Tibor Illes, Jiming Peng 2, Kees Roos, Tamas Terlaky Faculty of Information Technology and Systems Subfaculty of Technical Mathematics and Informatics

Transcription

1 DELFT UNIVERSITY OF TECHNOLOGY REPORT 98{5 A Strongly Polynomial Rounding Procedure Yielding A Maximally Comlementary Solution for P () Linear Comlementarity Problems T. Illes, J. Peng, C. Roos, T. Terlaky ISSN 922{564 Reorts of the Faculty of Technical Mathematics and Informatics 98{5 Delft June, 998

2 Tibor Illes, Jiming Peng 2, Kees Roos, Tamas Terlaky Faculty of Information Technology and Systems Subfaculty of Technical Mathematics and Informatics, Deartment of Statistics, Stochastic and Oerations Research Delft University of Technology, P.O. Box 53, 26 GA Delft, The Netherlands. e{mail: htt:// This author kindly acknowledges a one year research fellowshi at Delft University of Technology. His research is also suorted by the Hungarian National Research Fund OTKA No. T 432 and OTKA No. T On leave from: Deartment of Oerations Research, Eotvos Lorand University, Budaest, Hungary. 2 This author is suorted by the Dutch Organization for Scientic Research, NWO, through SWON-grant for the roject High Performance Methods for Mathematical Otimization. Coyright c998 by Faculty of Technical Mathematics and Informatics, Delft, The Netherlands. No art of this Journal may be reroduced in any form, by rint, hotorint, microlm or any other means without written ermission from Faculty of Technical Mathematics and Informatics, Delft University of Technology, The Netherlands.

3 Abstract We deal with Linear Comlementarity Problems (s) with P () matrices. First we establish the convergence rate of the comlementary variables along the central ath. The central ath is arameterized by the barrier arameter, as usual. Our elementary roof reroduces the known result that the variables on, or close to the central ath fall aart in three classes in which these variables are O(); O() and O( ), resectively. The constants hidden in these bounds are exressed in, or bounded by, the inut data. All this is rearation for our main result: a strongly olynomial rounding rocedure. Given a oint with suciently small comlementarity ga and close enough to the central ath, the rounding rocedure roduces a maximally comlementary solution in at most O(n 3 ) arithmetic oerations. The result imlies that Interior Point Methods (IPMs) not only converge to a comlementary solution of P () s but, when furnished with our rounding rocedure, they can roduce a maximally comlementary (exact) solution in olynomial time. Key words: linear comlementarity roblems, P () matrices, error bounds on the size of the variables, otimal artition, maximally comlementary solution, rounding rocedure. AMS Subject Classication : 9C33 Contents Introduction 2 Preliminaries 3 3 The otimal artition and two condition numbers 5 3. Otimal artition The rst condition number for ( ) The second condition number for ( ) Finding the otimal artition Otimal artition identication from aroximate centers 2 4. Finding the otimal artition from aroximate centers Comlexity of nding the otimal artition Rounding to a strictly comlementary solution 5 5. Rounding rocedure Comlexity of nding an exact solution Concluding remarks 7

4 Introduction In this aer we deal with a class of Linear Comlementarity Problems (s): ( ) s(x) := Mx + q ; x ; xs(x) = ; () where M is an n n real matrix, q 2 IR n and xs(x) denotes the coordinatewise roduct of the vectors x and s(x). We say that an algorithm solves ( ) if either it roduces a vector x satisfying the constraints of ( ) or it rovides a certicate that no such vector exist. In the rst case we say that x solves ( ). The vector x is a strictly comlementary solution of ( ) if it solves ( ) and x + s(x) > : Contrary to Linear Otimization (LO) [25], in general no strictly comlementary solution exists for ( ): there might exist airs of comlementary variables x i and s i (x) that are both zero in all solutions of the ( ). Comlementary solutions with the maximal number of nonzero coordinates will be referred to as maximally comlementary solutions. The existence of maximally comlementary solutions follows from the convexity of the solution set, roved by Cottle et al. in [4]. Kojima et al. [7] under some additional assumtions showed that solutions on the central ath converge to a maximally comlementary solution of ( ). All known algorithms for solving ( ) need some assumtion on the matrix M. So do Interior Point Methods (IPMs) as well. IPMs for solving ( ) are widely studied in the last decade. A survey on recent results is written by Yoshise [33]. Kojima, Mizuno and Yoshise [4] resented a olynomial time algorithm that roduces an exact solution for s where M is ositive semidenite. The same authors [5] established an O( nl) iteration bound for a otential reduction algorithm. Ji, Potra and Huang [] develoed a olynomial, O( nl) redictor-corrector method for ositive semidenite s under the assumtion that the sequence of iterates generated by their interior-oint algorithm converges to a strictly comlementary solution. Later, Ye and Anstreicher [3] roved the same iteration bound, O( nl) for redictor-corrector methods, removing the assumtion given in []. In 99, Kojima et al. [7] extended all the reviously known results to the wider class of so called P () s and unied the theory of s from the view oint of interior oint methods. Jansen, Roos and Terlaky [9] introduced a family of rimal-dual ane-scaling algorithms for ositive semidenite s. These results were recently extended to s with P () matrices by Illes, Roos and Terlaky [8]. The iteration bound of those algorithms are O(( + 4)n log x T s(x )=), where x is the initial iterate and is the comlementarity ga x T s(x) at termination. Interior-oint methods need an interior feasible oint to start with. Among others, Ji, Potra and Sheng [] studied the initialization roblem and roosed a redictor-corrector method for solving the P () s from infeasible starting oints. Kojima et al. [6, 7] gave a big-m construction that allows to solve the roblem in one hase. The aim of this aer is twofold. First we derive some bounds on the magnitude of the variables in the vicinity of the central ath 2, when the comlementarity ga is small enough. Second, a strongly olynomial rounding rocedure is resented that rovides a maximally comlementary (exact) solution from any interior oint solution that is in a certain neighborhood of the central ath and for which the comlementarity ga is suciently small. For deriving results on the magnitude of the variables in a given neighborhood of the central ath we use some known results from the theory of error bounds for systems of linear inequalities [2]. The theory of error bounds goes back to the early fties [7]; for recent develoments we refer to the survey aer [23] and the references therein. For s, a well-known local error bound is given by Robinson L is the binary inut length of the roblem [7]. 2 The central ath is dened in the usual way. See Section 2.

