Polynomial primal-dual affinescaling algorithms in semidefinite

programming

Report 96-42

E. de KlerkC. Roos

T. Terlaky

Technische Universiteit DelftDelft University of Technology

Faculteit der Technische Wiskunde en InformaticaFaculty of Technical Mathematics and Informatics

ISSN 0922-5641

Copyright c 1996 by the Faculty of Technical Mathematics and Informatics, Delft, TheNetherlands.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 andInformatics, Delft University of Technology, The Netherlands.

Copies of these reports may be obtained from the bureau of the Faculty of TechnicalMathematics and Informatics, Julianalaan 132, 2628 BL Delft, phone+31152784568.A selection of these reports is available in PostScript form at the Faculty’s anonymousftp-site. They are located in the directory /pub/publications/tech-reports at ftp.twi.tudelft.nl

DELFT UNIVERSITY OF TECHNOLOGYREPORT Nr. 96{42Polynomial primal{dual affine scalingalgorithms in semidefinite programmingE. de Klerk, C. Roos, T. Terlaky

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

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: e.deklerk@twi.tudelft.nl, c.roos@twi.tudelft.nl, t.terlaky@twi.tudelft.nl

Copyright c 1996 by Faculty of Technical Mathematics andInformatics, Delft, The Netherlands.No part of this Journal may be reproduced in any form, byprint, photoprint, micro�lm or any other means without writ-ten permission from Faculty of Technical Mathematics and In-formatics, Delft University of Technology, The Netherlands.ii

AbstractTwo primal{dual a�ne scaling algorithms for linear programming are extendedto semide�nite programming. The algorithms do not require (nearly) centeredstarting solutions, and can be initiated with any primal{dual feasible solution.The �rst algorithm is the Dikin-type a�ne scaling method of Jansen et al.[8] and the second the pure a�ne scaling method of Monteiro et al. [12].The extension of the former has a worst-case complexity bound of O(�0nL)iterations, where �0 is a measure of centrality of the the starting solution, andthe latter a bound of O(�0nL2) iterations.Key words: interior{point method, primal{dual method, semide�nite pro-gramming, Dikin steps.

iii

1 IntroductionThe 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 likelythat the computation was done with a variant of the a�ne scaling method, proposed byDikin nearly two decades earlier in 1967 [3]. The two algorithms are closely related, andmodi�cations of Karmarkar's algorithm by Vanderbei et al. [16] and Barnes [1] proved tobe a rediscovery of the a�ne scaling method. Interestingly enough, polynomial complexityof the a�ne scaling method in its original form has still not been proved.A primal{dual a�ne 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 theprimal{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, withoutattempting to stay centered. The worst{case iteration complexity for this method isO(nL2). Jansen et al. proposed a primal{dual Dikin-type a�ne scaling variant in [8] withimproved O(nL) polynomial complexity. This search direction minimizes the duality gapover the so-called Dikin ellipsoid in the primal{dual space. The advantage of this methodis that each step involves both centering and reduction of the duality gap.In this paper we generalize both these primal{dual a�ne scaling methods to semide�niteprogramming (SDP). The extension of interior point methods from LP to SDP is currentlyan active research area. The �rst algorithms to be extended were potential reductionmethods [13, 17], and recently much work has been done on primal{dual central pathfollowing algorithms [7, 10, 11, 15, 14, 5]. The methods here do not belong to any ofthese two classes, and as such constitute a new approach. In particular, a nearly centeredstarting solution is not required, although the worst case complexity bounds depend onthe degree of centrality of the starting solution.The importance of algorithms which can start from arbitrary feasible points is discussedby Goldfarb and Scheinberg in [6], where they study trajectories leading to the optimalset from arbitrary feasible starting points.The semide�nite programming problemWe will work with the following standard SDP formulation of the primal problem (P):minTr(CX)subject to Tr(AiX) = bi; i = 1; : : : ;mX � 0and of the dual problem (D): max bTy1

