality, we may assume x  s. Hence by applying (a) of Lemma 3.3 with H = (1 ? )I ? HP (XS ), we have kX ?1=2 UX ?1=2 kF p < 1; P (XS ) ? (1 ? )I kF  kH(1  p 1 ?  ?  )(1 ? ) which implies that I + X ?1=2 UX ?1=2  0, and therefore, X + = X + U  0. Thus (X +)1=2 exists. Using (3.26), (3.27), applying Lemma 3.2 with G = (X +)1=2 S +(X +)1=2 ? +I , M = P (X +)1=2 , and noting that B = MGM ?1 , we have k(X +)1=2 S +(X +)1=2 ? +I kF  21 kB + B T kF = 21 kPUV P ?1 + [PUV P ?1]T kF (from (3.27))  kPUV P ?1kF  + (from (3:28)): (3.29) The above inequality implies that

min((X +)1=2 S +(X +)1=2 )  (1 ? )+ > 0: Hence (X +)1=2 S +(X +)1=2  0, which gives S +  0. In view of (3.29), this shows that (3.22) holds for k + 1. Finally, (3.25) is an immediate consequence of (3.22).

16

4 Global convergence and iteration complexity

In this section we assume that F  is nonempty. Under this assumption we will prove that Algorithm 3.1, with  = 0, is globally convergent in the sense that lim  k!1 k

= 0; klim Rk = 0; !1 d

lim Rk k!1 i

= 0; i = 1; : : : ; m: In the sequel, we will frequently use the following well-known inequality:

kM1 M2 kF  minfkM1kkM2 kF ; kM1 kF kM2 kg; for any M1 ; M2 2 IR nn:

(4.1)

f ye; Se) 2 F , we have Lemma 4.1 (Potra-Sheng [14], Lemma 3.1) For any (X; f (S  X 0 + S 0  X ) = S  X + 2 S 0  X 0 + (1 ? )2Se  X f + Se  X 0 ) ? (1 ? )(S  X f + Se  X ): + (1 ? )(S 0  X Lemma 4.2 (Potra-Sheng[14], Lemma 3.2) Assume that F  is nonempty. Then for any (X ; y; S ) 2 F  and (X; y; S ) 2 N (;  ) we have X  S 0 + X 0  S  (2 +  +  )n0 ; (4.2a)   X  S + X  S  ((1 +  + )=(1 ? ) +  )n ; (4.2b)

where

 = (X 0  S  + X   S 0 )=(X 0  S 0):

(4.3)

Lemma 4.2 shows that the pair (X k ; S k ) generated by Algorithm 3.1 is bounded. More precisely, we have the following corollary, which can easily be deduced from Lemma 4.2 and Theorem 3.6.

Corollary 4.3 (Potra-Sheng [14], Corollary 3.3) Under the hypothesis of Lemma 4.2 we have

q

kX 1=2(S 0)1=2 k

 (2 +  +  )n0; q kS 1=2(X 0)1=2 kF  (2 +  +  )n0; q 1 = 2 0 ? 1 = 2 kX kF  k(S ) k (2 +  +  )n0; F

17

(4.4) (4.5) (4.6)

q

kS 1=2 kF  k(X 0)?1=2 k (2 +  +  )n0; kX 1=2 S 1=2 k2 = kX 1=2 SX 1=2k  (1 + ); kX ?1=2S ?1=2 k2 = kX ?1=2S ?1X ?1=2 k  (1 ?1) :

(4.7) (4.8) (4.9)

We note that the parameter  de ned in (3.3) must satisfy

p

  < 0:5 and  < 0:5; which will be frequently used in our analysis.

