feasibility model for linear programming developed by Goldman and Tucker 1, 10]. ... mention that Ye, Todd and Mizuno 12] proposed a homogeneous algorithm ...
REPORTS ON COMPUTATIONAL MATHEMATICS, NO. 82/1995, DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF IOWA
Homogeneous Interior{Point Algorithms for Semide nite Programming Florian A. Potra and Rongqin Shengy November, 1995
Abstract
A simple homogeneous primal-dual feasibility model is proposed for semide nite programming (SDP) problems. Two infeasible-interior-point algorithms are applied to the homogeneous formulation. The algorithms do not need big M initialization. If the original SDP problem has a solution, then both algorithms nd an -approximate solup tion (i.e., a solution with residual error less than or equal to ) in at most O( n ln( 0 =)) steps, where is the trace norm of a solution and 0 is the residual error at the (normalized) starting point. A simple way of monitoringppossible infeasibility of the original SDP problem is provided such that in at most O( n ln(0 =)) steps either an -approximate solution is obtained, or it is determined that there is no solution with trace norm less than or equal to a given number > 0.
Key Words: semide nite programming, homogeneous interior-point algorithm, polynomial complexity.
Abbreviated Title: Homogeneous algorithms for SDP.
y
This work was supported in part by NSF Grant DMS 9305760. Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA.
1
1 Introduction This paper is concerned with the semide nite programming (SDP) problem: (P ) minfC X : Ai X = bi ; i = 1; : : : ; m; X 0g;
(1.1)
and its associated dual problem: (D)
m X T maxfb y : yiAi + S i=1
= C; S 0g;
(1.2)
where C 2 nn; Ai 2 nn; i = 1; : : : ; m; b = (b1 ; : : : ; bm )T 2 m are given data, and X 2 S+n , (y; S ) 2 m S+n are the primal and dual variables, respectively. 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. We consider the primal{dual SDP problem: IR
IR
IR
IR
Ai X = bi; i = 1; : : : ; m; m X yiAi + S = C;
(1.3a) (1.3b)
X S = 0; X 0; S 0:
(1.3c)
i=1
Recently, Kojima, Shindoh and Hara [4], Nesterov and Todd [8], and Monteiro [5] extended some interior{point methods from LP to SDP. In the latter paper Monteiro developed a new formulation of the primal{dual search direction originally introduced in [4]. All above mentioned methods, with the exception of the infeasible{interior{point potential{ reduction method of Kojima, Shindoh and Hara [4], require a strictly feasible starting point and therefore are feasible-interior-point methods. Nesterov [7] proposed several infeasible{ interior{point algorithms for nonlinear conic problems, which include SDP, by trying to nd a recession direction of a shifted homogeneous primal{dual problem. More recently, Zhang [13], Kojima, Shida and Shindoh [3] and Potra and Sheng [9] independently proposed new infeasible-interior-point path{following algorithms for SDP. In the present paper, we propose a simple homogeneous primal-dual feasibility model for the problem (1.3). This formulation is an extension of the homogeneous and self{dual feasibility model for linear programming developed by Goldman and Tucker [1, 10]. We mention that Ye, Todd and Mizuno [12] proposed a homogeneous algorithm for linear programming, and their results were simpli ed by Xu, Hung and Ye [11] who developed an 2
ecient code by using Goldman and Tucker's homogeneous formulation for linear programming and an infeasible{interior{point method to solve it. Using a similar idea, we apply two interior{point algorithms to solve the homogeneous formulation of SDP from infeasible starting points. The algorithms do not need big M initialization. If the original SDP problem has a solution of trace norm , then both algorithms nd an -approximate solution in at p most O( n ln(0 =)) steps. We also provide a simplepway to monitor possible infeasibility of the original SDP problem such that in at most O( n ln(0 =)) steps we can either get a -approximate solution or determine that there is no solution with trace norm less than or equal to a given number > 0. This is better that the complexity of the algorithm considered in our previous paper [9] where we assumes that a solution exists and proved that the algorithm nds it inp O(n ln(0=) iterations if the starting point is large enough (big M initialization) or in O( n ln(0 =)) iterations if the starting point is almost feasible, where 0 denotes residual error at the starting point. The following notation and terminology are used throughout the paper:
p : the p-dimensional Euclidean space; p p IR + : the nonnegative orthant of IR ; p p IR ++ : the positive orthant of IR ; pq : the set of all p q matrices with real entries; IR 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 M , Tr(M ) = Ppi=1[M ]ii ; IR
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 pq : Frobenius norm of a matrix. IR
3
2 A homogeneous feasibility model Our homogeneous model is given by the following homogeneous system:
Ai X = bi ; i = 1; : : : ; m; m X yiAi + S = C;
(2.1a) (2.1b)
= ; X 0; S 0; 0; 0:
(2.1c) (2.1d)
i=1 bT y ? C X
It is easily seen that (2.1a), (2.1b) and (2.1c) imply
X S + = 0: (2.2) The above formulation is an extension of the homogeneous and self{dual feasibility model for linear programming developed by Goldman and Tucker [1, 10]. It is also closely related to the shifted homogeneous primal{dual model considered by Nesterov [6, 7]. The following theorem is easily checked.
