Symmetrization of Nonsymmetric Quadratic. Inequality. S.W. Hadley*, F. Rendl** and H. Wolkowicz* * Department of Combinatorics and Optimization

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1 Symmetrzaton of Nonsymmetrc Quadratc Assgnment Problems and the Homan-Welandt Inequalty S.W. Hadley*, F. Rendl** and H. Wolkowcz* * Department of Combnatorcs and Optmzaton Unversty of Waterloo Waterloo, NL 3G1, Canada ** Technsche Unverstat Graz Insttut fur Mathematk Kopernkusgasse 4, A{8010 Graz, Austra Aprl 4, 1996 Abstract A technque s proposed to transform a nonsymmetrc Quadratc Assgnment Problem (QAP) nto an equvalent one, consstng of (complex) Hermtan matrces. Ths technque provdes several new Homan- Welandt type egenvalue nequaltes for general matrces and extends the egenvalue bound for symmetrc QAPs to the general case. Keywords: Homan-Welandt nequalty, nonsymmetrc quadratc assgnment problems, egenvalue nequaltes. 1

2 1 Introducton The Quadratc Assgnment Problem (QAP) s one of the most dcult combnatoral optmzaton problems. It s dened as follows: QAP: For gven (real) nn matrces A and B, mnmze f() := trab t t over the set of permutaton matrces, where trace denotes trace. Ths problem s well known to be NP-hard. The QAP s surveyed n e.g. [, 5, 8]. Lower bounds on f() are nvestgated n [1, 3, 5, 7, 9, 1]. These consttute an essental ngredent n any Branch and Bound approach to solve the QAP. A connecton between the range of values of f() and the egenvalues of A and B has been establshed n [5, 1] for the case of symmetrc A and B. Ths resulted n the egenvalue bound for symmetrc QAPs. (A QAP s called symmetrc f both nput matrces A and B are symmetrc.) In [6] t s ponted out that ths egenvalue bound s equvalent to the Homan-Welandt Inequalty, see also [4], n the sense that each can be derved from the other. In ths paper the egenvalue approach for QAP s extended to the general (nonsymmetrc) case. Ths s acheved by transformng the quadratc form f() nto an equvalent quadratc form g() :=tr A+ ~ B ~ + wth Hermtan matrces A+ ~ and B+ ~. Ths allows us to apply the egenvalue bounds for symmetrc QAPs also n the general case. Moreover we show how the egenvalues of A and A ~ + are related through majorzaton. Fnally the equvalence between f() and g() leads to new Homan-Welandt type nequaltes for nonnormal matrces. The paper s organzed as follows. In Secton we revew the Homan- Welandt nequalty and the egenvalue bound for symmetrc QAPs. In Secton 3 we propose a nontrval symmetrzaton of QAPs, leadng to the man result of the paper, an egenvalue related bound for general QAPs. The secton s concluded by provdng majorzaton relatons between the egenvalues of A and the matrx A ~ +. (The matrx A ~ + s formed from the Hermtan and skew-hermtan parts of A.) Several new nequaltes of Homan-Welandt type for general matrces are derved n Secton 4. The Homan-Welandt Inequalty and Symmetrc QAPs The followng notaton wll be used throughout the paper. denotes the set of permutatons of f1; : : :; ng. For two vectors a; b < n we dene the

3 mnmal and maxmal scalar product of a and b by, respectvely, < a; b >? := mnf a b () : g; < a; b > + := maxf a b () : g: Note that < a; b > + = a t b f the components of a and b are both n nondecreasng order. The dstance d(a; b) between two (possbly complex) vectors a and b s dened by d(a; b) = mn ja? b () j : For a and b real ths smples to d(a; b) = kak + kbk? < a; b > +. If A s a square matrx, then (A) denotes the vector of egenvalues of A (n arbtrary order). We denote by kkk = p trkk the Frobenus norm of the matrx K, where denotes the conjugate transpose. In [10] Homan and Welandt prove the followng nequalty for the dstance between two normal matrces A and B, and the dstance between ther respectve egenvalues, d((a); (B)) ka? Bk : (1) Ths s commonly referred to as the Homan-Welandt (denoted H-W) Inequalty. Moreover, there exsts a permutaton such that ka? Bk j (A)? () (B)j : () The nequaltes can fal f A or B s nonnormal. For example, let A = ; B = : Then (1) fals snce d((a); (B)) = > ka? Bk = 1: Moreover, wth A as above and B the 0 matrx, we see that () fals snce ka? Bk = 1 > j (A)? () (B)j = 0; for all permutatons. But even though (1) and () may fal for general matrces, t s stll possble to extend the result to a larger class of matrces. 3

