Polynomial primal-dual affine scaling algorithms in semidefinite ...

Polynomial primal-dual affine scaling algorithms in semidefinite programming Report 96-42

E. de Klerk C. Roos T. Terlaky

Faculteit der Technische Wiskunde en Informatica Faculty of Technical Mathematics and Informatics Technische Universiteit Delft Delft University of Technology

ISSN 0922-5641

Copyright c 1996 by the Faculty of Technical Mathematics and Informatics, Delft, The Netherlands. No part of this Journal may be reproduced in any form, by print, photoprint, microfilm, or any other means without permission from the Faculty of Technical Mathematics and Informatics, Delft University of Technology, The Netherlands. Copies of these reports may be obtained from the bureau of the Faculty of Technical Mathematics and Informatics, Julianalaan 132, 2628 BL Delft, phone +31152784568. A selection of these reports is available in PostScript form at the Faculty’s anonymous ftp-site. They are located in the directory /pub/publications/tech-reports at ftp.twi.tudelft.nl

DELFT UNIVERSITY OF TECHNOLOGY

REPORT Nr. 96{42 Polynomial primal{dual affine scaling algorithms in semidefinite programming

E. de Klerk, C. Roos, T. Terlaky

ISSN 0922{5641 Reports of the Faculty of Technical Mathematics and Informatics Nr. 96{42 Delft, March 25, 1996 i

E. de Klerk, C. Roos and T. Terlaky, Faculty of Technical Mathematics and Informatics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands. e{mail: [email protected], [email protected], [email protected]

c 1996 by Faculty of Technical Mathematics and Copyright Informatics, Delft, The Netherlands. No part of this Journal may be reproduced in any form, by print, photoprint, micro lm or any other means without written permission from Faculty of Technical Mathematics and Informatics, Delft University of Technology, The Netherlands.

ii

Abstract Two primal{dual ane scaling algorithms for linear programming are extended to semide nite programming. The algorithms do not require (nearly) centered starting solutions, and can be initiated with any primal{dual feasible solution. The rst algorithm is the Dikin-type ane scaling method of Jansen et al. [8] and the second the pure ane scaling method of Monteiro et al. [12]. The extension of the former has a worst-case complexity bound of O( nL) iterations, where  is a measure of centrality of the the starting solution, and the latter a bound of O( nL ) iterations. Key words: interior{point method, primal{dual method, semide nite programming, Dikin steps. 0

0

0

2

iii

1 Introduction The introduction of Karmarkar's polynomial{time projective method for LP in 1984 [9] was accompanied by claims of some superior computational results. Later it seemed likely that the computation was done with a variant of the ane scaling method, proposed by Dikin nearly two decades earlier in 1967 [3]. The two algorithms are closely related, and modi cations of Karmarkar's algorithm by Vanderbei et al. [16] and Barnes [1] proved to be a rediscovery of the ane scaling method. Interestingly enough, polynomial complexity of the ane scaling method in its original form has still not been proved. A primal{dual ane scaling method is studied by Monteiro et al. in [12] where the primal{ dual search direction minimizes the duality gap over a sphere (steepest descent) in the primal{dual space. This algorithm may be viewed as a `greedy' primal{dual algorithm, which aims for the maximum reduction of the duality gap at each iteration, without attempting to stay centered. The worst{case iteration complexity for this method is O(nL ). Jansen et al. proposed a primal{dual Dikin-type ane scaling variant in [8] with improved O(nL) polynomial complexity. This search direction minimizes the duality gap over the so-called Dikin ellipsoid in the primal{dual space. The advantage of this method is that each step involves both centering and reduction of the duality gap. In this paper we generalize both these primal{dual ane scaling methods to semide nite programming (SDP). The extension of interior point methods from LP to SDP is currently an active research area. The rst algorithms to be extended were potential reduction methods [13, 17], and recently much work has been done on primal{dual central path following algorithms [7, 10, 11, 15, 14, 5]. The methods here do not belong to any of these two classes, and as such constitute a new approach. In particular, a nearly centered starting solution is not required, although the worst case complexity bounds depend on the degree of centrality of the starting solution. The importance of algorithms which can start from arbitrary feasible points is discussed by Goldfarb and Scheinberg in [6], where they study trajectories leading to the optimal set from arbitrary feasible starting points. 2