Theorem 2.1 The SDP problem (1.3) has a solution if and only if the homogeneous system (2.1) has a solution
(X ; y; S ; ; ) 2 S+n
IR
m S n IR IR ; + + +
such that = 0; > 0. We denote the solution set of the problem (1.3) by F . Also, let us de ne the solution set of (1.3) by H. Notice that H is not empty since (2.1) has a trivial zero solution. The residues of (2.1a){(2.1c) are denoted by:
Ri = bi ? Ai X; i = 1; : : : ; m; m X Rd = C ? yiAi ? S; i=1 T = ? b y + C X:
r Throughout the paper we will use the notation: = X nS++1 : 4
(2.3a) (2.3b) (2.3c) (2.4)
Let us de ne
n H++ = S++
IR
m S n IR IR : ++ ++ ++
Then we can de ne the infeasible central path of the homogeneous problem (2.1) by C = f(X; y; S; ; ) 2 H++ : XS = I; = ; Ri = (=0)Ri0; i = 1; : : : ; m; Rd = (=0)Rd0 ; r = (=0)r0g: In our algorithms, we will use the following neighborhood of this central path:
N ( ) = f(X; y; S; ; ) 2 H++ : kX 1=2 SX 1=2 ? I k2F + ( ? )2 = f(X; y; S; ; ) 2 H++ :
m X i=1
(i(XS ) ? )2 + ( ? )2
1=2
!1=2
g
g;
where is a constant such that 0 < < 1.
3 Search directions The search direction (U; w; V; ; ) of our algorithms is de ned by the following linear system: X ?1=2(XV + US )X 1=2 + X 1=2 (V X + SU )X ?1=2 = 2(I ? X 1=2 SX 1=2 ); (3.1a) + = ? ; (3.1b) Ai U ? bi = (1 ? )Ri; i = 1; : : : ; m; (3.1c) m X wiAi + V ? C = (1 ? )Rd; (3.1d) i=1
? bT w + C U = (1 ? )r; (3.1e) where is a parameter such that 2 [0; 1]. We note that equation (3.1a) can be written under the form 2XV X + USX + SXU = 2(X ? XSX ); so that the solution of the linear system (3.1) does not require computation of square roots of matrices (see [5]). Throughout the paper we will use the following notation: X 0 S 0 f e X= 0 ; S= 0 ; (3.2) 5
Ue = U0 0 ; Ve = V0 0 : Then it is easily seen that f e = Xn + S1 :
(3.3)
Lemma 3.1 Suppose (X; y; S; ; ) 2 H++. Then the linear system (3.1) has a unique solution
(U; w; V; ; ) 2 S n
IR
m S n IR IR :
Proof. Let us consider the following homogeneous linear system
X ?1=2(XV + US )X 1=2 + X 1=2 (V X + SU )X ?1=2 = 0 + = 0; Ai U ? bi = 0; i = 1; : : : ; m; m X wiAi + V ? C = 0;
(3.4a) (3.4b) (3.4c) (3.4d)
? bT w + C U = 0:
(3.4e)
i=1
In view of (3.4a), (3.4c), (3.4d) and the proof of Lemma 2.1 of Monteiro [5], it follows that U and V are symmetric. From (3.4c){(3.4e), we have U V + = 0, i.e., Ue Ve = 0 with e Ve given by (3.2) and (3.3). In view of (3.4a) and (3.4b), we have U;