4 One smple extenson for the H-W nequalty s to the matrces A = K A and B = KB, where K s postve dente and A and B are Hermtan. The valdty of the nequaltes follows from the fact that K has a square root and the egenvalues of A are the egenvalues of the Hermtan matrx K 1 1 AK. Note that A s normal f and only f K and A commute whch mples that A s Hermtan. In Secton 4 we wll present further generalzatons of the H-W nequalty to arbtrary square matrces. We now consder Hermtan A and B n order to show the close relaton between the untary relaxaton of the QAP and the H-W nequalty, see also [6]. Frst note that n the Hermtan case d((a); (B)) = kak + kbk? < (A); (B) > + : Expandng also shows that ka? Bk = kak + kbk? trab : (3) Therefore the H-W nequalty mples, usng (3) < (A); (B) >? trab < (A); (B) > + : (4) The followng theorem was proved n [5] and [1], and s the bass for the egenvalue bound of symmetrc QAPs. Theorem.1 [1] Let A and B be Hermtan matrces. Then maxftrab : untaryg = < (A); (B) > + ; mnftrab : untaryg = < (A); (B) >? : (5) Snce the permutaton matrces are contaned n the set of untary matrces, ths result ndeed provdes bounds on the range of values of a symmetrc QAP. Moreover, by comparng (4) and (5) we see that the equvalence of the H-W nequalty and the egenvalue bounds (5) becomes apparent, by observng that (B) can be assumed to be equal to (B ) for any untary. (The fact that trab <, even f the matrces nvolved are complex, follows from (3).) 4

5 3 Nonsymmetrc Quadratc Assgnment Problems For a square matrx A, let the matrces A + = A + A ; A? = A? A denote the Hermtan and skew-hermtan parts of A, respectvely. Consder a general real quadratc form x t Ax n the vector varable x. It s well known that x t Ax = x t A + x; for all x < n,.e. the quadratc form can be represented by an Hermtan matrx. Note that the egenvalues of A + majorze (see below) the real parts of the egenvalues of A, see [11]. The objectve functon f() = trab t t of a QAP wth (arbtrary) real matrces A and B can be vewed as a quadratc form n the matrx varable. It s natural to ask for a symmetrc representaton of f(), just as n the vector case. If we let x = vec() be the vector formed from unravellng rowwse, and we let K = A B be the Kronecker product of A and B, then t s easly vered that trab t t = x t Kx: Thus a trval way to symmetrze f() would be to use x t K + x nstead of f(). As a consequence we would have to work wth the n n matrx K + nstead of the two n n matrces A and B. Ths seems computatonally ntractable, e.g. even storng K + s nontrval for larger values of n. In the followng we propose a derent approach to symmetrze f(), that keeps the factored Kronecker product form of f(). Ths approach s based on the fact that trab t t = 0 f A s (real) symmetrc and B s skewsymmetrc. Lemma 3.1 Let A and B be real nn matrces wth A = A t and B =?B t. Then for any real n n matrx Proof. trab t t = 0: tra(b t t ) = tra t (B t ) =?trab t t : The rst equalty follows from trmn = trm t N t, the second from the propertes of A and B. Note that the lemma s wrong f we allow complex matrces A and B, or f s allowed to be complex. 5