The semide nite programming problem We will work with the following standard SDP formulation of the primal problem (P): minTr(CX ) subject to Tr(AiX ) = bi; i = 1; : : : ; m X  0 and of the dual problem (D): max bT y 1

subject to m X

yiAi + Z = C Z  0 where C and the matrices Ai are symmetric n  n matrices, b; y 2 IRm and `X  0' means X is symmetric positive semi{de nite. The matrices Ai are further assumed to be linearly independent. We will assume that a strictly feasible pair (X; Z ) exists. This ensures a zero duality gap (Tr(XZ ) = 0) at an optimal primal{dual pair [17]. These assumptions may be made without loss of generality: A general primal{dual pair of SDP problems may be embedded in a self{dual SDP problem with nonempty interior and known interior feasible solution [2]. The optimality conditions for the pair of problems (P) and (D) are Tr(AiX ) = bi; i = 1; : : : ; m m X y i Ai + Z = C i XZ = 0 X; Z  0: The system of relaxed optimality conditions: Tr(AiX ) = bi; i = 1; : : : ; m m X y i Ai + Z = C i XZ = I X; Z  0 has a unique solution fX (); y(); Z ()g which gives a parametric representation of the central path as a function of  [17]. i=1

=1

=1

Primal{dual ane scaling search directions Any set of feasible primal{dual search directions (
X;
y;
Z ) must satisfy Pm

i=1

Tr (Ai
X ) = 0; i = 1; : : : ; m
yiAi +
Z = 0:

(1)

Note that
X and
Z are orthogonal, i.e. Tr(
X
Z ) = 0: After a feasible step (X +
X; Z +
Z ), the duality gap becomes Tr(X +
X )(Z +
Z ). The search direction of the Dikin{type algorithm presented here minimizes this duality gap over the so-called 2

Dikin ellipsoid in the scaled primal{dual space, which will be de ned in the next section. We show that the computation of this search direction amounts to the solution of (2)
X + D
ZD = ? XZX 1 ; (Tr(XZ ) ) 2 subject to the conditions (1), where D is the scaling{matrix 2



1

D := Z ? 21 Z 12 XZ 12 2 Z ? 12 :

(3)

The primal{dual pure ane scaling search direction is the steepest descent direction for the duality gap Tr(XZ ). The computation of this direction involves the solution of the system
X + D
ZD = ?X; (4) subject to (1).

Measure of centrality The Dikin steps have the feature that the proximity to the central path is maintained, where this proximity is quanti ed by XZ ) (XZ ) = max((XZ (5) ) min with max (XZ ) the largest eigenvalue of XZ and min (XZ ) the smallest . The pure ane scaling steps may become increasingly less centered with respect to this measure which complicates the analysis somewhat. Note that (XZ )  1 and (XZ ) = 1 if and only if XZ = I for some  > 0, i.e. if the pair (X; Z ) is centered with parameter . 1

The Algorithms The two primal{dual ane scaling algorithms can be described in the following framework: The eigenvalues of X Z are real and positive if X; Z  0, since X Z  X ? 21 [X Z ]X 12 = X 12 Z X 21  0, where '' denotes the similarity relation. 1

3

Generic primal{dual ane scaling type algorithm Input: A strictly feasible pair (X 0; Z 0); A parameter 0 > 1 such that  (X 0Z 0)  0; An accuracy parameter  > 0.

begin

L := ln Tr X 0Z0 ;  := pn0 (Dikin steps);  := nL0 (Pure ane scaling steps); X := X ; Z := Z ; while Tr(XZ ) >  do (

)