Xf?1=2 (XfVe + Ue Se)Xf1=2 + Xf1=2(Ve Xf + SeUe )Xf?1=2 = 0; or equivalently, f1=2 Ve X f1=2 + [X f?1=2 Ue SeX f1=2 + X f1=2 SeUe X f?1=2 ] = 0: 2X
(3.5)
Therefore we obtain f1=2 Ve X f1=2 ] [X f?1=2 Ue SeX f1=2 + X f1=2 SeUe X f?1=2 ] = 0: [2X
(3.6)
Relations (3.5) and (3.6) imply the relation
Xf1=2 Ve Xf1=2 = 0; which yields Ve = 0 and therefore = 0. Then from (3.1b), we obtain = 0. Finally from Lemma 2.1 of Monteiro [5] it follows that U = V = 0. Consequently, the coecient 6
matrix of the linear system (3.1), which has 2n + m + 2 linear equations and 2n + m + 2 unknowns, is nonsingular, which implies the existence of a unique solution of (3.1). The symmetry of U and V determined by (3.1) can be deduced as in Lemma 2.1 of Monteiro [5].
Lemma 3.2 If (U; w; V; ; ) is a solution of the linear system (3.1), then, U V + = 0: Proof. Let us write (U 0 ; w0; V 0; 0 ; 0) = ((U + (1 ? )X; w + (1 ? )y; V + (1 ? )S; + (1 ? ) ; + (1 ? )): Then from (3.1c){(3.1e), we have Ai U 0 ? bi 0 = 0; i = 1; : : : ; m; m X wi0 Ai + V 0 ? 0 C = 0; i=1
and therefore
0 ? bT w0 + C U 0 = 0;
U 0 V 0 + 0 0 = 0: Hence, using the notation of (3.2) and (3.3), we get f) (Ve + (1 ? )Se) (Ue + (1 ? )X = (U + (1 ? )X ) (V + (1 ? )S ) + ( + (1 ? ) )( + (1 ? )) = 0: In view of (3.1a) and (3.1b), we obtain 1 [X f?1=2 (X fVe + Ue Se)X f1=2 + X f1=2 (Ve X f + SeUe )X f?1=2 ] = (I ? X f1=2 SeX f1=2 ): 2 By taking the trace of both sides of (3.8), we have e Xf Ve + Ue Se = ( ? 1)Xf S: Then by expanding (3.7) and using (3.9) we get Ue Ve = 0, i.e., U V + = 0: 7
(3.7) (3.8) (3.9)
Lemma 3.3 Let (X; y; S; ; ) 2 N ( ) for some 2 [0; 1). Suppose that (U; w; V; ; ) is a solution of the linear system (3.1). Then,
kXf?1=2 Ue Ve Xf1=2 k
kI ? Xf1=2 SeXf1=2 k2F : 2(1 ? )2
Proof. Using the notation introduced in (3.2) and (3.3) and Lemma 3.2, we have
Xf1=2Ve Xf1=2 + Xf?1=2 Ue SeXf1=2 + Xf1=2 SeUe Xf?1=2 = 2(I ? Xf1=2SeXf1=2 ); and
(3.10)
Ue Ve = 0:
(3.11)
kXf1=2 SeXf1=2 ? I kF :
(3.12)
Since (X; y; S; ; ) 2 N ( ), we obtain
Then the desired inequality follows from (3.10), (3.11), (3.12) and a manipulation similar to that used in the proof of Lemma 4.4 of Monteiro [5].