Lemma 4.4 Let (X; y; S ) 2 N (;  ); P = P (X; S ). Then, (i) maxfx ; s g  3; p (ii) for any M 2 IR nn ; kHP (M )kF  kPMP ?1kF  3kX ?1=2 MX 1=2 kF : Proof. Since P T P = X ?1=2 Jx X ?1=2 = S 1=2 JsS 1=2 , we have

Jx = [X 1=2 S 1=2 ]Js[X 1=2 S 1=2]T ; and Therefore we obtain,

Jx?1 = [(X 1=2 S 1=2 )?1]T Js?1[X 1=2 S 1=2 ]?1:

kJxk  kJskkX 1=2S 1=2 k2 = kJskmax(X 1=2 SX 1=2 )  (1 + ) kJsk; kJx?1k  kJs?1 kk(X 1=2S 1=2 )?1k2 = kJs?1 k=min(X 1=2 SX 1=2)  kJs?1 k=((1 ? ) ): Hence, Similarly we can prove

+  kJ kkJ ?1k  3k : kx = kJxkkJx?1 k  11 ? s  s s

(4.10)

ks  3kx:

(4.11)

18

Then (4.10) and (4.11) imply (i). (ii) can be proved by noting that kHP (M )kF  kPMP ?1kF = kQxJx1=2 X ?1=2 MX 1=2 Jx?1=2 QTx kF = kJx1=2 X ?1=2 MX 1=2 Jx?1=2 kF p kX ?1=2MX 1=2 k  p x F ? 1 = 2 1 = 2  3kX MX kF : f ye; Se) 2 F , and denote Lemma 4.5 Suppose (X; T = [X 1=2(S 0 ? Se)X 1=2 + X ?1=2 (X 0 ? Xf)SX 1=2 ] ? X 1=2 SX 1=2; Tx = X ?1=2 (X 0 ? Xf)X ?1=2 ; Ts = X 1=2 (S 0 ? Se)X 1=2 :

Then the quantity  de ned by (3.8) satis es the inequality:

p

   p p   32  kTxkF + 4 3kT kF kTskF + 3 3kT kF : f); w + (y 0 ? ye); V + (S 0 ? Se)) satis es Proof. It is easily seen that (U + (X 0 ? X (3.18) with H = HP (M ) where M = (X (S 0 ? Se) + (X 0 ? Xf)S ) ? XS: Hence, according to Corollary 3.4, we have H kF  kX ?1=2 (U + (X 0 ? Xf))X ?1=2kF  (1 ? pk )(1 ? )  4kH kF ; H kF  3kH k : kX 1=2(V + (S 0 ? Se))X 1=2kF  (11+?p)k F By (ii) of Lemma 4.4, we have p kH kF = kpHP (M )kF  3kX ?1=2 MX 1=2 kF = 3kT kF :

19

Therefore

p

 kX ?1=2 UX ?1=2 kF   kTxkF + 4kH kF   kTxkF + 4 3kT kF ;

p kX 1=2 V X 1=2kF  kTskF + 3kH kF  kTskF + 3 3kT kF :

Again, using (ii) of Lemma 4.4, we deduce

p

 = kpPUV P ?1k=  3kX ?1=2 UV X 1=2 kF =  p3[ kX ?1=2 UX ?1=2 kF ][kX 1=2 V X 1=2kF ]= 2    p p  32  kTx kF + 4 3kT kF kTskF + 3 3kT kF :

Lemma 4.6 Under the hypothesis of Lemma 4.5 we have 

p

2

 < 3:51:5 42:6(2:5 +  )nd0 + 7:8 n ;

(4.12)

where

d0 = max(k(X 0)?1=2 (X 0 ? Xf)(X 0)?1=2 kF ; k(S 0)?1=2(S 0 ? Se)(S 0)?1=2 kF ): Proof. Using the notation of Lemma 4.5, and Lemma 4.3, we can write