5 [24] which says that there exists a constant > and > such that dist(x;? ) k min (x; s(x)) k; (2) for all x satisfying k min (x; s(x)) k, where? denotes the solution set of ( ) in IR n +, dist(x;? ) = min y2? ky? xk and the minimum min(x; s(x)) is taken coordinatewise. By using the roerties of the central ath and some results on error bounds of Cook et al. [3] and Mangasarian and Shiau [3, 2], we derive some bounds on these constants in terms of the inut data if x is on or close to the central ath. To the best of our knowledge, this is the rst result yielding easy to calculate bounds for these constants in the study of s. The bounds on the magnitude of the variables along the central ath deend on the dimension n of the roblem, on the arameter and the barrier arameter that arameterizes the central ath, and on two condition numbers and of ( ). The condition number is closely related to that dened by Ye [32] and studied by Vavasis and Ye [28] for olyhedra with real number data and slightly modied by Roos, Vial and Terlaky [25] for the case of LO roblems. The second condition number,, will be introduced later. Other condition numbers for are dened in [8, 29]. It will be shown in a quite elementary way that in a given neighborhood of the central ath the variables fall aart in three classes and their magnitudes are O(); O() and O( ) resectively, rovided the arameter is suciently small. The rounding rocedure we describe for ( ) resembles the one resented in the aers [3, 2] and in the book [25]. We show that IPMs with a rounding rocedure terminate in a nite (olynomial) number of iterations and yield a maximally comlementary solution. There are some other methods [4, 8] in the literature that generate an exact solution to ( ) in O(n 3 L) iterations, but those are dierent from ours and do not generate a maximally comlementary solution. Kojima, Mizuno and Yoshise [4] in Aendix B of their aer, resented a method which leads to a basic solution of the s, thus not roviding a maximally comlementary solution. They comute a solution, z = (x; s) such that the variables can be slit into the comlementary sets I = fk : z k < 2?L g and I 2 = fk : z k 2?L g in such a way that the submatrix of (E;?M) corresonding to I 2 contains only linearly indeendent columns and by setting the variables for I equal to zero a comlementary solution of is obtained. Kojima et al. ([7], age 6) ointed out that: "Practically, however, it might be too comlicated to comute with the number 2?2L because it is too small." 3 The required comlementarity ga for our rounding rocedure is bigger in general (see Section 5), however, we do not claim that the accuracy theoretically needed to start our rounding rocedure is ractically reachable. Our rounding rocedure generates a maximally comlementary (exact) solution, while all the reviously known rounding rocedures roduce a comlementary basic solution, which in general, is not maximally comlementary. Having a maximally comlementary solution, a comlementary basic solution can be comuted in strongly olynomial time by using the basis identication rocedure described in Berkelaar et al. [2]. However, no strongly olynomial algorithm is known to generate maximally comlementary solution from a comlementary basic solution. The aer is organized as follows. Some reliminary results are discussed in Section 2. In Section 3 the otimal artition is dened and the related concet of maximally comlementary solutions. We introduce two condition numbers for ( ) and derive local bounds on the magnitude of the variables on the central ath. The main result in this section describes how the otimal artition can be determined if the barrier arameter is small enough. In Section 4 we generalize the results of Section 3 to oints that are close to the central ath, so-called aroximate centers, and we show that the otimal artition can be identied from x if x belongs to a certain neighborhood of the central ath and if x T s(x) is small 3 The value 2?2L is an uer bound on the required comlementarity ga that is sucient to identify a comlementary basic solution. 2

