1 1. Introuction The central path plays a funamental role in the interior point methoology, both for linear an semienite programming. Megio [10] showe

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1 Revise August, 1998 ON WEIGHTED CENTERS FOR SEMIDEFINITE PROGRAMMING Jos F. Sturm 1 an Shuzhong Zhang 1 ABSTRACT In this paper, we generalize the notion of weighte centers to semienite programming. Our analysis ts in the v-space framework, which is purely base on the symmetric primal-ual transformation an oes not make use of barriers. Existence an scale invariance properties are proven for the weighte centers. Relations with other primal-ual maps are iscusse. Key wors. semienite programming, symmetric primal{ual transformation, weighte center. 1 Econometric Institute, Erasmus University Rotteram, The Netherlans.

2 1 1. Introuction The central path plays a funamental role in the interior point methoology, both for linear an semienite programming. Megio [10] showe some highly interesting properties of the central path for linear programming. The fact that -centers are the minimizers of the logarithmic barrier function with parameter plays a crucial role in his analysis. With the introuction of primal-ual interior point methos by Kojima, Mizuno an Yoshise [6], it became more popular to ene -centers as the solutions of the perturbe KKT-systems x i z i = for i =1 ::: n where x < n enotes the primal nonnegative ecisionvariable an z < n is the corresponing ual slack variable. The analysis of the central path for semienite programming evolves in a similar way. Namely, early enitions of the central path are base on the logarithmic barrier (Nesterov an Nemirovsky [13], Vanenberghe an Boy [18]), whereas Kojima, Shinoh an Hara [7] propose a general framework for primal-ual path following algorithms for semienite programming, in which the-centers are ene as solutions of the perturbe KKT-systems XZ = I where X < nn (Z < nn ) is the primal (ual) positive enite ecision variable, an I is the orer n ientity matrix. With this enition, some basic properties of the central path were obtaine by Luo, Sturm an S. Zhang [8]. Inepenently, the central path was investigate by Golfarb an Scheinberg [] base on the logarithmic barrier enition. In linear programming, the concept of weighte centers has playe an increasingly important role in recent years, largely ue to the unifying works of Kojima et al. [5] an Jansen et al. [4]. The weighte center for linear programming can be ene as the minimizer of a W -weighte logarithmic barrier function or alternatively as the solution of the perturbe KKT system x i z i = w i for i =1 ::: n for some w < n +. Due to the existence of the weighte centers, it has been popular to consier the iterates in a homogenize prouct space of x an z, viz. the v-space [4, 5]. This proves to be an elegant an convenient way of analyzing various primal-ual interior point algorithms for linear programming. However, the generalization of this concept to semienite programming is not straightforwar. The major obstacle is the lack of a proper weighte barrier function for semienite programming.

3 In this paper, we propose to ene weighte centers in terms of the eigenvalues of the matrix XZ. It is shown that if the primal-ual Slater conition hols, then any set of n positive eigenvalues can be attaine by a feasible primal-ual solution pair (X Z). Moreover, this notion of weighte center is scale-invariant (see To, Toh an Tutuncu [17] for a enition). A ierent generalization of weighte centers towars semienite programming was recently propose by Monteiro an Pang [1]. Their weighte center has the property of uniqueness, but lacks some other esirable properties as we will iscuss later. This paper is organize as follows. In Section, we review some terminology an we ene the concept of weighte center for semienite programming. We will also iscuss how Newton's metho can be use to erive search irections that can be use in an interior point metho for semienite programming. The quality of these search irection is estimate in Section 3. We show in Section 4 that for any positive enite matrix W, there exists a W -weighte center uner the primal-ual Slater conition. In Section 5, we iscuss the new enition of weighte centers in relation to the Monteiro-Pang weighte centers [1]. Notation. Given X an Y in < nn, the stanar inner prouct is ene by X Y =trx T Y: The Eucliean norm an its associate operator norm, viz. the spectral norm, are both enote by kk. The Frobenius norm of a matrix X < nn is kxk F = p X X. It is well known that if 1 ::: n are the singular values of an n n matrix X, then X n kxk = F i kxk = max i: 1in i=1 Hence, kxk kxk F. We will also make use of the obvious inequality for n n matrices X Y Z that kxy Zk F kxkky k F kzk : For any X < nn an invertible Y < nn there hols eig(x) = eig(yxy ;1 ) (1) where eig(x) enotes the set of eigenvalues of X. The space of symmetric n n matrices is enote by S. Its orthogonal complement in< nn, viz. the space of skew symmetric n n matrices, is enote by S?. For given X < nn we let P S (X) := X + XT P S?(X) := X ; XT