6 Let ~A + = (A + + A? ); ~ A? = A +? A? (6) denote the postve and negatve Hermtan parts of A, respectvely. Note that both ~ A+ and ~ A? are Hermtan. Usng the postve Hermtan parts of A and B we can symmetrze f(). Theorem 3.1 Let A and B be two real n n matrces. For any real n n matrx trab = tr ~ A + ~ B + : (7) Proof. trab = tr(a + + A? )(B +? B? ) = tra + B +? tra? B? : The last equalty follows from the prevous lemma. tr ~ A+ ~ B + = tr(a + + A? )(B + + B? ) = tra + B +? tra? B? : The last equalty follows agan from the prevous lemma. As a consequence we can bound the range of an arbtrary QAP by the mnmal and maxmal scalar product of ( ~ A+ ) and ( ~ B+ ). Theorem 3. Let a QAP wth real matrces A and B be gven. Then for all permutaton matrces < ( ~ A+ ); ( ~ B+ ) >? trab t t < ( ~ A+ ); ( ~ B+ ) > + : Proof. By Theorem (3.1) we have for all permutaton matrces trab t t = trab = tr ~ A + ~ B + because A; B and are real. The bounds follow from Theorem (.1) by observng that permutaton matrces are untary. Relaton (3) also provdes a bound on the range of values of an arbtrary QAP. Theorem 3.3 Let a QAP wth real matrces A and B be gven. Then for all permutaton matrces?kak? kbk trab t t kak + kbk (8) 6

7 Proof. We have 0 ka B t k = kak + kbk trab t t for all permutaton matrces. It was already ponted out that the egenvalues of A + majorze the real parts of those of A. We conclude ths secton by provdng smlar majorzaton relatons for the egenvalues of A and ~ A +. Followng the notaton n [11] we denote by x [1] : : : x [n] the components of a gven vextor x = (x 1 ; : : :; x n ) < n n nonncreasng order. For gven x; y < n, we say that x majorzes y (denoted x y) f k =1 x [] k =1 y [] ; k = 1; : : :; n? 1; (9) x x n = y y n : (10) Theorem 3.4 Let A be an arbtrary n n matrx. Then Proof. ( ~ A + ) Re((A))? Im((A)): Let M = (1 + )A: Then M + = ~ A+. Usng see [11, p.37], we conclude (M + ) Re((M)) ( ~ A+ ) Re(((1 + )A)): Snce Re((1 + )z) = Re(z)? Im(z), the result follows. Theorem 3.5 Let A be an arbtrary n n matrx. Then (A + ) + Im((A? )) ( ~ A + ): Proof. Note that ~ A+ can be wrtten as the sum of the two Hermtan matrces A + and A?. Usng (M) + (N) (M + N) for Hermtan matrces M and N, see [11, p.41], the result follows. (In a slght abuse of notaton, we assumed here that for a Hermtan matrx M, (M) denotes the sequence of egenvalues of M n nonncreasng order.) Fnally we provde a majorzaton result between the sngular values of A and the egenvalues of ~ A +. Let k (A) denote the k th largest sngular value of A and k ( ~ A+ ) denote the k th largest egenvalue of ~ A+. 7

8 Theorem 3.6 Let A be an arbtrary n n matrx. Then k ( A ~ p + ) k (A); k = 1; : : :; n; and p (j 1 ( A+ ~ )j; : : :; j n ( A+ ~ )j) w (1 (A); : : :; n (A)); where w denotes weak majorzaton,.e."" replaces "=" n (10). Proof. In [11, p.40], t s shown that and The result follows usng k (M + ) k (M) (j 1 (M + )j; : : :; j n (M + )j) w ( 1 (M); : : :; n (M)): ~A + = ((1 + )A) + : It should be ponted out that smlar results as those above can be obtaned by usng the negatve Hermtan parts ~ A? and ~ B? nstead of the postve Hermtan parts. 4 New Homann-Welandt Type Inequaltes We conclude by provdng nequaltes between the dstance of two general matrces, based on the symmetrzaton derved n Secton 3. Frst we relate the dstance between two matrces to the dstance between the egenvalues of the respectve postve Hermtan parts. Theorem 4.1 Let A and B be two real n n matrces. Then d(( ~ A+ ); ( ~ B+ )) ka? Bk : Moreover, there exsts a permutaton such that ka? Bk ( ( ~ A + )? () ( ~ B + )) : 8