6 enough; such a vector x can be obtained in olynomial time by any interior oint method. Section 5 resents a strongly olynomial rounding rocedure that yields a maximally comlementary solution. Some concluding remarks close the aer in Section 6. Throughout, we shall use kk ( 2 [; ]) to denote the -norm on IR n, with kk denoting the Euclidean norm kk 2. E will denote the identity matrix, e will be used to denote the vector which has all its comonents equal to one. Given an n-dimensional vector x, we denote by X the n n diagonal matrix whose diagonal entries are the coordinates x j of x. If x, s 2 IR n then x T s denotes the dot roduct of the two vectors. Further, xs and x for 2 IR will denote the vectors resulting from coordinatewise oerations. For any matrix A 2 IR mn, A i ; A :j are the i-th row and the j-th column of A, resectively. Furthermore, (A) := ny j= ka :j k: For any index set J f; 2; : : : ; mg, jjj denotes the cardinality J and A J 2 IR jjjn the submatrix of A whose rows are indexed by elements in J. Moreover, if K f; 2; : : : ; ng, A JK 2 IR jjjjkj denotes the submatrix of A J whose columns are indexed by elements in K. 2 Preliminaries For further use we rst recall some well-known results and denitions. The reader may consult the aers [7] and [33] for roofs and details. where A matrix M 2 IR nn is a P () matrix if ( + 4) X i2i +(x) x i [Mx] i + X i2i?(x) x i [Mx] i ; 8x 2 IR n ; (3) I + (x) := f i n : x i [Mx] i > g; I? (x) := f i n : x i [Mx] i < g; and is a nonnegative real number. Note that the index sets I + (x) and I? (x) deend not only on x but on the matrix M as well. The matrix M is a P matrix if it is a P () matrix for some nonnegative : P = [ P (): One easily veries that M is a P () matrix if and only if M is ositive semidenite. Furthermore, if M is P () for some then M is P () for all. The class of sucient matrices (SU) was introduced by Cottle et al. [4]. A matrix M 2 IR nn is column sucient if for all x 2 IR n, X(Mx) ) X(Mx) = and row sucient if M T is column sucient. The matrix M is sucient if it is both row and column sucient. Recently, Valiaho [27] roved that P = SU. The sets of feasible and ositive feasible vectors are denoted resectively by? = fx : x ; s(x) g ;? = fx : x > ; s(x) > g ; 3

7 and the set of solutions of ( ) by? = fx : x ; s(x) ; xs(x) = g : It is known (cf. [7], Theorem 4.6.) that if M 2 P and? 6= ; then? 6= ;. Further, if? 6= ;, then? is comact, 4 moreover for every > there exists a unique x 2? such that xs(x) = e: In other words, assuming that? is nonemty the central ath C := fx 2? : xs(x) = e for some > g exists. Kojima et al. [7] showed that the assumtion? 6= ; can be made without loss of generality. Hence we may assume that the central ath C exists. The central ath C is a one-dimensional smooth curve that leads to a solution of ( ) when aroaches. 5 We insert here the following technical lemma that will be used at several laces below. Lemma 2. Let x be a solution of the equation Dx = d, where D is an integral and nonzero mn matrix and d an integral vector. If J denotes the suort of x and the columns of D :J are linearly indeendent, then (D) jx jj (D) kdk ; j 2 J: Here (D) denotes the largest absolute value of the determinants of the square submatrices of D. The right inequality holds also if d is not integral. Proof: For comleteness we include a roof here. Let x be as in the lemma and let the index set K be such that D KJ is a nonsingular square submatrix of D; such K exists because the columns in D :J are linearly indeendent. Now we have D KJ x J = d K, and hence, by Cramer's rule, 6 det D (j) KJ x j = ; 8j 2 J; (4) det (D KJ ) where D (j) KJ denotes the matrix arising when the j-th column in D KJ is relaced by d. Since the denominator in the above quotient is at least we obtain jx j j det D (j) KJ ; j 2 J: (5) By evaluating the last determinant to its j-th column, while using that each square submatrix is also a square submatrix of D, the right hand side inequality follows. For the left inequality we use that d is integral; since x j 6= this imlies that the numerator in (4) is at least one. 2 Corollary 2. If the columns of D are all nonzero then, under the assumtions of Lemma 2., The right inequality holds also if d is not integral. (D) jx jj (D) kdk ; j 2 J: 4 The key observation for this is Lemma 4.5 of Kojima et al. [7]. 5 For further details we refer to Chaters 2 and 4 in [7]. 6 The idea of using Cramer's rule in this way was alied rst by Khachiyan in [2]. 4

