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A semidefinite programming approach to the quadratic knapsack problem

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A semidefinite programming approach to the quadratic knapsack problem
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  Journal of Combinatorial Optimization, 4, 197–215, 2000c   2000 Kluwer Academic Publishers. Manufactured in The Netherlands. A Semidefinite Programming Approachto the Quadratic Knapsack Problem ∗ C. HELMBERG Konrad Zuse Zentrum f ¨ ur Informationstechnik Berlin, Heilbronnerstra ß e 10, D-10711 Berlin, Germany F. RENDL Technische Universit ¨ at Graz, Institut f ¨ ur Mathematik, Steyrergasse 30, A-8010 Graz, Austria R. WEISMANTEL Universit ¨ at Magdeburg, Institut f ¨ ur Mathematische Optimierung, Universitrits Platz 2, 0-39106 Magdeburg,Germany Received December 22, 1998; Revised January 18, 2000; Accepted January 19, 2000 Abstract.  In order to gain insight into the quality of semidefinite relaxations of constrained quadratic 0 / 1programming problems we study the quadratic knapsack problem. We investigate several basic semidefinite re-laxationsofthisproblemandcomparetheirstrengthintheoryandinpractice. Variouspossibilitiestoimprovethesebasic relaxations by cutting planes are discussed. The cutting planes either arise from quadratic representations of linear inequalities or from linear inequalities in the quadratic model. In particular, a large family of combinatorialcuts is introduced for the linear formulation of the knapsack problem in quadratic space. Computational resultson a small class of practical problems illustrate the quality of these relaxations and cutting planes. Keywords:  semidefinite programming, quadratic knapsack problem, cutting planes, 0 / 1 polytopes, relaxations 1. Introduction The quadratic knapsack problem(QK) Maximize  x t  Cx subject to  a t   x  ≤  b x  ∈ { 0 , 1 } n . with  a  ∈  N n , b  ∈  N is one of the standard combinatorial optimization problems (in someworks, the term quadratic knapsack problem has also been used for continuous versions of (QK)(Pardolosetal., 1991)). Typicalapplicationsofthediscreteproblemsuchasportfolioselection (Laughhunn, 1970) or site selection (Gallo et al., 1980) are of the following form.Afewitemshavetobeselectedfromagivengroundsetsubjecttobudgetaryconstraints. Theobjective, profit/quality maximization or risk minimization, is strongly influenced by thepairwiseinterrelationbetweentheitemsoftheselectedset. Besidesthesedirectapplications ∗ A preliminary version of this paper appeared in the Proceedings of IPCO ’96 (Helmberg et al., 1996).  198  HELMBERG, RENDL AND WEISMANTEL the quadratic knapsack problem also appears as a subproblem in several applications of constrained quadratic 0 / 1 programming, e.g. in VLSI- and compiler design (Ferreira et al.,1996; Johnson et al., 1993).Current practical approaches for solving (QK) are branch and bound algorithms. Thebounds are specially designed for the quadratic knapsack problem and often require specialproperties of the objective function such as nonnegativity of   c ij  (Hammer and Rader, 1997;Caprara et al., 1999). These approaches are of little use if (QK) appears as a subproblemwithin general constrained quadratic 0 / 1 programming. This drawback can be overcomeby resorting to a polyhedral approach. Typically, polyhedral methods work with linearrelaxations of (QK). However, these often exhibit a large gap even for very small instances( n  =  20). We design a polyhedral approach using semidefinite methods and show that thisleads to reasonable bounds.In recent years it became clear that for unconstrained quadratic 0 / 1 programming prob-lemssemidefiniteprogrammingyieldsstrongbounds, intheory(GoemansandWilliamson,1995; Nesterov, 1998) as well as in practice (Helmberg and Rendl, 1998). From a theoret-ical point of view there is a well established technique for including linear constraints inthis semidefinite relaxation (Balas, 1975a; Sherali and Adams, 1990; Lov´asz and Schrijver,1991; Balas et al., 1993). The main ingredients of this technique are the pairwise mul-tiplication of feasible inequalities and the substitution of products of