1

1

0

0

begin

Compute
X ,
Z from (2) and (1) (Dikin steps) or from (4) and (1) (Pure ane scaling steps); X := X + 
X ; Z := Z + 
Z ;

end end

We prove that the Dikin step algorithm computes a strictly feasible -optimal solution (X ; Z ) in O( nL) steps, and this solution satis es (X  Z )   . The pure primal{dual ane scaling algorithm converges in O( nL ) steps, and the solution satis es (X  Z )  3 . The paper is structured as follows. The Dikin step method is presented rst, and its simple analysis is then extended to the pure ane scaling method. In Section 2 is shown how the Dikin step direction is derived by working in a scaled primal{dual space. In Section 3 conditions to ensure a feasible steplength are derived, and convergence and the polynomial complexity result are proven in Section 4. In Sections 5 and 6 the analysis is extended to the pure primal-dual ane scaling algorithm. 0

0

2

0

0

Notation i (A) : ith eigenvalue of the n  n matrix A; max (A) = max  (A); if i(A) 2 IR 8i; i i 4

min (A) = min  (A); if i (A) 2 IR 8i; i i A  0 : A is symmetric positive semide nite; XX T kAk = Tr(AA ) = Aij (Frobenius norm) 2

2

=

i

X

j

i (A) if A symmetric; 2

i

kAk = (max(AT A)) (Spectral norm) = max(A) if A  0; (A) = max ji (A)j (spectral radius of A); i 1 2

2

(A) = max((AA)) if i(A) > 0 8i min = condition number of A if A  0; A  B : The matrices A and B are similar.

2 Dikin steps in semide nite programming For strictly feasible solutions X  0 and Z  0 to (P) and (D) respectively, the scaling matrix D de ned in (3) satis es D? X = ZD, or 1

D? 12 XD? 12 = D 12 ZD 12 := V: In other words, the matrix D may be used to scale the variables X and Z to the same symmetric positive de nite matrix V . Note that 2

V = D? 12 XZD 12  XZ; i.e. V has the same eigenvalues as XZ and is symmetric positive de nite. As a consequence the duality gap is given by X Tr(XZ ) = Tr(V ) = kV k = i (V ): 2

2

2

2

2

i

We can similarly scale the primal{dual search directions
X and
Z via

DX = D? 12
XD? 12 and

DZ = D 12
ZD 21 :

The matrix V is not unique and depends on the factorization of D [15]. We have used the symmetric square root factorization D = D 12 D 21 but any LLT factorization can be used. The factors only appear in the analysis though and not in the algorithm, the important point being that V 2 is obtained via a similarity transformation of X Z in all cases, and the analysis is concerned only with the eigenvalues of X Z . 2

5

The scaled directions DX and DZ are orthogonal by the orthogonality of
X and
Z , i.e. Tr(DX DZ ) = 0. In particular, we have

DZ = ?

m X i=1


yiD 12 AiD 21 ;

i.e. DZ must be in the span of matrices D 12 AiD 12 and DX in its orthogonal complement, i.e.   Tr D 12 AiD 21 DX = 0; i = 1; : : : ; m: The scaled Newton step in the V-space is denoted by DV = DX + DZ . After a feasible primal{dual step
X ,
Z the duality gap becomes Tr ((X +
X )(Z +
Z )) = Tr ((V + DX )(V + DZ )) = Tr(V + V DV ): The search direction of the algorithm is derived by minimizing the duality gap over the Dikin ellipsoid: kV ? DV k  1: In other words the search direction is the solution to min Tr(V + V DV ) D 2

1

2

subject to

V

kV ? DV k  1 1

DV = DX + DZ DZ 2 span fD 12 A D 12 ; : : : ; D 12 AmD 12 g DX 2 span fD 12 A D 12 ; : : : ; D 12 AmD 12 g?: It is easily veri ed that the optimal solution is given by (6) DV = DX + DZ = ? kVV k : The transformation back to the unscaled space is done by premultiplying and postmultiplying (6) by D 12 to obtain (7)
X + D
ZD = ?XZX 1 : (Tr(XZ ) ) 2 The primal{dual ane scaling direction (Dikin step direction) is obtained by solving (7) subject to the conditions (1). It is easily shown that (7) and (1) imply n X
yj Tr (AiDAj D) = ?Tr (AiXZX1 ) ; i = 1; : : : ; m: (8) (Tr(XZ ) ) 2 j The solution of this m  m linear system yields
y. (Note that the coecient matrix of Pm the system (8) is positive de nite [4]). Once
y is known,
Z follows from i
yiAi = ?
Z , and
X is subsequently obtained from (7). 1 1