8 Proof: If the columns of D are all nonzero, then the left inequality in Lemma 2. remains valid if we relace (D) by (D). This is immediate from the well-known Hadamard inequality for determinants and because D and d are integral. The inequality at the right follows by alying Hadamard's inequality to (5) The otimal artition and two condition numbers In the rest of the aer we assume that M 2 P () for some. This imlies that the matrix M is sucient. 3. Otimal artition Let us denote the index set f; 2; : : : ; ng by I and dene the sets B := fi 2 I : x i > for some x 2? g; N := fi 2 I : s i (x) > for some x 2? g; T := fi 2 I : x i = s i (x) = for all x 2? g: We show that these index sets are disjoint and B [ N [ T = I, i.e. they form the so-called otimal artition of the index set I with resect to ( ). Lemma 3. ([7]) The index sets B; N and T form a artition of the index set I. Proof: >From the denition of the sets B; N and T it is obvious that B \ T = ;, N \ T = ; and I = B [ N [ T. Let us assume that B \ N 6= ;. Then there exist x ; x 2? such that x j > ; s j(x ) = ; x j =, and s j(x ) >, for some j 2 I: Let us denote x := x? x ; s = s(x ); s = s(x ); s := s? s and X := diag(x). It is easy to see that and Xs = XMx x j s j = x j (Mx) j = (x j? x j )(s j? s j ) =?x js j < ; what contradicts the column suciency of the matrix M. 2 Corollary 3. Let x and x solve ( ), so x ; x 2?. Then x s(x ) = and x s(x ) =. Proof: The denition of the classes B and N imlies fi 2 I : x i > g B and fi 2 I : s i(x ) > g N. Since B \ N = ;, it follows that x and s(x ) are comlementary. The roof for x and s(x ) is analogous. 2 Corollary 3.2 The solution set? is convex. 7 The idea for deriving bounds from Hadamard's inequality is due to Klafszky and Terlaky [3] (in Hungarian). 5

9 Proof: Let x ; x 2? and 2 [; ]. If x := x +(?)x then x and s(x) = s(x )+(?)s(x ). Thus x 2?. Further, Corollary 3. gives that xs(x) =, whence x 2?. 2 A solution x 2? is called maximally comlementary, if x B > and s N (x) >. Since? is convex (and olyhedral) a maximally comlementary solution exists. 8 >From now on we assume that? 6= ;. If the i th column of M is zero then the P roerty imlies that the i th row is zero as well. Therefore s i (x) = q i in that case, for every x. Hence, if q i <, then ( ) is infeasible. If q i, then the constraint s i (x) is always satised and we may reduce the roblem by removing the i th row and column of M. Thus we will assume that all columns of M are nonzero. When q = then the () has a trivial solution (x = ). Therefore, without loss of generality we further assume that q 6=. 9 Our goal is to nd the otimal artition of the index set and, nally, to round o to a maximally comlementary solution. In fact, we will show that given x() we can nd the otimal artition rovided is small enough. To this end we need to give bounds for the size of the variables along the central ath. In the next two sections we obtain such bounds in terms of two condition numbers for ( ). 3.2 The rst condition number for ( ). In this section we introduce our rst condition number of ( ). This is done in a similar way as in Roos et al. [25] for LO roblems. Since? 6= ;,? is nonemty and comact (see Section 2), so the following two numbers are well dened. x := min max i2b x2? fx ig; s := min x2? fs i(x)g: i2n max By convention we take x = if B is emty and s = if N is emty, thus both x and s are ositive. If B is nonemty then x is nite and if N is nonemty then s is nite. Since q 6= it cannot haen that both B and N are emty, thus under the interior oint condition (? 6= ;), at least one of the two numbers is nite. As a consequence, the number := minf x ; s g is ositive and nite. One can easily verify that can also be written as := min max i2b[n x2? fx i + s i (x)g: In general, we have to solve a roblem without knowing its condition number. In such cases there is a chea way to get a lower bound for if the roblem data (M; q) are integer. We roceed by deriving such a lower bound. Lemma 3.2 If M and q are integral then (M). 2 8 The convexity of? is roved in another way in [4], (see Theorem 5-6, ages 24-24). Furthermore it is shown that? is a olyhedron. 9 It may be noted that in this aer we nd a strictly comlementary solution of ( ) under the assumtions? 6= ; and q 6=. If q = we have the trivial solution x =, but this solution will in general not be maximally comlementary. The case q = is interesting in itself. E.g., if M is skew-symmetric it covers LO and there exists a strictly comlementary solution [25]; the other extreme occurs if M is a ositive denite matrix (e.g., if M is the identity matrix): then B = N = ; and T = I. 6

10 Proof: For any vector x 2? we have, with s = s B s N A M BB M BN M BT M NB M NN M NT s T M T B M T N M T T x B x N x T A q B q N A : (6) q T Further, x 2? holds if and only if x N =, x T = s T =, s B =. This is equivalent M BB M NB M T B BN?E NN T N A x B = s B?q N?q T A ; xb ; s N : (7) Any maximally comlementary solution x yields a ositive solution of this system. In order to get a lower bound on we need to derive a lower bound on the maximal value of each coordinate of the vector z := (x B ; s N ) when this vector runs through all ossible solutions of (7). For each i we know that there exists a solution z with z i >. Hence there exists a basic solution z of (7) with z i >. Therefore, Corollary 2. yields the following lower bound on the biggest coordinate of z. max z x2? i (M :B ) : Since (M) (M :B ), the lemma follows. 2 Now we are ready to estimate the size of the variables x i ; s i (x) when x lies on the central ath, i.e. xs(x) = e, and i 2 B or i 2 N. We denote s() := s (x()). Theorem 3. For any ositive one has x i () ; i 2 B; n(+4) x i () n(+4) ; i 2 N; s i () n(+4) ; i 2 B; s i () n(+4) ; i 2 N: Proof: We rst consider the case i 2 N. Let us assume that x 2? and s := s(x). Taking into consideration that M 2 P (), and x(); s(); x; s we get (x()? x) T (s()? s) = (x()? x) T M(x()? x)?4 =?4 =?4?4 X X i2i +(x()?x) X i2i +(x()?x) X i2i +(x()?x) i2i +(x()?x) (x()? x) i [M(x()? x)] i (x()? x) i (s()? s) i ((x()s()) i? (x()s) i? (xs()) i + (xs) i ) (x()s()) i?4n: (8) The last inequality holds because x()s() = e and x 2?. On the other hand Combining this with (8) we have (x()? x) T (s()? s) = n? x() T s? x T s(): x() T s + s() T x n( + 4) 7