variables by linearvariables (the final projection step back into linear space is not needed in our context).Inthispaperwegotowardsapracticalrealizationofthistheoreticalapproach.InSection2wesetupvariousinitialsemidefiniterelaxationsof(QK)inthelineofBalas(1975a),Sheraliand Adams (1990), Lov´asz and Schrijver (1991) and Balas et al. (1993) and compare themwithrespecttoqualityandefficiency.Inparticularweintroducethesquarerepresentationof the knapsack inequality. For the practical examples tested, the semidefinte relaxation basedon this single constraint achieves almost the bound of the standard semidefinite relaxation.The latter bound is much more expensive to compute, because it involves  n  additionalinequalities. We show that in the special case of a linear cost function the semidefiniterelaxation formed by the square representation of the knapsack inequality still suffices tobeat the linear relaxation.In order to improve an initial semidefinite relaxation it is necessary to add further in-equalities that are valid for the linear formulation of the srcinal program in quadraticspace. Here there are two possibilities. One can derive valid inequalities that are valid forthepolyhedron associated with thefeasibleregion of thesrcinal problem and lift them intothe quadratic model (by the technique of Lov´asz and Schrijver or alternatives) or developa polyhedral understanding of the quadratic model itself. Some aspects of the quadraticknapsack polyhedron have been studied in Ferreira et al. (1996). In Section 3 we introducea new large family of combinatorial cuts for the linear formulation of the knapsack problemin quadratic space and analyze their strength in comparison to quadratic representations of certain linear cutting planes.Aclassofconstraintsthatappearsfrequentlywithinquadratic0 / 1programmingproblemsis the family of generalized upper bound constraints of the form   x i  ≤  k  . For constraintsof this form we construct a special quadratic representation, analyze its strength, and relateit to the hypermetric inequalities of the max-cut polytope. This is the topic of Section 4.  A SEMIDEFINITE PROGRAMMING APPROACH  199We discuss implementational issues and numerical results of the semidefinite program-mingcodeforthequadraticknapsackprobleminSection5.Itwillturnoutthatthesemidef-inite relaxations are of good quality. Computationally they are too expensive for solvinglargerinstancesof(QK)tooptimality. However, theapproachgivessomeinsightonimpor-tant classes of inequalities that may help to set up good initial relaxations of more complexproblems.Thisisofparticularimportanceincombinationwithnewmethods(Bensonetal.,1997; Helmberg and Rendl, 1997) that can compute approximate solutions to semidefiniterelaxations for large problem instances as well. 2. Modeling linear constraints for semidefinite relaxations A common approach for designing relaxations for quadratic 0 / 1 programs is to linearizethe quadratic cost function by switching to “quadratic space”. Variables  y ij  are introducedto model the products  x i  x  j  for  i  ≤  j . Stated differently, the dyadic product  xx t  is replacedby a (symmetric) matrix variable  Y  . We denote the diagonal of this matrix by  y . Usingthis notation the feasible set of matrices can be restricted to those satisfying  Y   −  yy t    0,i.e.  Y   −  yy t  must be positive semidefinite (Lov´asz and Schrijver, 1991; Balas et al., 1994).This condition is equivalent to  Y y y t  1     0 .  (1)The diagonal elements  y i  are obviously bounded by 0 and 1 and correspond to  x i . Lookingat the determinant of a 3 × 3 principal minor containing the last row we get  y i  y  j  −    y i  y  j ( 1 +  y i  y  j  −  y i  −  y  j )  ≤  y ij  ≤  y i  y  j  +    y i  y  j ( 1 +  y i  y  j  −  y i  −  y  j )  (2)which bounds the offdiagonal variables by  − 18  ≤  y ij  ≤  1. Note, that the nonnegativityconstraints  y ij  ≥  0 are not implied by this semidefinite relaxation.The easiest way to model a linear constraint  a t   x  ≤  b , with  a  ∈  N n and  b  ∈  N  on  Y   isto restrict the diagonal elements of   Y  , yielding the  diagonal representation  of   a t   x  ≤  b  andour first semidefinite relaxation,(SQK1) Maximize   C  , Y   subject to   Diag ( a ), Y   ≤  bY   −  yy t    0 . Can we do better than (SQK1) by choosing some other representation of the knapsack inequality? Let us first consider a particular case of the generic approach of Sherali andAdams (1990) and Lov´asz and Schrijver (1991).  