subject to mXi=1 yiAi + Z = CZ � 0where C and the matrices Ai are symmetric n�n matrices, b; y 2 IRm and `X � 0' meansX is symmetric positive semi{de�nite. The matrices Ai are further assumed to be linearlyindependent. We will assume that a strictly feasible pair (X;Z) exists. This ensures azero duality gap (Tr(XZ) = 0) at an optimal primal{dual pair [17]. These assumptionsmay be made without loss of generality: A general primal{dual pair of SDP problemsmay be embedded in a self{dual SDP problem with nonempty interior and known interiorfeasible solution [2].The optimality conditions for the pair of problems (P) and (D) areTr(AiX) = bi; i = 1; : : : ;mmXi=1 yiAi + Z = CXZ = 0X;Z � 0:The system of relaxed optimality conditions:Tr(AiX) = bi; i = 1; : : : ;mmXi=1 yiAi + Z = CXZ = �IX;Z � 0has a unique solution fX(�); y(�); Z(�)g which gives a parametric representation of thecentral path as a function of � [17].Primal{dual a�ne scaling search directionsAny set of feasible primal{dual search directions (�X;�y;�Z) must satisfyTr (Ai�X) = 0; i = 1; : : : ;mPmi=1�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 directionof the Dikin{type algorithm presented here minimizes this duality gap over the so-called2

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�X +D�ZD = � XZX(Tr(XZ)2) 12 ; (2)subject to the conditions (1), where D is the scaling{matrixD := Z� 12 �Z 12XZ 12� 12 Z� 12 : (3)The primal{dual pure a�ne scaling search direction is the steepest descent direction forthe duality gap Tr(XZ). The computation of this direction involves the solution of thesystem �X +D�ZD = �X; (4)subject to (1).Measure of centralityThe Dikin steps have the feature that the proximity to the central path is maintained,where this proximity is quanti�ed by�(XZ) = �max(XZ)�min(XZ) (5)with �max(XZ) the largest eigenvalue of XZ and �min(XZ) the smallest1. The pure a�nescaling steps may become increasingly less centered with respect to this measure whichcomplicates the analysis somewhat. Note that �(XZ) � 1 and �(XZ) = 1 if and only ifXZ = �I for some � > 0, i.e. if the pair (X;Z) is centered with parameter �.The AlgorithmsThe two primal{dual a�ne scaling algorithms can be described in the following framework:1The eigenvalues of XZ are real and positive if X;Z � 0, since XZ � X� 12 [XZ]X 12 = X 12ZX 12 � 0, where'�' denotes the similarity relation.3

Generic primal{dual a�ne scaling type algorithmInput:A strictly feasible pair (X0; Z0);A parameter �0 > 1 such that � (X0Z0) � �0;An accuracy parameter � > 0.beginL := ln Tr(X0Z0)� ;� := 1pn�0 (Dikin steps);� := 1nL�0 (Pure a�ne scaling steps);X := X0; Z := Z0;while Tr(XZ) > � dobeginCompute �X, �Z from (2) and (1) (Dikin steps)or from (4) and (1) (Pure a�ne scaling steps);X := X + ��X;Z := Z + ��Z;endendWe prove that the Dikin step algorithm computes a strictly feasible �-optimal solution(X�; Z�) inO(�0nL) steps, and this solution satis�es �(X�Z�) � �0. The pure primal{duala�ne scaling algorithm converges in O(�0nL2) steps, and the solution satis�es �(X�Z�) �3�0.The paper is structured as follows. The Dikin step method is presented �rst, and itssimple analysis is then extended to the pure a�ne scaling method. In Section 2 is shownhow the Dikin step direction is derived by working in a scaled primal{dual space. InSection 3 conditions to ensure a feasible steplength are derived, and convergence and thepolynomial complexity result are proven in Section 4. In Sections 5 and 6 the analysis isextended to the pure primal-dual a�ne scaling algorithm.Notation �i(A) : ith eigenvalue of the n� n matrix A;�max(A) = maxi �i(A); if �i(A) 2 IR 8i;4