11 which imlies x i ()s i x() T s n( + 4) 8 i 2 I: (9) Now if i 2 N and s such that s i is maximal, then by denition s i. Dividing by s i in (9) we obtain n( + 4) n( + 4) x i () : () s i Since x i ()s i () =, it also follows that s i () ; for all; i 2 N: n( + 4) This roves the second and fourth inequality in the lemma. The rst and third inequalities for i 2 B are obtained from (9) analogously The second condition number for ( ). In this section we derive bounds that will hel us to get control on the variables x i () and s i () if i 2 T. Before dealing with the main theorem in this section we review some results about systems of linear inequalities and equalities. Let A 2 IR mn and C 2 IR kn be two real matrices. For given b 2 IR m and d 2 IR k, consider the following system of linear inequalities Ax b; Cx = d: () Cook et al. [3] and Mangasarian and Shiau [2] studied the Lischitz continuity of solutions of () with resect to right-hand side erturbations of (). We will use a variant of those results. For comleteness, we give a simle roof, similar to that resented in [3]. Lemma 3.3 Let the system () have nonemty feasible sets? and? 2 for the right-hand side (b ; d ) and (b 2 ; d 2 ), resectively. For each x 2? there exists an x 2 2? 2 such that kx? x 2 k (A; C) b? b 2 d? d 2 ; (2) where (A; C) := max u;v 8 < : u v A T u + C T v = z? y; e T (z + y) = ; u; y; z ; the columns of (A T ; C T ) corresonding to nonzero elements of (u; v) are linearly indeendent 9 = ; : Proof: We are interested in nding t such that t = x? x, with x 2? 2, is minimal. This amounts to solving the linear minimization roblem min x t : Ax b 2 ; Cx = d 2 ; te + x x ; te? x?x : (3) Note that this roblem is feasible, since? 2 6= ;, and bounded. Hence, the otimal value t is equal to the otimal value of the dual roblem of (3). This gives t = max u T b 2 + v T d 2 + y T x? z T x : A T u + C T v + y? z = ; e T (z + y) = ; u; y; z : (4) This denition is a slight modication of the one given by Mangasarian and Shiau [2]. They simly require A T u + C T v =, not using the variables y and z. Our denition has the advantage that the feasible region of the otimization roblem dening (A; C) are the vertices of a olyhedral set. 8

12 Let (u; v; y; z) be an otimal solution of this roblem. Then we may write t = u T b 2 + v T d 2 + (y? z) T x = u T b 2 + v T d 2?? A T u + C T v T x = u T b 2 + v T d 2? u T Ax? v T Cx u T b 2 + v T d 2? u T b? v T d? u T b 2? b? + v T d 2? d u b2? b v : d 2? d Hence, the roof will be comlete if we show that (4) has an otimal solution (u; v; y; z) such that the columns of (A T ; C T ) corresonding to the nonzero comonents of (u; v) are linearly indeendent. This can be shown as follows. If, to the contrary, the columns corresonding to the nonzero coordinates of (u; v) are deendent then there exist vectors u and v, not both zero, such that A T u+c T v = and u i = if u i = and v i = if v i =. Note that z() := (u; v; y; z)+(u; v; ; ) will be feasible for (4) if is such that u + u. Due to the denition of u and v the set of for which this haens is a closed interval [; ] with < < (here we allow =? and = ). Hence we necessarily have u T b 2 + v T d 2 =, otherwise a contradiction with the otimality of z() would arise. As a consequence, z() is otimal for all 2 [; ]. Clearly, by choosing aroriately, we can obtain a solution of (4) with fewer nonzero coordinates. By reeating this rocedure we obtain a solution (u; v; y; z) of (4) for which the columns of (A T ; C T ) corresonding to the nonzero comonents of (u; v) are linearly indeendent. For such a solution we have u v (A; C); by the denition of (A; C). This comletes the roof. 2 We roceed by deriving a lower bound for (A; C). Lemma 3.4 One has (A; C) min i;j? kai k ; kc j k ; where a i runs through the rows of A and c j through the rows of C. Proof: Let a denote the i-th row of A. Then, if e i denotes the i-th unit vector, one has A T e i = kak : Hence, assuming a 6=, taking u = e i = kak ; v = ; z j = a j = kak if a j ; y j =?a j = kak if a j <, and all remaining entries of y and z equal to zero, the quadrule (u; v; y; z) is feasible for the maximization roblem dening (A; C). Therefore, (A; C) kuk = = kak. A similar argument yields that (A; C) = kck for each row c of C, and hence the lemma follows. 2 An uer bound for (A; C) can be derived if all the entries of A and C are integral. Lemma 3.5 For integer A; C one has (A; C) n(a T ; C T ) n(a T ; C T ): (5) Proof: Let (u; v; y; z) be a feasible solution for the maximization roblem in the denition of (A; C). Let w T = (u T ; v T ); A = (A T ; C T ). Then Aw = z? y. Since the columns of A corresonding to nonzero 9