4 3 enote the orthogonal projections of X on S an S? respectively. If X Sis positive (semi-) enite, we write X 0 (X 0). The cone of positive semi-enite matrices is enote by S + an the cone of positive enite matrices is S ++. For X S + welet min (X) enote the smallest eigenvalue of X. The orer n ientity matrix is enote by I.. The semienite programming problem an transformations.1. Problem statement Consier the primal-ual pair of semienite programming problems in conic formulation minfc X j X ; C 1 A X 0g () minfc 1 Z j Z ; C A? Z 0g (3) where C 1 S C S, A is a linear subspace of S with orthogonal complement A?. The semienite matrix X is the primal ecision variable, an Z is the ual ecision variable. The problems () an (3) satisfy the weak uality relation 0 X Z = C 1 Z + C X ; C 1 C for any feasible pair (X Z). The above relation shows that solving () an (3) is equivalent to minimizing the uality gap X Z over all feasible primal-ual pairs (X Z). We make the following assumption: Assumption 1. There exist solutions X 0 an Z 0 such that X ; C 1 Aan Z ; C A?. It is well known that with the above assumption (a primal-ual Slater conition), the pair ()-(3) is equivalent to ning a complementary solution pair, which isby enition a feasible solution pair (X Z) with X Z =0(i.e. no uality gap), see [1, 13, 9] among others. It is convenient to combine ()-(3) into the following formulation: (SDP) min X Z s:t: X ; C 1 A Z ; C A? X 0 Z 0: Notice that the set of complementary solution pairs for ()-(3) is the optimal solution set of (SDP).

5 4.. Primal-ual transformations Primal-ual transformations for semienite programming are base on the observation that given an invertible matrix L < nn,wehave X 0 if an only if L ;1 XL ;T 0: (4) The relation (4) implies that (SDP) is equivalent to the linearly transforme semienite programming problem Let minf X Z j L XL T ; C 1 A L ;T ZL ;1 ; C A? X 0 Z 0g: (5) A(L) :=fx SjLXL T Ag an remark that the orthogonal complement of A(L) in S is A(L)? = fz Sj L ;T ZL ;1 A? g: The set of feasible primal-ual pairs (X Z) for (5) is enote by F(L) :=f(x Z) S + S + j X ; L ;1 C 1 L ;T A(L) Z ; L T C L A(L)? g: We can now formulate (SDP) an (5) as minfx Z j (X Z) F(I)g =minf X Z j ( X Z) F(L)g =0: We let o F (L) enote the set of strictly feasible (or interior) solutions, i.e. o F (L) :=F(L) \ (S ++ S ++ ): Consier an interior solution pair (X Z) o F (I). It is shown in [16] that if L is an invertible n n matrix such that then L L T = Z ;1= (Z 1= XZ 1= ) 1= Z ;1= (6) L ;1 XL;T = L T ZL an vice versa. The subscript `' in L is reminiscent to the stanar notation use for primal-ual interior point methos in linear programming, where enotes a primal-ual scaling vector. The transformation X! L ;1 XL;T Z! L T ZL is known as a symmetric primal-ual transformation, since it maps both X an Z into the same positive enite matrix, viz. V = L ;1 XL;T = L T ZL (7) an we have (V V ) F(L ):