�min(A) = mini �i(A); if �i(A) 2 IR 8i;A � 0 : A is symmetric positive semide�nite;kAk2 = Tr(AAT ) =Xi Xj A2ij (Fr�obenius norm)= Xi �2i (A) if A symmetric;kAk2 = (�max(ATA)) 12 (Spectral norm)= �max(A) if A � 0;�(A) = maxi j�i(A)j (spectral radius of A);�(A) = �max(A)�min(A) if �i(A) > 0 8i= condition number of A if A � 0;A � B : The matrices A and B are similar.2 Dikin steps in semide�nite programmingFor strictly feasible solutions X � 0 and Z � 0 to (P) and (D) respectively, the scalingmatrix D de�ned in (3) satis�es D�1X = ZD, orD� 12XD� 12 = D 12ZD 12 := V:In other words, the matrix D may be used to scale the variables X and Z to the samesymmetric positive de�nite matrix2 V . Note thatV 2 = D� 12XZD 12 � XZ;i.e. V 2 has the same eigenvalues as XZ and is symmetric positive de�nite. As a conse-quence the duality gap is given byTr(XZ) = Tr(V 2) = kV k2 =Xi �2i (V ):We can similarly scale the primal{dual search directions �X and �Z viaDX = D� 12�XD� 12and DZ = D 12�ZD 12 :2The matrix V is not unique and depends on the factorization of D [15]. We have used the symmetric squareroot factorization D = D 12D 12 but any LLT factorization can be used. The factors only appear in the analysisthough and not in the algorithm, the important point being that V 2 is obtained via a similarity transformationof XZ in all cases, and the analysis is concerned only with the eigenvalues of XZ.5

The scaled directions DX and DZ are orthogonal by the orthogonality of �X and �Z,i.e. Tr(DXDZ) = 0. In particular, we haveDZ = � mXi=1�yiD 12AiD 12 ;i.e. DZ must be in the span of matrices D 12AiD 12 and DX in its orthogonal complement,i.e. Tr �D 12AiD 12DX� = 0; i = 1; : : : ;m:The scaled Newton step in the V-space is denoted by DV = DX +DZ . After a feasibleprimal{dual step �X, �Z the duality gap becomesTr ((X +�X)(Z +�Z)) = Tr ((V +DX)(V +DZ))= Tr(V 2 + V DV ):The search direction of the algorithm is derived by minimizing the duality gap over theDikin ellipsoid: kV �1DV k � 1:In other words the search direction is the solution tominDV Tr(V 2 + V DV )subject to kV �1DV k � 1DV = DX +DZDZ 2 span fD 12A1D 12 ; : : : ;D 12AmD 12gDX 2 span fD 12A1D 12 ; : : : ;D 12AmD 12g?:It is easily veri�ed that the optimal solution is given byD�V = D�X +D�Z = � V 3kV 2k: (6)The transformation back to the unscaled space is done by premultiplying and postmulti-plying (6) by D 12 to obtain �X +D�ZD = �XZX(Tr(XZ)2) 12 : (7)The primal{dual a�ne scaling direction (Dikin step direction) is obtained by solving (7)subject to the conditions (1). It is easily shown that (7) and (1) implynXj=1�yjTr (AiDAjD) = �Tr (AiXZX)(Tr(XZ)2) 12 ; i = 1; : : : ;m: (8)The solution of this m�m linear system yields �y. (Note that the coe�cient matrix ofthe system (8) is positive de�nite [4]). Once �y is known, �Z follows fromPmi=1�yiAi =��Z, and �X is subsequently obtained from (7).6

A note on the centrality measureThe centrality measure �(XZ) de�ned in (5) satis�es0 � 1� 1�(XZ)with equality holding on the central path. A di�erent measure for proximity to the centralpath was used by Sturm and Zhang [15]:�(XZ) := I � 1�V 2 ;where � = Tr(XZ)=n = Tr(V 2)=n. Note that one may have �(XZ) = O(pn) if �(XZ) =O(1). This shows that �(XZ) � � de�nes a larger neighbourhood of the central paththan �(XZ) � � . The following relation between the two measures is readily proved:�(XZ) = I � nTr(V 2)V 2! � pn I � nTr(V 2)V 2! 2= pn 1� n�min(V 2)Pi �i(V 2) !� pn 1� �min(V 2)�max(V 2)! = pn 1� 1�(XZ)! :3 Feasibility of the Dikin stepHaving computed the Dikin step direction (�X;�Z), a feasible steplength must be es-tablished. Denoting X(�) := X + ��X; Z(�) := Z + ��Z;we establish a value �� > 0 such that X(��) � 0 and Z(��) � 0. The following lemma givesa su�cient 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 di�erentiable if � 2 (0; ��) and � 2 IR and is zero if � is an eigenvalue ofX(�). The implicit function theorem therefore implies that the eigenvalues of X(�) (andsimilarly of Z(�)) are continuous functions of �.Since det(X(�)Z(�)) = det(X(�)) det(Z(�)), one hasdet(X(�)Z(�)) =Yi �i(X(�))Yi �i(Z(�)) (9)The left hand side of eq. (9) is strictly positive on [0; ��]. This shows that the eigenvaluesof X(�) and Z(�) remain positive on [0; ��]. 2In order to derive bounds on � which are su�cient to guarantee a feasible steplength, weneed the following three technical results.Lemma 3.2 The spectral radius of (DXDZ +DZDX) is bounded by�(DXDZ +DZDX) � 12kDX +DZk2:Proof:It is trivial to verify thatDXDZ +DZDX = 12 h(DX +DZ)2 � (DX �DZ)2iwhich implies �12(DX �DZ)2 � DXDZ +DZDX � 12(DX +DZ)2:It follows that �12kDX �DZk2I � DXDZ +DZDX � 12kDX +DZk2I:Since DX and DZ are orthogonal the matrices (DX +DZ) and (DX �DZ) have the samenorm. Consequently�12kDX +DZk2I � DXDZ +DZDX � 12kDX +DZk2Ifrom which the required result follows. 28