kT kF  kX 1=2 (S 0 ? Se)X 1=2 kF + kX ?1=2(X 0 ? Xf)SX 1=2kF + kX 1=2 SX 1=2kF = kX 1=2 (S 0)1=2 (S 0)?1=2 (S 0 ? Se)(S 0 )?1=2(S 0 )1=2X 1=2 kF f)(X 0 )?1=2 (X 0 )1=2 S 1=2 S 1=2 X 1=2 k + kX ?1=2S ?1=2 S 1=2 (X 0)1=2 (X 0)?1=2 (X 0 ? X F 1 = 2 1 = 2 +kX SX kF  kX 1=2 (S 0)1=2 k2k(S 0)?1=2 (S 0 ? Se)(S 0 )?1=2kF f)(X 0 )?1=2 k + kX ?1=2S ?1=2 kkS 1=2(X 0 )1=2 k2 kX 1=2S 1=2 kk(X 0)?1=2 (X 0 ? X F p + nkX 1=2 SX 1=2k  (2 +  +  )n0d0(1 + 11 ?+  ) + pn(1 + ) p =  [2(2 +  +  )nd0=(1 ? ) + (1 + ) n] p <  [2(5 + 2 )nd0 + 1:5 n]:

20

Also, kTxkF 

kX ?1=2 S ?1=2S 1=2 (X 0)1=2 (X 0)?1=2 (X 0 ? Xf)(X 0)?1=2 (X 0)1=2 S 1=2 S ?1=2 X ?1=2kF  kX ?1=2 S ?1=2k2 kS 1=2(X 0)1=2 k2 d0  (2 +(1?+))n0 d0

and

= (2 +  +  )n d0  (5 + 2 )nd0; 1?

kTskF    

kX 1=2 (S 0)1=2 (S 0)?1=2 (S 0 ? Se)(S 0)?1=2 (S 0)1=2 X 1=2 k kX 1=2 (S 0)1=2 k2d0

(2 +  +  )n0d0 = (2 +  +  )nd0 (2:5 +  )nd0: Then (4.12) follows from Lemma 4.5. According to Lemma 3.5 and Lemma 4.6, it follows that if F  is not empty, then the step length k de ned by (3.9) is bounded away from 0. This implies global convergence as shown in the following theorem. Theorem 4.7 If F  is not empty, then Algorithm 3.1 is globally convergent at a linear rate. Moreover, the iteration sequence (X k ; y k ; S k ) is bounded and every accumulation point of (X k ; yk ; S k ) belongs to F  (i.e., is a primal dual optimal solution of the SDP problem). Using Lemma 4.6, we can easily deduce the following result. Theorem 4.8 Suppose that F  is nonempty and that the starting point is chosen such that there is a constant  independent of n satisfying the inequality f)(X 0 )?1=2 k ; k(S 0 )?1=2 (S 0 ? Se)(S 0 )?1=2 k )  n?1=2  : (2:5 +  ) max(k(X 0)?1=2(X 0 ? X F F p Then Algorithm 3.1 terminates in at most O( n ln(0 =)) iterations, where 0 = maxfX 0  S 0; kRd0k; jRi0j; i = 1; : : : ; mg: (4.13) Corollary 4.9 Suppose that F  is nonempty and that the startingppoint is feasible, i.e., (X 0; y0; S 0) 2 F . Then Algorithm 3.1 terminates in at most O( n ln(0 =)) iterations, where 0 is de ned by (4.13). 21

Theorem 4.10 Suppose X 0 = S 0 = I , where  > 0 is a constant such that kX k  ; kS  k   for some (X ; y; S ) 2 F . Then the step length k de ned by (3.9) satis es

the inequality

1 k > 95n=p ! + 1;

p

(4.14)

where ! is a constant such that ?   != . Proof. According to Lemma 4.2, we have

(Tr(X ) + Tr(S ))  (2 +  +  )n0 = (2 +  +  )n2; i.e.,

n X i=1

(i(X ) + i(S ))  (2 +  +  )n:

In the sequel we will frequently use the fact that  < 0:5. Since X   S  = 0 we get the relation

 = (S   X 0 + X   S 0)=(X 0  S 0 ) = (Tr(X  ) + Tr(S ))=(n)  1; which implies

n X

kX 1=2k2F + kS 1=2k2F = (i(X ) + i(S ))  (3 + )n:

It is easily seen that

i=1

(4.15)

kX 0 ? X k   and kS 0 ? S  k  :

(4.16) f ye; Se) = (X  ; y ; S  ) we have Applying (4.15), (4.16), Corollary 4.3, and Lemma 4.5 with (X;

kX 1=2(S 0 ? S  )X 1=2kF  kX 1=2 k2F kS 0 ? S k  3:52 n;

(4.17)

kX ?1=2(X 0 ? X  )SX 1=2kF = k(X ?1=2S ?1=2 )S 1=2 (X 0 ? X )S 1=2 (S 1=2 X 1=2)kF  kX ?1=2 S ?1=2kkS 1=2 X 1=2kkS 1=2 k2F kX 0 ? X k = kX ?1=2 S ?1X ?1=2k1=2 kX 1=2 SX 1=2k1=2 kS 1=2 k2F kX 0 ? X k  6:12n: (4.18) 22

In view of (4.17), (4.18) and Corollary 4.3, we obtain kT kF  kX 1=2 (S 0 ?pS )X 1=2 kF + kX ?1=2 (X 0 ? X )SX 1=2kF + kX 1=2SX 1=2 kF <  [9:6n + 1:5 n]  11:1n: (4.19)

 kTx kF =  kX ?1=2 (X 0 ? X )X ?1=2kF =  k(X ?1=2 S ?1=2)S 1=2 (X 0 ? X )S 1=2 (S ?1=2 X ?1=2)kF   kS 1=2 k2F kX ?1=2 S ?1=2k2 kX 0 ? X  k  7n:

(4.20)

kTskF = kX 1=2 (S 0 ? S )X 1=2 kF  kX 1=2 k2F kS 0 ? S k  3:5n:

(4.21)

Therefore,

Consequently,

p

   p p   32  kTxkF + 4 3kT kF kTskF + 3 3kT kF < 8890:61:5n2 :

2 1 + 4=( ? ) + 1 1 > q =( ? ) + 1 1 > 95n=p ! + 1:

  ^ = q

In the following corollary we summarize the complexity results for standard starting point of the form X 0 = S 0 = I . Corollary 4.11 Assume that in Algorithm 3.1 we choose a starting point of the form X 0 = S 0 = I , where  > 0 is a constant. Let 0 be given by (4.13) and let  > 0 be arbitrary. Then the following statements hold: 23

(i) If F  6= ;, then the algorithm terminates with an -approximate solution (X k ; y k ; S k ) 2 F in a nite number of steps k = K < 1. (ii) If   maxfkX  k; kS kg, for some (X  ; y ; S  ) 2 F  then K = O(n ln(0 =)). c = O(n ln(0 =)) such that either (iii) For any choice of  > 0 there is an index k = K (iiia) (X k ; y k ; S k ) 2 F, or, p (iiib)   1=(95n= ! + 1), and in the latter case there is no solution (X ; y ; S  ) 2 F  with   maxfkX  k; kS kg.

5 Local convergence

De nition 5.1 A triple (X ; y; S ) 2 F  is called a strictly complementary solution of (1.3) if X  + S   0. In this section we investigate the asymptotic behavior of Algorithm 3.1. We will prove the superlinear convergence of Algorithm 3.1 under the following two assumptions. Assumption 1. The SDP problem has a strictly complementary solution (X ; y; S ). k k

p X S = 0: Assumption 2. The iteration sequence satis es klim !1 X k  S k

Let Q = (q1 ; : : : ; qn) be an orthogonal matrix such that q1; : : : ; qn are eigenvectors of X  and S , and de ne IB = fi : qiT X  qi > 0g;

IN = fi : qiT S  qi > 0g:

It is easily seen that IB [ IN = f1; 2; : : : ; ng. For simplicity, let us assume that     QT X Q = 0B 00 ; QT S Q = 00 0 ; N where B and N are diagonal matrices. Here and in the sequel, if we write a matrix M in the block form   M M 11 12 M= M M ; 21 22 then we assume that the dimensions of M11 and M22 are jIB jjIB j and jIN jjIN j, respectively. 24