4 A generic homogeneous algorithm Generic Homogeneous Algorithm
Let (X 0; y0; S 0; 0 ; 0) = (I; 0; I; 1; 1): Repeat until a stopping criterion is satis ed: Choose 2 [0; 1] and compute the solution (U; w; V; ; ) of the linear system (3.1). Compute steplength such that
X + U 0; S + V 0; + > 0; + > 0:
Update the iterates (X +; y+; S +; +; +) = (X; y; S; ; ) + (U; w; V; ; ): We will de ne a stopping criterion later in this section. 8
Theorem 4.1 The iterates generated by the Generic Homogeneous Algorithm satisfy: + + = X S + + + = (1 ? (1 ? )); (4.1a) +
n+1 = (1 ? (1 ? ))Ri ; i = 1; : : : ; m; (4.1b) + Rd = (1 ? (1 ? ))Rd ; r = (1 ? (1 ? ))r: (4.1c) Proof. By the de nition of the Generic Homogeneous Algorithm, we have X + S + = (X + U ) (S + V ) = X S + (X V + U S ) + U V = X S + (n ? X S ) + U V; and ++ = ( + )( + ) = + ( + ) + = + ( ? ) + : Therefore, from Lemma 3.2, we obtain X + S + + ++ = (1 ? (1 ? ))(n + 1); which implies (4.1a). Finally, (4.1b) and (4.1c) can be easily deduced from (3.1c){(3.1e). By using the above theorem and the fact that 0 = 1, we deduce the following corollary. Corollary 4.2 Let (X k ; yk; S k ; k ; k ) be generated by Generic Homogeneous Algorithm. Ri+
Then,
Rik = k Ri0; i = 1; : : : ; m; Rdk = k Rd0; rk = k r0: Let us de ne a linear manifold H0 = f(X 0; y0; S 0; 0; 0 ) 2 S n m S n : Ai X 0 ? 0 bi = 0; i = 1; : : : ; m; m X yi0Ai + S 0 ? 0 C = 0; IR
i=1
0 ? bT y0 + C X 0 = 0g 9
IR
IR
Then it is easily seen that if (X 0; y0; S 0; 0; 0) 2 H0 , then
X 0 S 0 + 0 0 = 0: The following lemma shows that the iterates (X k ; yk ; S k ; k ; k ) are bounded. Lemma 4.3 If (X k ; yk; S k ; k ; k ) are generated by the Generic Homogeneous Algorithm, then Tr(X k + S k ) + k + k = (n + 1)(1 + ): (4.2) Proof. For simplicity, we omit the index k. From Corollary 4.2, we have (X ? X 0; y ? y0; S ? S 0; ? 0; ? 0) 2 H0: Therefore,
(S ? S 0) (X ? X 0) + ( ? 0 )( ? 0) = 0: (4.3) The desired result follows by expanding (4.3) and using the fact that X 0 = S 0 = I and 0 = 0 = 1.
Theorem 4.4 Let (X k ; yk ; S k ; k ; k ) be generated by the Generic Homogeneous Algorithm and let > 0 be a given number. Suppose that the following two conditions are satis ed: (a) there exists a solution (X ; y ; S ) of the original problem (1.3) such that Tr(X +S ) ; (b) there exists a constant ! 2 (0; 1) such that k k !k ; 8k: Then,
! ; 8 k: k + 1
Proof. Let = 1; = 0. Obviously, (X ; y ; S ; ; ) is a solution of (2.1). From the proof of Lemma 4.3, for any real number , we have
(X ? X 0 + X ; y ? y0 + y; S ? S 0 + S ; ? 0 + ; ? 0 + ) 2 H0 : Therefore, we get (S ? S 0 + S ) (X ? X 0 + X ) + ( ? 0 + )( ? 0 + ) = 0: By expanding the above equality and noting that is an arbitrary real number and that X S + = 0, we obtain
S X + S X + + = (S 0 X + S X 0 + 0 + 0 ) = [Tr(X + S ) + 1]: 10
Then, from condition (a) and = 1, we deduce that
[Tr(X + S ) + 1] ( + 1):
(4.4)
Hence, by (4.4) and condition (b), we have ! : ! +1
Corollary 4.5 Under condition (b) of Theorem 4.4 suppose that k ! 0 as k ! 1. Then (i) the primal{dual problem (1.3) has a solution if and only if k ! 0 as k ! 1 and there exists a constant > 0 such that k ; 8k; (ii)(1.3) has no solution if and only if k ! 0 as k ! 1. Proof. Part (i) is an immediate consequence of Theorem 4.4. From Lemma 4.3, the sequence f k g is bounded. Let us observe that if (1.3) has no solution then f k g does not have a positive accumulation point (otherwise, there exists a subsequence f(X k = k ; yk= k ; S k = k )g converging to a solution of (1.3)), and therefore k ! 0. Conversely, if k ! 0, then by Theorem 4.4 (1.3) has no solution. Let us de ne the stopping criterion of the Generic Homogeneous Algorithm: Stopping criterion.