Corollary 3.1 For the Dikin step DX +DZ = �V 3=kV 2k, one has�(DXDZ +DZDX) � 12�(V 2):Proof:By Lemma 3.2 one has2� (DXDZ +DZDX) � kDX +DZk2= kV 3kkV 2k!2= Tr(V 6)(kV 2k)2� �(V 2) Tr(V 4)(kV 2k)2 = �(V 2)which is the required result. 2Lemma 3.3 Let Q be an n� n real symmetric matrix, S an n� n real skew{symmetricmatrix. One has det(Q + S) > 0 if Q � 0. Moreover, if it is known that �i(Q + S) 2IR; 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 sincexT (Q+ S)x = xTQx > 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 whichis 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 thenfollows that Q � �I � 0. By the same argument as above we then have (Q + S) � �Inonsingular, or det ((Q+ S)� �I) 6= 0:This implies that � cannot be an eigenvalue of (Q+ S). Similarly, (Q+ S) cannot havean eigenvalue smaller than �min(Q). This gives the required result. 29

We are now in a position to �nd a step size � which guarantees that the Dikin stepwill be feasible. To simplify the analysis we introduce a parameter � > 1 such that�(XZ) = �(V 2) � � . This implies the existence of numbers �1 and �2 such that�1I � V 2 � �2I; �2 = �1�: (10)Lemma 3.4 The steps X(�) = X + ��X and Z(�) = Z + ��Z are feasible if the stepsize � satis�es � � �� where �� = min(kV 2k2�2 ; 4�1kV 2k) :Furthermore � (X(��)Z(��)) � �:Proof:We show that the determinant of X(�)Z(�) remains positive for all � � ��. One then hasX(��); Z(��) � 0 by Lemma 3.1.To this end note thatX(�)Z(�) � (V + �DX)(V + �DZ )= V 2 + �DXV + �V DZ + �2DXDZ= V 2 � �V 4kV 2k + 12�2(DXDZ +DZDX )+ �12�2(DXDZ �DZDX) + 12�(DXV + V DZ � V DX �DZV )� ;since DX +DZ = �V 3=kV 2k. The matrix in square brackets is skew{symmetric. Lemma3.3 therefore implies that the determinant of [X(�)Z(�)] will be positive if the matrixM(�) := V 2 � �V 4kV 2k + 12�2(DXDZ +DZDX)is positive de�nite. Note that M(0) = V 2 � 0 and � (M(0)) � � . We proceed to provethat � (M(�)) remains bounded by �(M(�)) � � for 0 � � � �� . This is su�cient toprove 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 matrixX(��)Z(��) therefore has positive eigenvalues and we can apply the second part of Lemma3.3 to obtain � (X(��)Z(��)) � � (M(��)) � �:We start the proof by noting that if � is an eigenvalue of V 2 then (� � ��2=kV 2k) is aneigenvalue of [V 2 � �V 4=kV 2k]. The function�(t) := t� � t2kV 2k10