9 Proof. Note that by Theorems (3.1) and (3.) we have ka? Bk = traa + trbb? trab = tr ~ A+ ~ A+ + tr ~ B+ ~ B+? tr ~ A+ ~ B+ : ( ~ A + ) + ( ~ B + )? < ( ~ A + ); ( ~ B + ) > + : = d(( ~ A+ ); ( ~ B+ )): The remanng part of the theorem s proved smlarly usng the mnmal scalar product of the egenvalues. Fnally we provde a lower bound on the dstance between the egenvalues of two arbtrary matrces. Theorem 4. Let A and B be two arbtrary n n matrces. Then (Re(trA)? Re(trB)) (Im(trA)? Im(trB)) + d((a); (B)): (11) n n Proof. Let a = Re((A)), b = Re((B)), c = Im((A)), d = Im((B)), e = a? b, and f = c? d. Then a lower bound on the dstance of the egenvalues of A and B s gven by the (global) mnmum of the followng program. e + f such that mn e;f e = Re(trA)? Re(trB) f = Im(trA)? Im(trB): The objectve functon s convex, and Re(trA)? Re(trB) e = n and Im(trA)? Im(trB) f = n satsfy the rst and second order sucent optmalty condtons. Substtuton nto the objectve functon yelds the result. We leave t as an open problem to derve good upper bounds on d((a); (B)). 9

10 5 Dscusson and Summary We have shown that an arbtrary QAP can be expressed usng (possbly complex) Hermtan matrces. Ths allowed us to derve egenvalue related bounds on the range of values of general QAPs. We do not clam that these bounds, taken as they are, wll be compettve wth exstng boundng rules for general QAPs. To make these bounds better, further work, as n the symmetrc case s necessary. In [1] the concept of "reductons" s used to mprove the egenvalue bound for symmetrc problems. Ths nvolved nonsmooth optmzaton and turned out to be very successful. Snce "reductons" can also be appled n the general case, the mprovement technques apply here as well. On the other hand, a projecton technque s used n [9] to mprove the egenvalue bound of Theorem (.1) by constranng the set of untary matrces to an ane subspace. A smlar technque can be appled also for general QAPs. Future research wll have to demonstrate the practcal qualty of the bounds proposed n ths paper. The close connecton between the Homan-Welandt nequalty and the egenvalue bound for symmetrc QAPs on one hand, and the symmetrzaton of general QAPs on the other hand suggested several extensons of the Homan-Welandt nequalty for general matrces. The key role s played here by A+ ~, the postve Hermtan part of A. References [1] A.A. Assad and W. u, On lower bounds for a class of (0,1) programs, Operatons Research Letters 4, 175{180, [] R.E. Burkard, Locatons wth spatal nteractons: the quadratc assgnment problem, Dscrete Locaton Theory (eds.: P.B. Mrchandan and R.L. Francs), Wley Intersecton Seres n Dscrete Mathematcs and Optmzaton, 387{437, [3] P. Carrares and F. Malucell, A new lower bound for the quadratc assgnment problem, Techncal Report TR-7/88, Psa, [4] W.E. Donath and A.J. Homan, Lower bounds for the parttonng of graphs, IBM J. Res. Dev. 17, 40{45, [5] G.Fnke, R.E.Burkard and F.Rendl, Quadratc assgnment problems, Annals of Dscrete Mathematcs 31, 61{8,

11 [6] G. Fnke and E.B. Medova-Dempster, Approxmaton approach to combnatoral optmzaton problems, Techncal Report, Techncal Unversty of Nova Scota, Halfax, [7] A.M. Freze and J. Yadegar, On the quadratc assgnment problem, Dscrete Appled Mathematcs 5, 89{98, [8] S.W. Hadley, Contnuous optmzaton approaches for the quadratc assgnment problem, Ph.D. thess, Unversty of Waterloo, Canada, [9] S.W. Hadley, F. Rendl and H. Wolkowcz, A new lower bound usng projecton for the quadratc assgnment problem, Report 1989, TU- Graz. [10] A.J. Homan and H.W. Welandt, The varaton of the spectrum of a normal matrx, Duke Mathematcs 0, 37{39, [11] A.W. Marshall and I. Olkn, Inequaltes: Theory of Majorzaton and ts Applcatons, Academc Press, New York, [1] F.Rendl and H.Wolkowcz, Applcatons of parametrc programmng and egenvalue maxmzaton to the quadratc assgnment problem, Report TU Graz, 1989, to appear n: Mathematcal Programmng. 11