6 5.3. Denition of weighte centers Consier (X Z) o F (I) an let XZ be a iagonal matrix with the eigenvalues of the matrix XZ on its iagonal. Since X an Z are both positive enite, XZ must be a positive iagonal matrix, see (1). Moreover, the uality gap is the sum of the eigenvalues of XZ, X Z = tr XZ : If we want to approach the optimal solution set of (SDP) from the interior solution set o F (I), we see that the corresponing eigenvalue matrix XZ has to approach the origin (the matrix of all zeros) from the set of positive iagonal matrices. A -center [18] is a pair (X() Z()) o F (I) such that XZ () =I. It can easily be verie, e.g. by using (1), that XZ () =I if an only if X()Z() =I which means that X() an Z() commute in this case. It is known that the central path, ene as the set f(x() Z()) j >0g is a smooth curve [7]. Path-following algorithms generate a sequence of approximate -centers with # 0. A W -weighte center can now be ene as a pair (X Z) F o (I) such that 1= XZ some positive iagonal matrix W. Unlike the-center case however, X an Z are in general not commutable an 1= XZ = W oes not imply XZ = W. Moreover, there is no unique path f(x(t) Z(t)) o F (I) j XZ (t) =tw t>0g (8) as can be seen from the following example, with n =,where the set (8) is a -imensional surface. Example.1. Consier the semienite programming problem with ata C 1 = C = I A = A? = S: We are intereste in trajectories satisfying (8) with W = : For any smooth function : < +! [;1 1], the trajectory X(t) =I Z(t) =t is such a trajectory. 4 +p 3 1 ; (t) p (t) 5 (t) ; 1 ; (t)

7 6 However, if W = p I, i.e. the -center case, then the pair (X() Z()) is unique. This fact follows from the strict convexity of the barrier function (see [, 7]), but can also easily be shown using the symmetric primal-ual transformation: Lemma.1. Suppose ( p I p I) F o o (L ). If ( X Z) F (L ) an X Z = I then X = Z p = I. Proof: Let D X := 1 ( X ; p I) DZ := 1 ( Z ; p I): Then I = X Z = p ( I +DX )( p I +D Z ) = ( p I + D X + D Z +(D X ; D Z ))( p I + D X + D Z ; (D X ; D Z )) = ( p I + D X + D Z ) ; (D X ; D Z ) +(D X ; D Z )( p I + D X + D Z ) ; ( p I + D X + D Z )(D X ; D Z ): Using the fact that the above matrix is symmetric, it follows that I =( p I + D X + D Z ) ; (D X ; D Z ) ( p I + D X + D Z ) : Because D X + D Z ; p I it follows from the above inequality that D X + D Z 0: (9) However, using D X?D Z an X Z = I, wehave n = tr( X Z)=n p + tr(dx + D Z ) i.e. tr(d X + D Z ) = 0. Together with (9), this implies that D X + D Z = 0. From the orthogonality D X?D Z,we further obtain D X = D Z =0. This conclues the proof..4. The V -space Consier an interior solution pair (X Z) o F (I). From (7) we have V =(L ;1 XL;T )(L T ZL )=L ;1 XZL which shows that the eigenvalues of V are ientical to the eigenvalues of XZ, see (1). In fact, we maychoose L in such away that V = 1= XZ. Namely, suchanl can be compute by the following proceure (see To, Toh an Tutuncu [17]):

8 7 1. Compute Cholesky factorization L X, X = L X L T X :. Compute eigenvector-eigenvalue ecomposition (Q XZ ), L T XZL X = Q T XZ Q: 3. Let L = L X Q T ;1=4 XZ V =1= XZ : The weighte center with respect to a positive enite matrix W can now be characterize as a pair (X Z) o F (I) with V = L ;1 XL;T = L T ZL = W (10) for some nonsingular L. In Section 4 we show that such weighte centers inee exist for every W 0. With this characterization, we can ene weighte centers for any positive enite W, not necessarily iagonal. Similarly, having XZ approach 0 from the set of positive iagonal matrices correspons to letting V approach 0 from the cone of positive enite matrices. In this paper, we are intereste in computing a W -center (10) for given W S ++. However, the Newton equation for solving (10) is uneretermine, ue to the nonuniqueness of W - centers as illustrate by Example.1. well-ene by aing restrictions on the choice of L. In the sequel, we will make the Newton irection Consier a trajectory ( X(t) Z(t)) o F (L ), with X(0) = V Z(0) = V an let G(t) be the positive enite matrix ene by We let X(t) =G(t) Z(t)G(t) : V (t) =G(t) Z(t)G(t) =G(t) ;1 X(t)G(t) ;1 : (11) Obviously, G(0) = I an V (0) = V. Noticing that (V (t) V(t)) F(L G(t)) we see that V (t) is both primal an ual feasible if we use the transformation L G(t). Let D X = 1 X(t) j t=0 D Z = 1 Z(t) j t=0 : t t