b − a t   x  ≥  0 implies ( b − a t   x )( b − a t   x )  =  b 2 − 2 ba t   x  + a t   xx t  a  ≥  0 .  200  HELMBERG, RENDL AND WEISMANTEL So a possible representation for the knapsack inequality could read b 2 − 2 ba t   y  + a t  Ya  ≥  0 . However, this inequality is already implied by the semidefinite constraint  Y   −  yy t    0.Because of the integrality of the coefficients  a i  and  b  we can employ a combinatorialargument to sharpen this inequality. Observe that  | ( 2 b  − 1 ) − 2 a t   x |  is at least one for all0 / 1 vectors  x  and therefore the quadratic representation of   ( 2 b  −  1  −  2 a t   x ) 2 ≥  1 yieldsa valid inequality for the boolean quadric polytope which is best formulated with respectto (1), ( − 2 a t  2 b − 1 )  Y y y t  1   − 2 a 2 b − 1   ≥  1 .  (3)Thisinequalityisnomoreimpliedbythecondition Y  −  yy t    0. Indeed,itcanbeworkedoutthat(3)belongstotheclassofhypermetricinequalities 1 andthatforspecialchoicesof  a  and b  it defines a facet of the boolean quadric polytope (for definition see Section 3). Notice,that this inequality does not exclude 0 / 1 solutions that violate the knapsack constraint.However, it is tight for all  x  which satisfy  a t   x  =  b  or  a t   x  =  b − 1 and might therefore turnout to be a useful cutting plane.Toachieveourgoalofcuttingoffalargerpartofthebooleanquadricpolytopeweexploitthe fact that  a t   x  ≥ − b  on the feasible set. Squaring both sides of   a t   x  ≤  b  yields a t   xx t  a  ≤  b 2 . Replacing  xx t  by  Y   we call this the  square representation  of the inequality  a t   x  ≤  b  and useit to form a second relaxation(SQK2) Maximize   C  , Y   subject to   aa t  , Y   ≤  b 2 Y   −  yy t    0 . Lemma 2.1.  ( SQK  2 )  is tighter than  ( SQK  1 ) . Proof:  With  Z   =  Y   −  yy t  we get a t   Za + ( a t   y ) 2 ≤  b 2 (4)which implies  a t   y  ≤  b  by the positive semidefiniteness of   Z  .   This proof suggests the following corollary. Corollary 2.2.  If a t   y  =  b for some Y satisfying   aa t  , Y   ≤  b 2 and Y   −  yy t    0 ,  then ais in the null space of Z   =  Y   −  yy t  .  A SEMIDEFINITE PROGRAMMING APPROACH  201Another possibility to construct quadratic representations is to multiply the inequality byeither  x i  or  ( 1 −  x i )  (Balas, 1975a; Sherali and Adams, 1990; Lov´asz and Schrijver, 1991;Balas et al., 1993). If, for some fixed  i , we sum up the two inequalities n   j = 1 a  j  y ij  ≤  by i  (5) n   j = 1 a  j (  y  j  −  y ij )  ≤  b ( 1 −  y i )  (6)we get  a t   y  ≤  b . Lemma 2.3  (Balas, 1975a; Sherali and Adams, 1990; Lov´asz and Schrijver, 1991) .  Therelaxation obtained by replacing    Diag ( a ), Y   ≤  b of   ( SQK  1 )  with a pair of inequalities ( 5 )  and   ( 6 )  for some i is tighter than  ( SQK  1 ) . By including all  n  inequalities of type (5) and one additional inequality of type (6) weget(SQK3) Maximize   c , r   subject to n   j = 1 a  j  y ij  ≤  by i  i  =  1 ... n n   j = 1 a  j (  y  jj  −  y 1  j )  ≤  b ( 1 −  y 1 ) Y   −  yy t    0 . Lemma 2.4.  ( SQK  3 )  is tighter than  ( SQK  2 ) . Proof:  By multiplying inequality  i  of type (5) with  a in   j = 1 a i a  j  y ij  ≤  ba i  y i and summing up over all  n  inequalities, we obtain  a t  Ya  ≤  ba t   y  ≤  b 2 . The right hand sideinequality follows from Lemma 2.3.   Itisclearthat(SQK3)canbeimprovedattheexpenseofaddingtheremaininginequalitiesof the form (6) to the relaxation. The improvement of the latter relaxation towards (SQK3)is, in all our test examples, neglectable. This is the reason why we work with (SQK3).Our computational experiments have also revealed that among the relaxations (SQK2) and(SQK3), the first one has the best trade-off between running time and quality. In fact, it isusually more efficient to start with (SQK2) and to add Inequalities (5) and (6) in case of violation only, see Section 5.
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