is monotonically increasing on t 2 [0; �2] if � � ��, since �� � kV 2k=(2�2). Thus�(�1)I � V 2 � �V 4kV 2k � �(�2)I 8 0 � � � ��or�(�1)I + 12�2(DXDZ +DZDX) �M(�) � �(�2)I + 12�2(DXDZ +DZDX) 8 0 � � � ��:We will therefore certainly have �(M(�)) � � if� [�(�1)I + 12�2(DXDZ +DZDX)] � �(�2)I + 12�2(DXDZ +DZDX):This matrix inequality can be simpli�ed using �2 = ��1 and subsequently dividing by �.This yields � 22 � �� 21kV 2k ! I + �(� � 1)�12(DXDZ +DZDX)� � 0:This may be further simpli�ed using� 22 � �� 21 = (� � 1)�1�2to obtain �1�2kV 2k! I + 12�(DXDZ +DZDX ) � 0which will surely hold if �1�2kV 2k! I � ���12(DXDZ +DZDX)� I � 0:Substituting the bound ��12(DXDZ +DZDX )� � 14�(V 2) � 14�2from Corollary 3.1 yields �1�2kV 2k � 14��2 � 0;or � � 4�1kV 2kwhich is the second bound in the lemma. This completes the proof. 211

4 Convergence and Complexity analysisA feasible Dikin step of length � reduces the duality gap by at least a factor (1 � �pn).Formally we haveLemma 4.1 Given a feasible primal{dual pair (X;Z) and a steplength � such that theDikin step is feasible, i.e. X(�) := X + ��X � 0, and Z(�) := Z + ��Z � 0. It holdsthat Tr (X(�)Z(�)) � 1� �pn!Tr(XZ):Proof:The duality gap after the Dikin step is given byTr (X(�)Z(�)) = Tr ((V + �DX)(V + �DZ))= Tr �V 2 + �V (DX +DZ)�= Tr V 2 � � V 4kV 2k!= kV k2 � �kV 2k= 1� �kV 2kkV k2!Tr(XZ):By the Cauchy-Schwarz inequality one haskV k2 = Tr �IV 2� � kIkkV 2k = pn V 2 ;which gives the required result. 2We are now in a position to prove a worst-case iteration complexity bound.Theorem 4.1 Let � > 0 be an accuracy parameter, �0 = �(X0Z0), L = ln (Tr(X0Z0)=�),and � = 1�0pn . The Dikin Step Algorithm requires at most �0nL iterations to compute afeasible 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, notethat � = 1�0pn = �1�2pn � �1pn2�2 = k�1Ik2�2 � kV 2k2�2 ;12

since 0 � �1I � V 2. This shows that � meets the �rst condition of Lemma 3.4. Moreover,it holds that kV 2k � �2pn, which implies4�1kV 2k � 4�1�2pn = 4�0pn > �:The default choice of � therefore meets the conditions of Lemma 3.4 and ensures a feasibleDikin step.The initial duality gap is Tr(X0Z0) 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�1 � 1n�0�k Tr(X0Z0) � �:Taking logarithms yieldsk ln�1� 1n�0�+ ln �Tr(X0Z0)� � ln(�):Since � ln�1� 1n�0� � � 1n�0� ;this will certainly be satis�ed ifkn�0 � ln �Tr(X0Z0)�� ln � = ln Tr(X0Z0)�which implies the required result. 2The O(�0n) complexity bound is a factor pn worse than the best known bound forprimal{dual algorithms, but this is due to the use of large neighbourhoods of the centralpath.5 The pure primal{dual a�ne scaling methodThe analysis of the pure primal{dual a�ne scaling algorithm is analogous to that of theDikin step method, but there is one signi�cant di�erence: Whereas the Dikin steps stayin the same neighbourhood of the central path, the same is not true of the pure a�nescaling 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 beshortened to � = 1nL�0 ; (11)and the worst case iteration complexity bound becomes O(�0nL2).We need to modify the analysis of the Dikin step algorithm with regard to the following:13

� We allow for an increase in the distance �(XZ) from the central path by a constantfactor t > 1 at each step;� The steplength � in (11) is shown to be feasible for �0nL2 iterations, provided that:� We choose the factor t in such a way that the distance from the central path stayswithin the bound �(XZ) < 3�0 for O(�nL2) iterations { the convergence criterionis met before the deviation from centrality becomes worse than 3�0.6 The pure primal{dual a�ne search directionThe search direction is the steepest descent direction of the functionf(DV ) = Tr(V DV + V 2)which gives the duality gap after a step DV in the scaled V -space, i.e. D�V = �rf(DV ) =�V . As before, we can write this in terms of X and Z to obtain�X +D�ZD = �X:The primal{dual a�ne scaling direction is the solution of this equation subject to thefeasible search direction conditions (1). A feasible step along this direction gives thefollowing reduction in the duality gap:Lemma 6.1 Given a feasible primal{dual pair (X;Z) and assume that the a�ne scalingstep 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:Analogous to the proof of Lemma 4.1. 2As with the Dikin step analysis, we will also need the following bound:Lemma 6.2 For the primal{dual a�ne scaling step DV = DX +DZ = �V , one has�(DXDZ +DZDX) � 12kV k2:Proof:Follows from Lemma 3.2. 214