13 elements of w are linearly indeendent, we may aly Lemma 2., which yields kwk? A kz? yk? A : The last inequality follows since kz? yk kz + yk =. Since kwk nkwk the rst inequality in the lemma follows from this. The rest of the lemma follows from the Hadamard inequality for determinants. Hence the roof is comlete. 2 We now are going to aly Lemma 3.5 to a second condition number for ( ) which enables us to bound the variables along the central ath. This second condition number, denoted as, deends on the inut matrix M and the otimal artition (B; N; T ). It is dened as follows. Denition 3. Let I ; I 2 be a artition of the index set I such that B I and N I 2. Let us dene: 2 := max 4@ M?E 3 E I2 A?EI ; 5 : I +I 2=I?E E I2 I If the matrix M is integral, then we can give a lower bound and an easily comutable uer bound for. Lemma 3.6 If M is integral, then max I +I 2=I n 2 6 M?E E I2 E I T A?EI ; 3 7 T?E I2 5 = n(m) n(m): Proof: The rst inequality is immediate from Lemma 3.4, the second inequality follows from Lemma 3.5, the equality is obvious and the last inequality is Hadamard's inequality. 2 Now we are ready to state our main theorem in this section. Theorem 3.2 If then < 2 n 2 ( + 4) 2 ; (6) x i (); s i () ; i 2 T: Proof: When (6) holds, one can easily verify that and Letting x i () s i () n( + 4) > s i (); 8i 2 B; n( + 4) n( + 4) > x i (); 8i 2 N: n( + 4) I = fi : x i () s i ()g; I 2 = fi : x i () < s i ()g;

14 we have B I and N I 2 if (6) holds. Hence, dening we have H i (x()) = H(x) := min (x; s(x)) ; si () if i 2 I ; x i () if i 2 I 2 : >From the fact that H(x()) = min(x(); s()) and x() i s i () = we conclude that H i (x()). Consider the following linear system: Mx? s =?q x I2 = s I =?x I?s I2 : (7) It is easy to see that the feasible set of the system (7) is the solution set? of ( ). Let this set lay the role of? 2 in the Lemma 3.3. Further, let the solution set of the following linear system lay the role of? : Mx? s =?q x I2 = H I2 (x()) s I = H I (x())?x I?s I2 : Clearly? is not emty, because x() satises (8). Now it follows from Lemma 3.3 that there exists a solution x of (7), i.e. x 2?, such that x? x() s? s() 2 M?E E I2 E I Using the denition of and H i (x()) it follows that x? x() s? s() A ;?EI?E I2 3 5 kh(x())k : : Since x i =, for i 2 T; we conclude that for all i 2 T \ I one has s i () x i () : Similarly for all i 2 T \ I 2, it holds x i () s i () : This roves the theorem. 2 (8) 3.4 Finding the otimal artition In Table we collected the results of the last two theorems (Theorem 3. and Theorem 3.2). These results have an imortant consequence. If is so small that n( + 4) <

15 i 2 B i 2 N i 2 T x i () n(+4) n(+4) x i () s i () n(+4) s n(+4) i () Table : Local bounds for the variables on the central ath. and < n( + 4) then we have a comlete searation of the variables. Both inequalities give the same bound on, namely < 2 2 n2 ( + 4) 2 : (9) This means that if a oint on the central ath is given such that (9) holds, then we can determine the otimal artition (B; N; T ) of ( ). Unfortunately, in ractice we may not assume that we can calculate oints on the central ath exactly. Practical algorithms generate oints in the vicinity of the central ath. Therefore, in the next section we deal with the situation that a oint x is given in an aroriate neighborhood of the central ath. We will show that if x is close enough to x(), with small enough, we also have a comlete searation of the variables into the three dierent classes B; N and T. This will imly that all ath-following IPMs eventually roduce iterates that are suitable to identify the otimal artition of ( ). 4 Otimal artition identication from aroximate centers In this section we generalize the results of the revious section to the case where a oint x is given in a secic neighborhood of the central ath. On the central ath all the coordinates of the vector xs(x) are equal. This suggests that a good measure of centrality could be the ratio of the smallest and largest coordinate. If we bound this ratio, then a neighborhood of the central ath is obtained. We therefore use the following measure of centrality c (x) := max(xs(x)) min(xs(x)) ; where max(xs(x)) denotes the largest coordinate of xs(x) and min(xs(x)) denotes the smallest one. 4. Finding the otimal artition from aroximate centers We rst generalize the results of Theorem 3. and Theorem 3.2 to the case where x is not on the central ath C. This measure of centrality is introduced by Ling [9] and used in [8, 9]. The same measure of centrality is used throughout the book of Roos et al. [25]. 2