3

2

2

2

=1

=1

6

A note on the centrality measure The centrality measure (XZ ) de ned in (5) satis es 1 0  1 ? (XZ ) with equality holding on the central path. A dierent measure for proximity to the central path was used by Sturm and Zhang [15]:



1

(XZ ) :=

I ?  V

; 2

p where  = Tr(XZ )=n = Tr(V )=n. Note that one may have (XZ ) = O( n) if (XZ ) = O(1). This shows that (XZ )   de nes a larger neighbourhood of the central path than (XZ )   . The following relation between the two measures is readily proved:

!

n

(XZ ) =

I ? Tr(V ) V

!

p n

 n

I ? Tr(V ) V

! p n min (V ) = n 1 ? P  (V ) i i ! ! p p  1 min (V )  n 1 ?  (V ) = n 1 ? (XZ ) : max 2

2

2

2

2

2

2

2

2

2

3 Feasibility of the Dikin step Having computed the Dikin step direction (
X;
Z ), a feasible steplength must be established. Denoting X () := X + 
X; Z () := Z + 
Z; we establish a value  > 0 such that X ()  0 and Z ()  0. The following lemma gives a sucient condition for a feasible steplength .

Lemma 3.1 If one has det (X ()Z ()) > 0 8 0     ; then X ()  0 and Z ()  0.

Proof:

7

The function

f (; ) := det [(X + 
X ) ? I ] is continuously dierentiable if  2 (0;  ) and  2 IR and is zero if  is an eigenvalue of X (). The implicit function theorem therefore implies that the eigenvalues of X () (and similarly of Z ()) are continuous functions of . Since det(X ()Z ()) = det(X ()) det(Z ()), one has Y Y det(X ()Z ()) = i(X ()) i(Z ()) (9) i

i

The left hand side of eq. (9) is strictly positive on [0;  ]. This shows that the eigenvalues of X () and Z () remain positive on [0;  ]. 2 In order to derive bounds on  which are sucient to guarantee a feasible steplength, we need the following three technical results.

Lemma 3.2 The spectral radius of (DX DZ + DZ DX ) is bounded by (DX DZ + DZ DX )  21 kDX + DZ k : 2

Proof: It is trivial to verify that

i h DX DZ + DZ DX = 21 (DX + DZ ) ? (DX ? DZ ) 2

which implies

? 21 (DX ? DZ )  DX DZ + DZ DX  21 (DX + DZ ) : 2

It follows that

2

2

? 12 kDX ? DZ k I  DX DZ + DZ DX  21 kDX + DZ k I: 2

2

Since DX and DZ are orthogonal the matrices (DX + DZ ) and (DX ? DZ ) have the same norm. Consequently ? 21 kDX + DZ k I  DX DZ + DZ DX  21 kDX + DZ k I from which the required result follows. 2 2

2

8

Corollary 3.1 For the Dikin step DX + DZ = ?V =kV k, one has (DX DZ + DZ DX )  21 (V ): 3

2

2

Proof: By Lemma 3.2 one has 2 (DX DZ + DZ DX )  kDX + DZ k ! k V k = kV k V ) = (Tr( kV k) V ) = (V )  (V ) (Tr( kV k) which is the required result. 2

2

3

2

6

2

2

4

2

2

2

2

2

Lemma 3.3 Let Q be an n  n real symmetric matrix, S an n  n real skew{symmetric matrix. One has det(Q + S ) > 0 if Q  0. Moreover, if it is known that i (Q + S ) 2 IR; i = 1; : : : ; n then it holds that i(Q + S ) > 0; i = 1; : : :; n and (Q + S )  (Q):