Ek maxfkRdk k= k ; jRik j= k ; i = 1; : : : ; m; (X k S k + k k )= k2 g ;
(4.5)
where > 0 is the tolerance. Let us de ne the set of -approximate solutions of (1.3) by
F = f(X; y; S ) 2 S+n m S+n : X S ; jbi ? A X j ; i = 1; : : : ; m; kC ? Pmi=1 yiAi ? S k g: IR
Obviously, if (4.5) is satis ed, then (X k = k ; yk = k ; S k = k ) is an -approximate solution of the original problem (1.3). Let us observe that if our algorithm starts at a point that is a multiple of (I; 0; I; 1; 1), say (I; 0; ; ; ), then we obtain an iteration sequence (X k ; yk; Sk ; k ; k ) = (X k ; yk ; S k ; k ; k ): However, we note that (X k = k ; yk= k ; Sk = k ) = (X k = k ; yk = k ; S k = k ): 11
Therefore, we see that the sequence (X k = k ; yk = k ; S k = k ) does not depend on the magnitude of the starting point. This explains why we do not need a big M initialization. If we de ne the residual error at the point (X; y; S; ; ) by res((X; y; S; ; )) = maxfX S + ; kRd kF ; jRij; i = 1; : : : ; mg; (4.6) then for our starting point we have 0 res((I; 0; I; 1; 1)) = maxfn + 1; kC ? I kF ; jbi ? Tr(Ai )j; i = 1; : : : ; mg; while for a ( + 1) multiple of our starting point we have () res(( + 1)(I; 0; I; 1; 1)) (4.7) = maxf( + 1)2(n + 1); ( + 1)kC ? I kF ; ( + 1)jbi ? Tr(Ai)j; i = 1; : : : ; mg 0 ( + 1)2: Using the above introduced quantity we obtain the following result. Lemma 4.6 If k !=( + 1), then then the following estimate holds: Ek k ()=!2; where () is the residual error de ned by (4.7). Proof. From (4.5) and Corollary 4.2, we get Ek = maxfk kRd0kF = k ; k jRi0j= k ; i = 1; : : : ; m; k (n + 1)= k2g = k maxfkRd0 kF = k ; jRi0 j= k ; i = 1; : : : ; m; (n + 1)= k2g k maxfkRd0 kF =(!=( + 1)); jRi0 j=(!=( + 1)); i = 1; : : : ; m; (n + 1)=(!=( + 1))2g (k =!2) maxf( + 1)kRd0kF ; ( + 1)jRi0j; i = 1; : : : ; m; ( + 1)2(n + 1)g = k ()=!2: From Theorem 4.4 and Lemma 4.6, we see that the quantity k is a useful certi cate for monitoring possible infeasibility of the problem (1.3). Corollary 4.7 Under condition (b) of Theorem 4.4 if k < !=( + 1); for some k; (4.8) then the primal{dual problem (1.3) has no solution (X ; y ; S ) such that Tr(X + S ) . In the next two sections, pwe will describe two examples of the Generic Homogeneous Algorithm, which achieves O( n ln(0 =)){iteration complexity. 12
5 A predictor{corrector algorithm In this section, we apply the recent infeasible{interior{point predictor{correctorpalgorithm of Potra and Sheng [9] to the homogeneous system (2.1) and show that it has O( n ln(0 =))iteration complexity. Let ; be two positive constants satisfying the inequalities
< 1: < < 2(1 ? )2 1? 2
(5.1)
For example, = 0:25; = 0:41 verify (5.1).