9 8 Using (11), we cannow linearize V (t) as follows: V (t) = 1 h i G(t) ;1 X(t)G(t) ;1 + G(t) Z(t)G(t) = V + t(d X + D Z )+o(t): (1) o From the requirement ( X(t) Z(t)) F (L ), we further obtain D X A(L ) D Z A? (L ): (13) The linearization of V (t) will be enote by F (t). Letting D v = D X + D Z (14) it follows from (1) that F (t) :=V + td v : (15) Remark that (D X D Z ) is the orthogonal ecomposition of the matrix D v onto A(L ) an A(L )? respectively. We have erive that the Newton irection for solving the nonlinear equation V (1) = W is the unique solution (D X D Z ) of (13)-(14) with D v = W ; V. From now on, we let ( X(t) Z(t)) enote the solution pair in o F (L ) that is obtaine by a partial Newton step: X(t) =V +td X Z(t) =V +tdz : In the next section, we will estimate the error term kf (t) ; V (t)k F that emerges by linearizing V (t). 3. A boun on higher orer terms In this section, we will estimate the error term kf (t) ; V (t)k F V (t). We see that X(t) = X(t)+ Z(t) X(t) + ; Z(t) Z(t) = X(t)+ Z(t) of the linearization F (t) of ; X(t) ; Z(t) an X(t)+ Z(t) X(t) ; Z(t) = F (t) = t(d X ; D Z ) (17) where F (t) is ene by (15).Since D X?D Z,wehave kd X ; D Z k F = kd v k F : (18) For t 0 with F (t) 0, we ene (t) := (16) t kd vk F min (F (t)) : (19) The next lemma shows that X(t) an Z(t) are interior feasible solutions if (t) < 1.

10 9 Lemma 3.1. Let t 0 be such that F (t) 0. Then X(t) (1 ; (t)) min (F (t))i an Z(t) (1 ; (t)) min (F (t))i for all t [0 t]. If (t) < 1 then V (t) exists an is positive enite for t [0 t]. Proof: By enition (15), we have F (t) =F (t) ; (t ; t)d v where D v = F (1) ; V. Suppose 0 t t. Using (19), it follows that min (F (t)) (1 ; (1 ; t t )( t)) min (F (t)): Using (16)-(18), we have X(t) =F (t)+t(d X ; D Z ) F (t) ; t kd v k F I (1 ; (t)) min (F (t))i an similarly Z(t) (1 ; (t)) min (F (t))i: This shows that if (t) < 1 then X(t) an Z(t) are positive enite an hence V (t) is well ene. The above lemma provies a sucient conition for the existence of V (t). examine the ierence between V (t) an its linearization F (t). We will now Lemma 3.. Let t 0 be such that F (t) 0 an (t) < 1. There hols F (t) ; V (t) = 1 h i P S (I ; G(t))( Z(t) ; V (t))(i + G(t)) : Proof: First remark using (11) that Z(t) ; V (t) = Z(t) ; G(t) Z(t)G(t) =PS h (I ; G(t)) Z(t)(I + G(t)) i an X(t) ; V (t) =G(t)V (t)g(t) ; V (t) =;P S h (I ; G(t))V (t)(i + G(t)) i :

11 10 Aing the above two relations yiels X(t)+ Z(t) ; V (t) = 1 P S h (I ; G(t))( Z(t) ; V (t))(i + G(t)) i : Base on the above lemma, it is natural to boun kf (t) ; V (t)k F by eriving estimates for ki ; G(t)k an Z(t) ; V (t) F. Such estimates are given by Lemma 3.3 an Lemma 3.4 below. Lemma 3.3. Let t 0 be such that F (t) 0 an (t) < 1. There hols 1 ; (t) 1+(t) I G(t)4 1+(t) 1 ; (t) I an therefore kg(t) ; Ik h 1+(t) 1;(t) i 1=4 ; 1: Proof: Remark that ( Z(t) 1= G(t) Z(t) 1= ) = Z(t) 1= X(t) Z(t) 1= = Z(t)(I +t Z(t) ;1= (D X ; D Z ) Z(t) ;1= ) Z(t) (0) where the last equation follows from (17). From Lemma 3.1 an relation (18) we have Z(t) ;1= (D X ; D Z ) Z(t) ;1= Hence, using (0), ( Z(t) 1= G(t) Z(t) 1= ) (1 + which implies that G(t) 4 1+(t) 1 ; (t) I: kd X ; D Z k (1 ; (t)) min (F (t)) kd v k F (1 ; (t)) min (F (t)) : t kd v k F (1 ; (t)) min (F (t)) ) Z(t) = 1+(t) 1 ; (t) Z(t) In the above erivation we use the fact that if A 0 then BAB A implies kbk. This fact is a special case of Stein's theorem [15], but it is also easily prove irectly. Namely, let be an eigenvector of B corresponing to its eigenvalue which ismaximum in absolute value, B =. Then, pre- an post-multiplying T an on the both sies of BAB A yiels T A T A