Lemma 5.2 If Assumption 1 is satis ed, then we have   (p k )  ; QT (X k )?1=2Q =  O(1) O(1) p p QT (X k )1=2 Q = OO(p(1) ) O O(1) O(1= k ) ; k O ( k ) p )  p ) O(1)   p  O (  ) O ( O (1 = k k k T k 1 = 2 T k ? 1 = 2 Q (S ) Q = O(p ) O(1) ; Q (S ) Q = O(1) O(1) : k

(5.1) (5.2)

As in [14], we de ne a linear manifold:

M  f(X 0; y0; S 0) 2 S n  IR m  S n : Ai  X 0 = bi; i = 1; : : : ; m; m X yi0 Ai + S 0 = C; i=1 qiT X 0qj = 0 if i or j 2 IN ; qiT S 0qj = 0 if i or j 2 IB :g

(5.3)

It is easily seen that if (X 0; y0; S 0) 2 M, then

QT X 0Q =



MB 0  ; Q T S 0 Q =  0 0  : 0 0 0 MN

Lemma 5.3 (Potra-Sheng [14], Lemma 4.5) Under Assumption 1, F   M. Lemma 5.4 (Potra-Sheng [14], Lemma 4.6) Under Assumption 1, every accumulation point of (X k ; y k ; S k ) is a strictly complementary solution of (1.3).

In the next theorem, we propose a sucient condition for the superlinear convergence of Algorithm 3.1. Let us de ne k = k (?) = 1 k(X k )?1=2 (X k ? X k )(S k ? Sk )(X k )1=2 kF ; (5.4) k where (X k ; yk ; Sk ) is the solution of the following minimization problem: minfk(X k )?1=2 (X k ? X 0)(S k ? S 0)(X k )1=2 kF : (X 0; y0; S 0) 2 M; k(X 0; S 0)kF  ?g; (5.5) and ? is a constant such that k(X k ; S k )kF  ?; 8k. Note that every accumulation point of (X k ; yk ; S k ) belongs to the feasible set of the above minimization problem and the feasible set is bounded. Therefore (X k ; Sk ) exists for each k. 25

Theorem 5.5 Under Assumption 1, if k ! 0 as k ! 1, then Algorithm 3.1 is superlin-

early convergent. Moreover, if there exists a constant  > 0 such that k = O(k ), then the  convergence has Q-order at least 1 +  in the sense that k+1 = O(1+ k ). Proof. By Lemma 3.5, it remains to prove that k ! 0 as k ! 1. For simplicity, let us  w + y ? y; V + S ? S) satis es (3.18) omit the index k. It is easily seen that (U + X ? X; with H = HP (X (S ? S) + (X ? X )S ? XS ) = HP ((X ? X )(S ? S)) (since X S = 0): Let
= X ?1=2 (X ? X )(S ? S)X 1=2 : (5.6) Then, according to (ii) of Lemma 4.4, p p (5.7) kH kF  3k
kF = 3: Denoting
x = X ?1=2 (U + X ? X )X ?1=2;
s = X 1=2 (V + S ? S)X 1=2 ; and applying Corollary 3.4, we obtain

which implies Similarly,

p H kF  k
x kF  (1 ? pk  4 3; )(1 ? ) p k
xkF  4 3:

(5.8)

H kF  3p3: k
skF  (11+?p)k

(5.9)

 y; S) 2 M, we have By Lemma 5.2 and the fact that (X;  ?1=2 kF kX ?1=2(X ? X )X ?1=2 kF = kI ? X ?1=2 XX  ?1=2kF  kI kF + kX ?1=2 XX p  T X ?1=2 QkF = n + kQT X ?1=2QQT XQQ p  (xc1 ; : : : ; xcn)T kF = n + k(xc1 ; : : : ; xcn)QT XQ p  j )c = n + k Pi;j2IB (qiT Xq xi xcj T kF = O(1): 26