Now let � = �(XZ) and �0 = �(X0Z0) for the current pair of iterates (X;Z) and startingsolution (X0; Z0) respectively, and let �1; �2 satisfy (10). We also de�ne the ampli�cationfactor t := 1 + 1nL2�0 ;which is used to bound the deviation from centrality in a given iteration.Lemma 6.3 If � � 3�0t , then the steps X(�) = X + ��X and Z(�) = Z + ��Z arefeasible for the step size �� = 1nL�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 ofX(�)Z(�) remains positivefor all � � ��, which ensures X(��); Z(��) � 0 by Lemma 3.1.As before, note thatX(�)Z(�) � (V + �DX)(V + �DZ )= V 2 + �DXV + �V DZ + �2DXDZ= (1 � �)V 2 + 12�2(DXDZ +DZDX)+ �12�2(DXDZ �DZDX) + 12�(DXV + V DZ � V DX �DZV )� ;since DX + DZ = �V . The matrix in square brackets is skew{symmetric. Lemma 3.3therefore implies that the determinant of [X(�)Z(�)] will be positive if the matrixM(�) := (1� �)V 2 + 12�2(DXDZ +DZDX)is positive de�nite. Note that M(0) = V 2 � 0 and � (M(0)) = � . We proceed toprove that � (M(�)) remains bounded by �(M(�)) � t� for 0 � � � ��, for the �xedampli�cation factor t. This is su�cient to prove that M(�) � 0; 0 � � � ��, andtherefore that a step of length �� is feasible.Moreover, after such a feasible step we will have X(��) � 0, Z(��) � 0. The matrixX(��)Z(��) therefore has positive eigenvalues and we can apply the second part of Lemma3.3 to obtain � (X(��)Z(��)) � � (M(��)) � t�:To start the proof, note that�1(1��)I+12�2(DXDZ+DZDX) �M(�) � �2(1��)I+12�2(DXDZ+DZDX) 8 0 � � � ��:15

We will therefore certainly have �(M(�)) � t� ift� [�1(1� �)I + 12�2(DXDZ +DZDX)] � �2(1� �)I + 12�2(DXDZ +DZDX):Using �2 = ��1 the last relation becomes�2(1 � �)(t� 1)I + 12�2(t� � 1)(DXDZ +DZDX) � 0: (12)Since one has �(DXDZ + DZDX ) � 12kV k2 � 12�2n by Lemma 6.2, inequality (12) willhold if (1� �)(t� 1)� 14�2(t� � 1)n � 0: (13)Using the assumption that t� � 3�0, it follows that (13) will surely hold if(1 � �)� 1nL2�0�� 14�2(3�0 � 1)n � 0;which is sati�ed by �� = 1nL�0 . 2We now investigate how many iterations can be performed while still satisfying the as-sumption �(XZ) � 3�0=t of Lemma 6.3.Lemma 6.4 One has �(XZ) � 3�0for the �rst nL2�0 iterations of the pure primal{dual a�ne scaling algorithm.Proof:By Lemma 6.3 one has �(XZ) � �0tk after k iterations,provided that k is su�ciently small to guarantee �0tk � 3�0. Using t = 1 + 1nL2�0 , weobtain �0tk = �0 �1 + 1nL2�0�k < 3�0 if k � nL2�0;which gives the required result. 2It only remains to prove that nL2�0 iterations are su�cient to guarantee convergence.This is easily proved, analogously to the proof of Theorem 4.1. Formally we haveTheorem 6.1 Let � > 0 be an accuracy parameter, �0 = �(X0Z0), L = ln (Tr(X0Z0)=�)and � = 1�0nL . The pure primal{dual a�ne scaling algorithm requires at most �0nL2iterations to compute a feasible primal{dual pair (X�; Z�) satisfying �(X�Z�) � 3�0 andTr(X�Z�) � �. 16

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