12 11 an therefore kbk =. Using the primal-ual symmetry, it follows also that G(t) ;4 1+(t) 1 ; (t) I: Lemma 3.4. Let t 0 be such that F (t) 0 an (t) < 4=5. There hols p Z(t) ; V (t) F 1 ; (t) p p 3 1 ; (t) ; 1+(t) t kd vk F : Proof: We have Z(t) ; V (t)+g(t)( Z(t) ; V (t))g(t) = Z(t) ; X(t) =t(dz ; D X ) from which we obtain ( Z(t) ; V (t)) = t(dz ; D X )+P S h (G(t) ; I)(V (t) ; Z(t)) i +(G(t) ; I)(V (t) ; Z(t))(G(t) ; I): Applying the triangle inequality, it further follows that Z(t) ; V (t) F t kd v k F +(kg(t) ; Ik + kg(t) ; Ik ) Z(t) ; V (t) F or, equivalently, (3 ; (1 + kg(t) ; Ik) ) Z(t) ; V (t) F t kd v k F : (1) Remark now that Lemma 3.3 implies (1 + kg(t) ; Ik) 4 1+(t) 1 ; (t) : () Combining (1) an () we obtain for (t) < 4=5 that Z(t) ; V (t) F p 1 ; (t) 3 p 1 ; (t) ; p 1+(t) t kd vk F : Combining Lemma 3., 3.3 an 3.4, we obtain the following estimate for the error term kf (t) ; V (t)k F. Lemma 3.5. Let t 0 be such that F (t) 0 an (t) < =3. Then kf (t) ; V (t)k F (t)t kd vk F ; 3(t) :

13 1 Proof: From Lemma 3.3 we have (1 + (t))1=4 + (t))1=4 ki + G(t)kkI ; G(t)k (1 + )((1 ; 1) (1 ; (t)) 1=4 (1 ; (t)) 1=4 = Combining Lemma 3. an Lemma 3.4 with (3) we obtain p 1+(t) p 1 ; (t) ; 1: (3) kf (t) ; V (t)k F 1 ki + G(t)k Z(t) ; V (t) F ki ; G(t)k p p 1+(t) ; 1 ; (t) p p 3 1 ; (t) ; 1+(t) t kd vk F p 1+(t)=(1 ; (t)) ; 1 = p 3 ; 1+(t)=(1 ; (t)) t kd vk F : (4) Remark now that s s 1+ (t) 1 ; (t) 1+ (t) 1 ; (t) + (t) (t) =1+ (1 ; (t)) 1 ; (t) which together with (4) implies for (t) < =3 that kf (t) ; V (t)k F (t)t kd vk F ; 3(t) : As a corollary to Lemma 3.5, we obtain a quaratic convergence result: Corollary 3.1 (q-quaratic convergence). Suppose F (1) 0. If (1) < =3 then kf (1) ; V (1)k F 4. Weighte centers (1) ; 3(1) kf (1) ; V k F = kf (1) ; V k F ( ; 3(1)) min (F (1)) : We have alreay seen that given (X (0) Z (0) ) F o (I), one can compute L (0) such that (V (0) V (0) ) F(L (0) ) for some positive enite matrix V (0). We willnow show thatfor any positive enite matrix W S ++, there exists an invertible matrix L such that (W W) F(L). This fact will be shown by construction. To be more precise, we willshow that the following algorithm prouces a sequence L (k) if it is properly initialize. such that L =lim k!1 L (k) an W =lim k!1 V (k),