Proof: First note that Q + S is nonsingular since xT (Q + S )x = xT Qx > 0 80 6= x 2 IRn; using the skew symmetry of S . We therefore know that (t) := det[Q + tS ] 6= 0 8t 2 IR; since tS remains skew{symmetric. One now has that is a continuous function of t which is nowhere zero and strictly positive for t = 0 as det(Q) > 0. This shows det(Q + S ) > 0. To prove the second part of the lemma, assume  > 0 is such that  > max(Q). It then follows that Q ? I  0. By the same argument as above we then have (Q + S ) ? I nonsingular, or det ((Q + S ) ? I ) 6= 0: This implies that  cannot be an eigenvalue of (Q + S ). Similarly, (Q + S ) cannot have an eigenvalue smaller than min (Q). This gives the required result. 2 9

We are now in a position to nd a step size  which guarantees that the Dikin step will be feasible. To simplify the analysis we introduce a parameter  > 1 such that (XZ ) = (V )   . This implies the existence of numbers  and  such that  I  V   I;  =  : (10) 2

1

2

1

2

2

2

1

Lemma 3.4 The steps X () = X + 
X and Z () = Z + 
Z are feasible if the step size  satis es    where ( ) k V k 4   = min 2 ; kV k : 2

1

2

2

Furthermore

 (X ()Z ())  :

Proof:

We show that the determinant of X ()Z () remains positive for all   . One then has X (); Z ()  0 by Lemma 3.1. To this end note that X ()Z ()  (V + DX )(V + DZ ) = V + DX V + V DZ +  DX DZ = V ? V + 1  (DX DZ + DZ DX ) kV k 2   1 + 2  (DX DZ ? DZ DX ) + 21 (DX V + V DZ ? V DX ? DZ V ) ; since DX + DZ = ?V =kV k. The matrix in square brackets is skew{symmetric. Lemma 3.3 therefore implies that the determinant of [X ()Z ()] will be positive if the matrix 1  (D D + D D ) M () := V ? kV + X Z Z X V k 2 is positive de nite. Note that M (0) = V  0 and  (M (0))   . We proceed to prove that  (M ()) remains bounded by (M ())   for 0     . This is sucient to prove that M ()  0; 0    , and therefore that a step of length  is feasible. Moreover, after such a feasible step we will have X ()  0, Z ()  0. The matrix X ()Z () therefore has positive eigenvalues and we can apply the second part of Lemma 3.3 to obtain  (X ()Z ())   (M ())  : 2

2

4

2

2

2

2

3

2

2

4

2

2

2

We start the proof by noting that if  is an eigenvalue of V then ( ?  =kV k) is an eigenvalue of [V ? V =kV k]. The function (t) := t ?  kVt k 2

2

4

2

2

2

10

2

2

is monotonically increasing on t 2 [0;  ] if   , since   kV k=(2 ). Thus ( )I  V ? kV V k  ( )I 8 0     or ( )I + 12  (DX DZ + DZ DX )  M ()  ( )I + 12  (DX DZ + DZ DX ) 8 0    : We will therefore certainly have (M ())   if  [( )I + 12  (DX DZ + DZ DX )]  ( )I + 21  (DX DZ + DZ DX ): This matrix inequality can be simpli ed using  =  and subsequently dividing by . This yields !  ?  I + ( ? 1)  1 (D D + D D )  0: kV k 2 X Z Z X 2

2

1

4

2

1

2

2

2

2

2

2

1

2

2

2

2 2

2

1

2 1

2

This may be further simpli ed using  ?  = ( ? 1)  to obtain !   I + 1 (D D + D D )  0 kV k 2 X Z Z X 2 2

2 1

1 2

1 2 2

which will surely hold if

!

  I ?   1 (D D + D D ) I  0: kV k 2 X Z Z X 1 2 2

Substituting the bound



 1  2 (DX DZ + DZ DX )  41 (V )  41  from Corollary 3.1 yields   ? 1   0; kV k 4 or   k4V k which is the second bound in the lemma. This completes the proof. 2

1 2

2

2

2

1

2

11

2

4 Convergence and Complexity analysis A feasible Dikin step of length  reduces the duality gap by at least a factor (1 ? pn ). Formally we have

Lemma 4.1 Given a feasible primal{dual pair (X; Z ) and a steplength  such that the Dikin step is feasible, i.e. X () := X + 
X  0, and Z () := Z + 
Z  0. It holds that

!