Algorithm 5.1
(X; y; S; ; ) (I; 0; I; 1; 1); Repeat until stopping criterion (4.5) is satis ed or is suciently small:
(Predictor step)
Solve the linear system (3.1) with = 0; Compute = maxf~ 2 [0; 1] : n X i=1
(i(X ()S ()) ? (1 ? ))2 + ( ()() ? )2
!1=2
(1 ? ); 8 2 [0; ~]g;
where
(X (); S (); (); ()) = (X; S; ; ) + (U; V; ; ); (X; y; S; ; ) X; y; S; ; ) + (U; w; V; ; );
(Corrector step)
Solve the linear system (3.1) with = 1; (X; y; S; ; ) X; y; S; ; ) + (U; w; V; ; ):
Using a proof similar to that of Lemma 2.5 of Potra and Sheng [9], we get the following lower bound for the steplength : 2 ; (5.2) q 1 + 4=( ? ) + 1 where
1 kXf?1=2Ue Ve Xf1=2 kF : 13
(5.3)
Then, in view of Lemma 3.3, we obtain f1=2 SeX f1=2 k2 k X 2(1 ? )22 F 2 + kI ? X f1=2 SeX f1=2 k2 k I k F F = 2(1 ? )22 + 1)2 + 22 = n + 1 + 2 = O(n); (n 2(1 ? ) 2 2 2(1 ? )2 which implies
Here we have used the relation
1 : 1 ? 1 ? O (p n)
f1=2 SeX f1=2 ] = 0: (I ) [I ? X Then, from Theorem 4.1, we have 1 ) : k+1 = (1 ? k )k (1 ? O(p n) k
(5.4)
Also, we can prove that
(X k ; yk; S k ; k ; k ) 2 N ( ); 8k: Hence, condition (b) of Theorem 4.4 is satis ed with ! =p1 ? . Therefore, in view of (5.4), Theorem 4.4 and Lemma 4.6, we see that in at most O( n ln(0 =)) iterations, either the p inequality k != holds all the time or it is violated for some k O( n ln(0 =)). In the former case, we get an -approximate solution of (1.3) while in the later case (1.3) has no solution (X ; y; S ) such that Tr(X + S ) . To summarize, we obtain the following polynomial complexity result.
Theorem 5.2 (i) If F is not empty, then Algorithm 5.1 terminates with an -approximate solution (X k = k ; yk = k ; S k = k ) 2 F in a nite number of steps k = K < 1. p (ii) If = Tr(X + S ) for some (X ; y ; S ) 2 F , then K = O( n ln( 0 =)). c = O(pn ln(0 =)) such that either (iii) For any choice of > 0 there is an index k = K (iiia) (X k = k ; y k = k ; S k = k ) 2 F, or, (iiib) k < !=( + 1), and in the latter case there is no solution (X ; y ; S ) 2 F with Tr(X + S ).
14
6 A short{step algorithm In this section, we extend a recent short{step feasible path{following algorithm by Kojima, Shindoh and Hara [4] and simpli ed by Monteiro [5] for solving the homogeneous system from infeasible starting points. Algorithm 6.1 (X; y; S; ; ) (I; 0; I; 1; 1); Repeat until the stopping criterion (4.5) is satis ed p or is suciently small: Solve the linear system (3.1) with = 1 ? 0:3= n + 1; (X; y; S; ; ) X; y; S; ; ) + (U; w; V; ; ): By an analogous analysis to that used in Theorem 4.1 of Monteiro [5], we can prove that (X k ; yk ; S k ; k ; k ) 2 N (0:3); 8k. From Theorem 4.1, we get
p
k+1 = (1 ? 0:3= n + 1)k : Obviously, condition (b) of Theorem 4.4 is satis ed with ! = 0:7. Hence, similar to Theorem 5.2, we obtain the following complexity result for Algorithm 6.1.