(5.10)

Similarly, Let us observe that

kS ?1=2 (S ? S)S ?1=2 kF = O(1): 



(5.11)



X ?1=2 UV X 1=2 = X ?1=2UX ?1=2 X 1=2 V X 1=2    =
x ? X ?1=2 (X ? X )X ?1=2
s ? X 1=2(S ? S)X 1=2   =
x
s ? X ?1=2(X ? X )X ?1=2
s   ?
x X 1=2S 1=2 S ?1=2 (S ? S)S ?1=2 S 1=2 X 1=2 +
: Then from (5.7), (5.8), (5.9), (5.10), (5.11) and Corollary 4.3, we have kX ?1=2UV X 1=2 kF  k
xkF k
skF + kX ?1=2(X ? X )X ?1=2kF k
skF +kX 1=2S 1=2 k2 k
xkF kS ?1=2(S ? S)S ?1=2kF + k
kF = O( ): Hence, according to statement (ii) of Lemma 4.4 we get,

 = kpPUV P ?1kF =  3kX ?1=2UV X 1=2 kF = = O(): Therefore, k ! 0 if k ! 0. Finally, if k = O(k ) for some constant  > 0, then we have k = O(k ). From Lemma 3.5, 1?  1? q 2 1 + 1 + 4=( ? ) q 1 + 4=( ? ) ? 1 = q 1 + 4=( ? ) + 1 = q 4=( ? ) 2 ( 1 + =( ? ) + 1)  =( ? ) = O(  ):  Therefore, k+1 = (1 ? k )k = O(k1+ ). Recalling (3.25), we obtain k+1 = O(1+ k ). Because of Theorem 5.5, the local convergence analysis established in [15] also applies to Algorithm 3.1. Therefore we can state the main result of this section. 27

Theorem 5.6 Under Assumptions 1 and 2, Algorithm 3.1 is superlinearly convergent. Moreover, if X k S k = O(k0:5+ ) for some constant  > 0, then the convergence has Q-order at least 1 + minf; 0:5g.

6 Further remarks We have shown that the class of predictor-corrector algorithms de ned by Algorithm 3.1 shares the same global and local convergence properties with the infeasible-interior-point algorithm of Kojima, Shida and Shindoh and Potra and Sheng. This result suggests that the practical performance of these algorithms should be similar. The iteration complexity of Algorithm 3.1 depends on the spectral condition number of Jxk or Jxk . If P = X ?1=2 , then Jx = I and  = 1. If P = S 1=2 , then Js = I and  = 1. For the Nesterov-Todd direction,pwe have Jx = (X 1=2SX 1=2 )1=2 and Js = (S 1=2XS 1=2 )?1=2 , therefore q   (1 + )=(1 ? ) < 3, and for example, the choice of = 0:31;  = 0:19 works. It is interesting to consider Jx = (X 1=2 SX 1=2 ) ; or Js = (S 1=2 XS 1=2 ) for  2 IR : Then,   ((1 + )(1 ? ))jj  3jj: For instance, the choice of = 0:41  3?jj=2;  = 0:25  3?jj=2 works. From a computational point of view, the choice of Js = I (where P T P = S ) seems to be preferable (cf. Zhang [19]). However more computational experiments are necessary before a de nitive conclusion is reached (see also [17] for a comparison between the performance of several Mehrotra predictor-corrector algorithms for SDP).

References [1] F. Alizadeh, J.-P. A. Haeberly, and M. L. Overton. Primal-dual interior point methods for semide nite programming. Working paper, 1994. [2] C. Helmberg, F. Rendl, R.J. Vanderbei, and H. Wolowicz. An interior-point method for semide nite programming. Technical report, Program in Statistics and Operations Research, Princeton University, 1994. [3] M. Kojima, M. Shida, and S. Shindoh. Global and local convergence of predictor{ corrector infeasible{interior{point algorithms for semide nite programs. Research Reports on Information Sciences B-305, Department of Information Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152, Japan, October 1995. 28

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