16 Lemma 4. Let x 2? and s = s(x). If c (x), for some >, and := xt s(x) n then one has x i s i n( + 4) ; i 2 B; ( + 4)n ; i 2 B; x i s i ( + 4)n ; i 2 N; n( + 4) ; i 2 N: If, further then 2 n 2 ( + 4) 2 ; (2) x i ; s i ; i 2 T: Proof: The roof uses essentially the same arguments as the roofs of Theorem 3. and Theorem 3.2. The arguments leading to () in the roof of Theorem 3. are still valid, so x i ( + 4)xT s = ( + 4)n for i 2 N: (2) The rest of the roof is a little comlicated by the fact that x is not on, but only in a certain neighborhood of the central ath. If c (x) then there are, 2 (; ) such that e xs e; with = : (22) These inequalities relace the identity x i ()s i () = used in the roof of Theorem 3.. Due to the left inequality in (22) we also have x i s i for all i. Hence using (2) we must have s i The right inequality in (22) gives x T s n, thus s i ( + 4)x T s : n( + 4) = n( + 4) : This roves the second and fourth inequality in the lemma. The roof of the rst and third inequalities can be obtained in the same way, therefore their roof is left to the reader. To rove the last statement of the lemma, we notice that for the current oint (x; s), it obviously holds and x i s i x i s i Let H(x) = min(x; s), the above two inequalities give = nx is i x T s n min(xs) n max(xs) ; 8i = ; 2; : : : ; n; = nx is i x T s n max(xs) ; 8i = ; 2; : : : ; n: n min(xs) [H(x)] i ; 8i = ; 2; : : : ; n: Following similar arguments as in the roof of Theorem 3.2, one can easily derive the conclusion. 2 3

17 = xt s(x) n i 2 B i 2 N i 2 T x i n(+4) (+4)n x i s i (x) (+4) n(+4) s i Table 2: Local estimates for variables belonging to index sets B; N and T if c (x). In Table 2 we collected the results of the above lemma. We conclude that the artition (B; N; T ) can be identied if x T s(x) is so small that ( + 4)n < ; and < n( + 4) : It is easy to verify that both inequalities give the same bound for thus for comlete searation of the variables we need 2 < : (23) n ( + 4)2 Therefore we may state without further roof our main result. Theorem 4. Let x 2? be such that c (x), for some >, and = xt s(x) n. If (23) is true, then, with s = s(x), the otimal artition of ( ) follows from T = fi : x i ; s i g; B = fi 62 T : x i > s i g and N = fi 62 T : x i < s i g: Comlexity of nding the otimal artition In this section we assume that we have given a oint x () 2? close to the central ath (i.e. c (x () ) for some > ). We dene by n =? x () T s (). Starting at x interior oint methods for solving ( ) need O( n log(n =)) iterations (see, e.g., [, 5, 7, 3]), or O(n log(n =)) (see, e.g., [8]) iterations to generate a oint x such that c (x) and x T s(x). The rst bound holds for methods with small udates of the barrier arameter whereas the second bound is tyical for methods using large udates, and also for methods using a Dikin-tye ane-scaling direction. Hence, by substituting the value of according to Theorem 4., we can get iteration bounds to identify the otimal artition. The above will be illustrated below for the Dikin ane-scaling algorithm resented in [8]. If n 4, this algorithm, with = 2, requires at most 3( + 4)n log n iterations to generate a oint x such that c (x) 2 and x T s(x). (24) 4

18 Theorem 4.2 Starting at a oint x () 2? with c? x () 2, and n 4, the Dikin ane-scaling algorithm reveals the otimal artition after at most iterations. 3( + 4)n log 8n2 ( + 4) ( + 4)n log? 8n 4 ( + 4) 2 (M) 4 Proof: The exression (24) gives the number of iterations to reach an -solution. With as in Theorem 4. and = n we obtain the rst bound. The inequality follows by using the uer bound for in Lemma 3.6 and the lower bound for in Lemma Similar results can be derived for other olynomial IPMs. 5 Rounding to a strictly comlementary solution We just established that the otimal artition of ( ) can be found after a olynomial number of iterations with any known ath-following IPMs for P () s. The required number of iterations deends on the starting oint x (), the arameter, and on the condition numbers and. Our ultimate goal is not only to nd the otimal artition but to nd an exact and maximally comlementary solution of ( ). Assuming that the otimal artition (B; N; T ) has been determined, with B nonemty, 2 we describe a rounding rocedure that can be alied to any suciently centered ositive vector x with x T s(x) small enough, and the rounding rocedure yields a vector ~x such that (7) is satised and ~x B > ; s N (~x) >. As might be exected, the accuracy that was sucient to nd the otimal artition is not enough to erform the rounding rocedure. In Theorem 5. we will give a bound on the comlementary ga that rovides sucient accuracy for our rounding rocedure. The rounding rocedure yields a maximally comlementary solution in strongly olynomial time. Finally, the number of iterations, required to reach the necessarily small comlementarity ga, is bounded by Theorem Rounding rocedure Let x 2? ; s = s(x) be given and assume that the otimal artition (B; N; T ) is known. Now we want to comute x B ; s N such that s N + s N x B + x B > and s N + s N > ; (25) A M BB M BN M BT M NB M NN M NT M T B M T N M T T x B + x B A q B q N A ; (26) q T because then ~x := (x B + x B ; ; ) 2?, and this solution is maximally comlementary. Since x and s satisfy (6), we may subtract (6) from (26), leading to the following B s N?s T A M BB M BN M BT M NB M NN M NT M T B M T N M T T x B?x N?x T A ; (27) 2 If B = ;, then x = and s = q is the only ossible solution. The vector (; q) solves the roblem if and only if q. 5