Theorem 6.2 (i) If F is not empty, then Algorithm 6.1 terminates with an -approximate solution (X k = k ; yk = k ; S k = k ) 2 F in a nite number of steps k = K < 1. p (ii) If = Tr(X + S ) for some (X ; y ; S ) 2 F , then K = O( n ln( 0 =)). c = O(pn ln(0 =)) such that either (iii) For any choice of > 0 there is an index k = K (iiia) (X k = k ; y k = k ; S k = k ) 2 F, or, (iiib) k < !=( + 1), and in the latter case there is no solution (X ; y ; S ) 2 F with Tr(X + S ).
7 Further remarks For simplicity of analysis, we have chosen the starting point (I; 0; I; 1; 1). Indeed, we can use any starting point (X 0; y0; S 0; 0 ; 0) 2 H++ near the infeasible central path because we are actually interested in the sequence (X k = k ; yk = k ; S k = k ) which does not depend on the magnitude of the starting point. 15
As a nal comment we note that we can also extend the long{step algorithm for linear programming by Kojima Mizuno and Yoshise [2] to the homogeneous system (2.1). By using the proof techniques and the results of Monteiro [5] we can prove that the iteration complexity of this extended algorithm is O(n1:5 ln(0 =)).
References [1] A. J. Goldman and A. W. Tucker. Polyhedral convex cones. In H. W. Kuhn and A. W. Tucker, editors, Linear Inequalities and Related Systems. Princeton University Press, Princeton, NJ, 1956. p. 19. [2] M. Kojima, S. Mizuno, and A. Yoshise. A primal{dual interior point algorithm for linear programming. In N. Megiddo, editor, Progress in Mathematical Programming, pages 29{47. Springer Verlag, New York, 1989. [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. [4] M. Kojima, S. Shindoh, and S. Hara. Interior-point methods for the monotone linear complementarity problem in symmetric matrices. Research Reports on Information Sciences B-282, Department of Information Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152, Japan, April 1994. [5] R. D. C. Monteiro. Primal-dual path following algorithms for semide nite programming. Working paper, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA, September 1995. [6] Y. E. Nesterov. Long{step strategies in interior point potential{reduction methods. Working paper, Department SES COMIN, Geneva University, 1993. to appear in Mathematical Programming. [7] Y. E. Nesterov. Infeasible start interior point primal{dual methods in nonlinear programming. Working paper, Center for Operations Research and Econometrics, 1348 Louvain-la-Neuve, Belgium, 1994.
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[8] Y. E. Nesterov and M. J. Todd. Primal{dual interior{point methods for self{scaled cones. Technical Report 1125, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853{3801, USA, 1995. [9] F. A. Potra and R. Sheng. A superlinearly convergent primal{dual infeasible{interior{ point algorithm for semide nite programming. Reports on Computational Mathematics 78, Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA, October 1995. [10] A. W. Tucker. Dual systems of homogeneous relations. In H. W. Kuhn and A. W. Tucker, editors, Linear Inequalities and Related Systems. Princeton University Press, Princeton, NJ, 1956. p. 3. [11] X. Xu, P. Hung, and Y. Ye. A simpli cation of the homogeneous and self-dual linear programming algorithm and its implementation. Working paper, Department of Management Sciences, The University of Iowa, Iowa City, Iowa 52242, USA, 1993. to appear in Annals of Operations Research. p [12] Y. Ye, M. J. Todd, and S. Mizuno. An O( nL)-iteration homogeneous and self-dual linear programming algorithm. Mathematics of Operations Research, 19(1):53{67, 1994. [13] Y. Zhang. On extending primal{dual interior{point algorithms from linear programming to semide nite programming. TR 95-20, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21228{5398, USA, October 1995.
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