 Tr (X ()Z ())  1 ? p Tr(XZ ): n

Proof: The duality gap after the Dikin step is given by Tr (X ()Z ()) = Tr (( V + DX )(V + DZ))  = Tr V + V (DX + DZ ) ! V = Tr V ?  kV k = kV k ? kV !k = 1 ?  kkVV kk Tr(XZ ): 2

4

2

2

2

2

2

2

By the Cauchy-Schwarz inequality one has

kV k = Tr IV  kI kkV k = pn

V

; 2



2



2

2

2

which gives the required result. We are now in a position to prove a worst-case iteration complexity bound.

Theorem 4.1 Let  > 0 be an accuracy parameter,  = (X Z ), L = ln (Tr(X Z )=), 0

0

0

0

0

and  = 0 p1 n . The Dikin Step Algorithm requires at most 0 nL iterations to compute a feasible primal{dual pair (X  ; Z ) satisfying (X  Z  )  0 and Tr(X Z )  .

Proof:

We rst prove that the default choice of  always allows a feasible step. To this end, note that pn k I k kV k   1  =  pn =  pn  2 = 2  2 ; 1

0

1

2

2

12

2

1

2

2

since 0   I  V . Thisp shows that  meets the rst condition of Lemma 3.4. Moreover, it holds that kV k   n, which implies 4  4p  = p 4 > : kV k  n  n The default choice of  therefore meets the conditions of Lemma 3.4 and ensures a feasible Dikin step. The initial duality gap is Tr(X Z ) which is reduced at each iteration by at least a factor (1 ? 1=n ) (Lemma 4.1). After k iterations the duality gap will be smaller than  if  k 1 1 ? n Tr(X Z )  : Taking logarithms yields     k ln 1 ? n1 + ln Tr(X Z )  ln(): 1

2

2

2

1

1

2

0

2

0

0

0

0

0

0

0

0

Since



   1 1 ? ln 1 ? n  n ; this will certainly be satis ed if k  ln Tr(X Z ) ? ln  = ln Tr(X Z ) n  which implies the required result. 0

0

0

0

0

0

0

2

The O( n) complexity bound is a factor pn worse than the best known bound for primal{dual algorithms, but this is due to the use of large neighbourhoods of the central path. 0

5 The pure primal{dual ane scaling method The analysis of the pure primal{dual ane scaling algorithm is analogous to that of the Dikin step method, but there is one signi cant dierence: Whereas the Dikin steps stay in the same neighbourhood of the central path, the same is not true of the pure ane scaling steps. The deviation from centrality at each step can be bounded at each iteration, though, and polynomial complexity can be retained at a price: the step length has to be shortened to 1 ; (11)  = nL and the worst case iteration complexity bound becomes O( nL ). We need to modify the analysis of the Dikin step algorithm with regard to the following: 0

0

13

2

 We allow for an increase in the distance (XZ ) from the central path by a constant

factor t > 1 at each step;  The steplength  in (11) is shown to be feasible for  nL iterations, provided that:  We choose the factor t in such a way that the distance from the central path stays within the bound (XZ ) < 3 for O(nL ) iterations { the convergence criterion is met before the deviation from centrality becomes worse than 3 . 0

2

2

0

0

6 The pure primal{dual ane search direction The search direction is the steepest descent direction of the function f (DV ) = Tr(V DV + V ) which gives the duality gap after a step DV in the scaled V -space, i.e. DV = ?rf (DV ) = ?V . As before, we can write this in terms of X and Z to obtain
X + D
ZD = ?X: The primal{dual ane scaling direction is the solution of this equation subject to the feasible search direction conditions (1). A feasible step along this direction gives the following reduction in the duality gap: 2

Lemma 6.1 Given a feasible primal{dual pair (X; Z ) and assume that the ane scaling step with steplength  is feasible, i.e. X () := X + 
X  0, and Z () := Z + 
Z  0. It holds that Tr (X ()Z ())  (1 ? )Tr(XZ ): Proof: 2

Analogous to the proof of Lemma 4.1. As with the Dikin step analysis, we will also need the following bound:

Lemma 6.2 For the primal{dual ane scaling step DV = DX + DZ = ?V , one has (DX DZ + DZ DX )  21 kV k : 2

Proof: 2

Follows from Lemma 3.2. 14

Now let  = (XZ ) and  = (X Z ) for the current pair of iterates (X; Z ) and starting solution (X ; Z ) respectively, and let  ;  satisfy (10). We also de ne the ampli cation factor t := 1 + nL1  ; which is used to bound the deviation from centrality in a given iteration. 0

0

0

0

0

1

2

2

Lemma 6.3 If  

 t

3 0

feasible for the step size

0

, then the steps X () = X + 
X and Z () = Z + 
Z are

1 ;  = nL 0

and the deviation from centrality is bounded by  (X ()Z ())  t:

Proof: As in the proof of Lemma 3.4, we show that the determinant of X ()Z () remains positive for all   , which ensures X (); Z ()  0 by Lemma 3.1. As before, note that X ()Z ()  (V + DX )(V + DZ ) = V + DX V + V DZ +  DX DZ = (1 ? )V + 21  (DX DZ + DZ DX )   1 1 + 2  (DX DZ ? DZ DX ) + 2 (DX V + V DZ ? V DX ? DZ V ) ; since DX + DZ = ?V . The matrix in square brackets is skew{symmetric. Lemma 3.3 therefore implies that the determinant of [X ()Z ()] will be positive if the matrix M () := (1 ? )V + 21  (DX DZ + DZ DX ) is positive de nite. Note that M (0) = V  0 and  (M (0)) =  . We proceed to prove that  (M ()) remains bounded by (M ())  t for 0     , for the xed ampli cation factor t. This is sucient to prove that M ()  0; 0     , and therefore that a step of length  is feasible. Moreover, after such a feasible step we will have X ()  0, Z ()  0. The matrix X ()Z () therefore has positive eigenvalues and we can apply the second part of Lemma 3.3 to obtain  (X ()Z ())   (M ())  t: 2

2

2

2

2

2

2

2

To start the proof, note that  (1?)I + 21  (DX DZ +DZ DX )  M ()   (1?)I + 12  (DX DZ +DZ DX ) 8 0    : 1

2

2

15

2

We will therefore certainly have (M ())  t if t [ (1 ? )I + 21  (DX DZ + DZ DX )]   (1 ? )I + 21  (DX DZ + DZ DX ): Using  =  the last relation becomes (12)  (1 ? )(t ? 1)I + 12  (t ? 1)(DX DZ + DZ DX )  0: Since one has (DX DZ + DZ DX )  kV k   n by Lemma 6.2, inequality (12) will hold if (1 ? )(t ? 1) ? 41  (t ? 1)n  0: (13) Using the assumption that t  3 , it follows that (13) will surely hold if   (1 ? ) nL1  ? 14  (3 ? 1)n  0; 2 which is sati ed by  = nL0 . 2

1

2

2

2

1

2

2

1 2

1 2 2

2

2

0

2

2

0

0

1

We now investigate how many iterations can be performed while still satisfying the assumption (XZ )  3 =t of Lemma 6.3. 0

Lemma 6.4 One has

(XZ )  3

0

for the rst nL 0 iterations of the pure primal{dual ane scaling algorithm. 2

Proof:

By Lemma 6.3 one has

(XZ )   tk after k iterations, provided that k is suciently small to guarantee  tk  3 . Using t = 1 + nL2 0 , we obtain  k 1 k  t =  1 + nL  < 3 if k  nL  ; which gives the required result. 2 0

0

0

0

2

2

0

0

1

0

0

It only remains to prove that nL  iterations are sucient to guarantee convergence. This is easily proved, analogously to the proof of Theorem 4.1. Formally we have 2

0

Theorem 6.1 Let  > 0 be an accuracy parameter,  = (X Z ), L = ln (Tr(X Z )=) 0

0

0

0

0

and  = 01nL . The pure primal{dual ane scaling algorithm requires at most 0 nL2 iterations to compute a feasible primal{dual pair (X  ; Z  ) satisfying (X  Z  )  30 and Tr(X  Z )  . 16

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