19 which is thus equivalent to (26). We can rewrite this M BB M NB M T B BN?E NN T N A x B = s M BNx N + M BT x T? s B M NN x N + M NT x T M T N x N + M T T x T? s T A : (28) We conclude that we can round x to a maximally comlementary solution ~x if we can nd a solution (x B ; s N ) of (28) that satises (25). We show below that if x T s(x) = n is small enough and x is close enough to the central ath, then such (x B ; s N ) can be found by Gaussian elimination. It may be useful to oint out that the analysis below works out well because the variables x T ; x N ; s B and s T that occur in the right hand side of (28) are `small' if is small enough and x is close enough to the central ath. These variables are bounded above by Lemma 4.. Since x B and s N are `large', by the same lemma, it is therefore not surrising that (28) admits a solution such that (25) holds. Theorem 5. Let x 2? be such that c (x) = 2. If < 2 8n 3 ( + 4) 2 2 kmk2 (M)2 (29) then the rounding rocedure yields a maximally comlementary solution in at most O(n 3 ) arithmetic oerations. Proof: To kee the exressions simle we introduce the following notations: A M BB M NB M T B Then equation (28) becomes BN?E NN T N A xb ; z := and r := s M BNx N + M BT x T? s B M NN x N + M NT x T M T N x N + M T T x T? s T A : Az = r: (3) When solving (3) by Gaussian elimination, which needs O(n 3 ) arithmetic oerations, we obtain a solution such that the columns of A corresonding to its suort are linearly indeendent. Hence, using Corrollary 2., kzk (A) krk = (M :B ) krk (M) krk : (3) We roceed by estimating krk. We use the trivial inequality krk nkrk and krk! B M T N M T T?E M BN M BT?E B M NN M NT x N xt s B st C A : (32) Observe that the value of given by (29) satises the hyothesis of Theorem 4.. Therefore, we have a comlete searation of the variables. As a consequence, all entries in the vectors x N ; x T ; s B and s T are bounded above by. Hence, the innity norm of the concatenation of these vectors, that aears at the right in (32), is bounded above by this number. Obviously the innity norm of the matrix in (32) is bounded above by the innity norm of M. Thus we nd krk 2n n ( + 4) kmk : Substitution in (3) yields kzk n kmk (M) : (33) 6

20 Using the lower bound of Lemma 4. (with = 2) for the entries of x B and s N, we conclude that the rounding rocedure certainly yields a maximally comlementary solution if This inequality is equivalent to 2n kmk (M) < < 2n( + 4) : 2 2n n ( + 4) kmk (M) : which yields the bound for in the theorem. This comletes the roof Comlexity of nding an exact solution We aly the results of the revious section to estimate the number of iterations required by the Dikin ane-scaling algorithm to reach the state where the rounding rocedure yields a maximally comlementary solution. Without further roof we may state our nal result. Theorem 5.2 Starting at a oint x () 2? with c? x () 2, and n 4, the Dikin ane-scaling algorithm requires at most 3( + 4)n log 8n3 ( + 4) 2 2 kmk2 (M)2 2 iterations to generate a oint x at which the rounding rocedure roduces a maximally comlementary solution. 2 6 Concluding remarks The aim of this aer was to show that one can determine a maximally comlementary solution of ( ) in olynomial time, thus extending a well-known result for LO (cf. Roos et al. [25]). We assumed that? 6= ;, q 6= and that a starting oint x () 2? is given. Under these assumtion we could derive the desired result. A crucial oint in the analysis is the convergence rate along the central ath of the variables in the index set T, which is O( ). All known roofs of this result use a corollary of Robinson [24] related to the theory of olyhedral multifunctions. In Section 3 we resented a new and relatively simle roof. In the analysis we need two condition numbers for P () s, both of which aear in the achieved iterations bound. Both numbers were bounded by exressions in the inut data. Using Theorem 3.2 and Theorem 4. we showed that if x 2? is suciently close to the central ath and x T s(x) suciently small then we can identify the otimal artition and comute a maximally comlementary solution by using Gaussian elimination (Theorem 5.). Similar bounds were resented by Kojima et al. [7, 4] to generate a comlementary basic solution of ( ). The number of iterations to obtain the accuracy necessary to run the rounding rocedure is comuted for Dikin ane-scaling algorithm [8] in Theorem 5.2. Similar results for other known IPMs can be obtained, as well. 7

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