14 13 Algorithm 1 (Weighte center). Input: weight W S ++ an initial invertible L (0) an positive enite V (0) such that (V (0) V (0) ) F o (L (0) 1 F 3 min(w ). Step 0 Let k =0. Step 1 Let ) an W ; V (0) D (k) v = W ; V (k) an ecompose it into D (k) X an D (k) Z A(L(k) ). + D(k) Z = D(k) v such that D (k) X A? (L (k) ) Step Compute X (k) (1) = V (k) +D (k) X Z (k) (1) = V (k) +D (k) Z : Step 3 Let G (k) be positive enite an satisfy X (k) (1) = (G (k) ) Z (k) (1)(G (k) ) : Let L (k+1) = L (k) G(k). Compute V (k+1) = G (k) Z (k) (1)G (k) : Step 4 Set k = k +1 an return to Step 1. In each iteration k =0 1 ::: of Algorithm 1, a full Newton step is mae towars W, i.e. F (k) (1) = V (k) + D (k) v = W: (5) Lemma 4.1. Consier Algorithm 1. If W ; V (0) F 1 3 min(w ) (6) then for any k f0 1 :::g there hols W ; V (k+1) F an hence lim k!1 V (k) = W: 1 min (W ) W ; V (k) F 1 3 W ; V (k) F (7)

15 14 Proof: By enition (19), we have D (k) v F (k) (1) = min (F (k) (1)) = W ; V (k) F min (W ) : Relation (6) is therefore equivalent to (0) (1) 1=3. Applying Corollary 3.1, we obtain (7). We can now prove the existence of all W -weighte centers in a neighborhoo of V (0). Lemma 4.. Let W S ++ an (V (0) V (0) ) F(L (0) < nn. If W ; V (0) F 1 3 min(w ) then there exists an invertible matrix L < nn such that Proof: (W W) F(L): Initialize Algorithm 1 with L (0) this en, we notice that L (k+1) = L (k) G(k) for all k 0. Now wehave from Lemma 3.3, ) for some invertible matrix L(0) an V (0). We woul like toshow that lim k!1 L (k) exists. To an kg (k) ; Ik [ 1+(k) 1 ; (k) ]1=4 ; 1 (k+1) 1 3 (k) for k =0 1 :::. This shows that Hence, 1X k=0 (k) < 1: (8) kl (k+1) k = kl (k) G(k) k = kl (0) G(0) G (k) k kl (0) k(1 + kg(0) ; Ik) (1 + kg (k) ; Ik) Y kl (0) k k [ 1+(i) 1 ; (i) ]1=4 (9) i=0

16 15 for all k 0. Now, let B := kl (0) k 1Y i=0 [ 1+(i) 1 ; (i) ]1=4 : From (8) we have B<1 an from (9) kl (k+1) kb for all k 0. This implies that kl (k+1) ; L (k) k = kl(k) (G(k) ; I)k BkG (k) ; Ik: Since P 1 k=0 kg(k) ; Ik < 1 it follows that fl (k) j k =0 1 :::g isaconvergent sequence. Let L = lim k!1 L(k) : Similarly, we can show that f(l (k) );1 j k =0 1 ::g is a convergent sequence an L ;1 = lim k!1 (L(k) );1 : Moreover, by Lemma 4.1 we have W = lim k!1 V (k) : Now consier (X (k) Z (k) ) F(I). We have lim k!1 X(k) = lim an similarly, k!1 L(k) lim k!1 Z(k) = L ;T WL ;1 : Since F(I) is close, it follows that an hence (LW L T L ;T WL ;1 ) F(I) (W W) F(L): V (k) (L (k) )T = LW L T With an inuctive argument, wecannow prove that there exists a W -center for any W S ++. Theorem 4.1. For any W S ++ there exists an invertible matrix L < nn such that (W W) F(L) an (LW L T L ;T WL ;1 ) F(I):

17 16 Proof: Due to Assumption 1, we know that there exists a pair (X Z) o F (I) an hence (V V ) o F (L ) where V an L are ene by (7) an (6) respectively. The lemma is prove for the case W = V. Now supposev 6= W an let := min( min (V ) min (W )): Since V W S ++,wehave >0. Now consier W () :=(1; )V + W for [0 1]. Remark that for any [0 1] we have min (W ()) (1 ; ) min (V )+ min (W ) : Now ene 0 1 ::: by j := min(1 Remark that j 3 kw ; V k F )forj =0 1 :::: W = W (1) = W ( j )forj 3 kw ; V k F : We willprove by inuction that for any j, there exists L (j) such that (W ( j ) W( j )) F(L (j) ): (30) Since W ( 0 )=V, the statement (30) hols for j =0withL (0) = L. We have kw ( j+1 ) ; W ( j )k F =( j+1 ; j ) kw ; V k F 3 min(w ( j+1 )) : 3 Together with Lemma 4., this implies that if the statement (30) hols for W ( j ) then there exists L (j+1) such that (W ( j+1 ) W( j+1 )) F(L (j+1) ): Thus the theorem is prove by inuction.

18 17 5. Discussion In this paper, we stuie a possible way of generalizing the concept of weighte centers from linear towars semienite programming. A ierent approach was recently propose by Monteiro an Pang [1]. In this section, we will compare the properties of the Monteiro-Pang weighte center with the weighte center that we propose in this paper. Monteiro an Pang [1] stuie the map 1 (XZ + ZX) base on the theory of local homeomorphic maps. One of their results states that for any positive enite matrix W there exists a unique matrix pair (X Z) such that X is primal feasible an Z is ual feasible an the symmetric part of XZ equals W, i.e. P S (XZ)= 1 (XZ + ZX)=W However, the converse is not true: given an interior feasible pair (X Z) o F (I), the matrix P S (XZ) is in general not positive enite. The following example illustrates this fact. (See also page 464 of Horn an Johnson [3]). Example 5.1. Let X := Z := 4 5 ; ; then P S (XZ)= ;3 3 5 : For the weighte center that we propose in this paper, we know from Theorem 4.1 that for any positive enite matrix W, there exists a, possibly not unique, pair (X Z) F o (I) an an invertible matrix L such that L ;1 XL ;T = L T ZL = W: (31) Conversely, we know from (7) that for any pair (X Z) F o (I) there exists a matrix L an a positive enite matrix W such that (31) hols. Another issue is scale-invariance, see To, Toh an Tutuncu [17]. The weighte center propose in this paper is scale invariant in the sense that if the pair (X Z) is a W -weighte center an L is an invertible n n matrix, then (L ;1 XL ;T L T ZL)isaW-weighte center for the transforme problem (5). However, the next example shows that the Monteiro-Pang weighte center is not scale-invariant.

19 18 Example 5.. Let X := Z := 4 3 ;1 ;1 3 5 L := then but P S (XZ)= P S (L ;1 XZL)= =4 3= : As a consequence, we seethat the eigenvalues of the symmetric part P S (XZ) are in general ierent from the eigenvalues of the matrix prouct XZ, which enes the weight use in this paper. References [1] Alizaeh, F., \Interior point methos in semienite programming with applications to combinatorial optimization," SIAM Journal on Optimization 5 (1995) [] Golfarb, D. an Scheinberg, K., \Interior point trajectories in semientie programming," manuscript, Department of Inustrial Engineering an Operations Research, Columbia University, NewYork, [3] Horn, R. an Johnson, C., Matrix Analysis, Cambrige University Press, Cambrige, [4] Jansen, B., Roos, C., Terlaky, T. an Vial, J.-Ph., \Primal-ual target-following algorithms for linear programming," Annals of Operations Research 6 (1996) [5] Kojima, M., Megio, N., Noma, T. an Yoshise, A., A Unie Approach to Interior Point Algorithms for Linear Complementarity Problems, Springer-Verlag, Berlin, [6] Kojima, M., Mizuno, S. an Yoshise, A., \A primal-ual interior point algorithm for linear programming," in Progress in Mathematical Programming: Interior point an relate methos pp. 9-37, (e. Megio, N.), Springer Verlag, New York, [7] Kojima, M., Shinoh, S. an Hara, S., \Interior-point methos for the monotone linear complementarity problem in symmetric matrices," SIAM Journal on Optimization 7 (1997) [8] Luo, Z.-Q., Sturm, J.F. an Zhang, S., \Superlinear convergence of a symmetric primalual path following algorithm for semienite programming," SIAM Journal on Optimization 8 (1998)

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