Y. Nesterov. Quality of semidefinite relaxation for nonconvex quadratic optimization. CORE Discussion Paper 9719, Louvain-La-Neuve, Belgium, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[1] we show how a good (and sometimes optimal) binary solution can be recovered from the presented scheme. Some simulations results are proposed in the final section. The main advantage of our eigenvalue relaxation over the standard Semi Definite Programming (SDP) relaxation of [1] or [3] is that the variables to be optimized lie in Ii n i instead of Ii ( for SDP, resulting in high reduction of the computational effort. Notations. Transposition is denoted by . The set of real symmetric matrices of order n is denoted by n. The maximum eigenvalue of X n is denoted by ,kmax(X) ....

....bundle methods, e.g. the L Newton algorithm of Oustry [4] 3 Worst case bound and randomized approxi mation The previous section provided an upper bound to the transformed problem (3) Our next question is: how good is this approximation The answer is given by a theorem of Yu. Nesterov [3]. Nesterov s bound is obtained as follows. Problem (4) is in fact equivalent to the matrix optimization problem max trace(BX) such that X 0, diagX I, rankX i (11) To see this, realize that the rank condition implies that X xx t for some x 6 , which implies that diagX (x, 2 t = x) ....

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Nesterov Yu., Quality of semi-definite relaxation for nonconvex quadratic optimization, Core discussion paper 9719 Center for Operations Research & Econometrics, Louvain-la-Neuve (1997)

....problem. Hence, this method does not suffer from local maxima. 2. The relaxed problem is a semi definite programming problem, which is known to be efficiently solvable [ll] 3. The SD relaxation algorithm has a theoretical guarantee that the approximation accuracy is, at worst, moderate [10]. Moreover, the performance of this algorithm in practice is substan tially better than that of the worst case. In addition to SD relaxation, we also consider two other relaxation methods called unconstrained relaxation and bound relaxation for the MLD problem. It will be shown that these two ....

....the dimensionality of the problem. Since the original and relaxed problems have different problem dimensions, some special techniques are required to convert the SD relaxation solution to an approximate Boolean QP solution. A randomization method has been proposed for this conversion process [9, 10]. To gain an intuitive understanding of the randomization, we consider alternative expressions of the Boolean QP and SD relaxation problems. The Boolean QP problem in (4.1) can be expressed as: max xixjQij (4.6) i 1, n Define x to be the solution of (4.6) Notice that x is also the ....

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Y.E. Nesterov, "Quality of semidefinite relaxation for nonconvex quadratic optimization," CORE Discussion Paper 9719, Belgium, March 1997.

....the solution x i = sgn(#r, v i #) The expected value of this solution can be compared to the value of the relaxation, and this leads to randomized # approximation algorithms. This technique has been applied, with modifications, to several other problems in combinatorial optimization (e.g. [3, 4, 8, 9, 17, 22, 27, 30]) Almost any randomized approximation algorithm based on the hyperplane technique applied to a semidefinite relaxation can be derandomized, as was shown by Mahajan and Ramesh [21] In the meantime, researchers in mathematical programming have shown that the interior point methods that extend ....

Y. Nesterov. Quality of semidefinite relaxation for nonconvex quadratic optimization. Technical report, CORE, Universite Catholique de Louvain, Belgium, 1997.

....r from the n dimensional normal distribution, and deriving the solution x i = sgn(hr; v i i) The expected value of this solution can be compared to the value of the relaxation. This technique has been applied, with modifications, to several other problems in combinatorial optimization (e.g. [2, 3, 6, 7, 12, 14, 21, 23]) In the meantime, researchers in mathematical programming have shown that the interior point methods that extend polynomial time solvability from linear programming to semidefinite programming also extend polynomial time solvability to the class of symmetric cones. This includes the second order ....

Y. Nesterov. Quality of semidefinite relaxation for nonconvex quadratic optimization. Technical report, CORE, Universite Catholique de Louvain, Belgium, 1997.

....quantity over y 2 f Gamma1; 1g n . Thus we have rewritten MAX 2 SAT as a Boolean quadratic programming problem: unsat : min y ae 1 8 i y T A T Ay Gamma 2e T Ay j j y 2 f Gamma1; 1g n oe : Such problems have standard semidefinite relaxations with provable quality bounds [16]. The first step in deriving the relaxation is to remove the linear term in the objective by adding an additional Boolean variable y n 1 2 f Gamma1; 1g to obtain unsat = min y ae 1 8 i y T A T Ay Gamma 2y n 1 e T Ay j j (y; y n 1 ) 2 f Gamma1; 1g n 1 oe : 2) Note that the ....

Yu. Nesterov. Quality of semidefinite relaxation for nonconvex quadratic optimization. CORE Discussion paper 9719, Belgium, March 1997. 16

....1g is equivalent to x 2 i = 1, for example. Lov asz and Schrijver [35] considered the generic combinatorial problem q max = max Phi x T Qx : x i 2 f Gamma1; 1g (8i) Psi (1) and suggested the relaxation q = maxfTr (QX) diag (X) e; X 0g : 2) For this general relaxation Nesterov [42] recently proved that q Gamma q q max Gamma q min 4 Gamma Gamma q Gamma q Delta where (q min ; q max ) is the range of feasible objective values in (1) and (q; q) is the range of feasible values in the relaxation problem (2) Moreover, a random feasible solution x to (1) ....

Yu. Nesterov. Quality of semidefinite relaxation for nonconvex quadratic optimization. CORE Discussion paper 9719, Belgium, March 1997.

....technique. Indeed, Karloff [31] has shown that the analysis of the random hyperplane technique is tight, namely there exists a family of graphs for which the expected weight E[w(ffi(S) of the cut produced is arbitrarily close to ff SDP . Instead of comparing (13) and (11) term by term, Nesterov [52] recently proposed a different analysis proving that E[w(ffi(S) 2 ( 1 4 L(G) ffl Y ) Even though the resulting bound of 2= 0:63661 Delta Delta Delta is weaker than 0:87856, the analysis only assumes that L(G) 0 and not the stronger requirement that the weights are nonnegative. ....

....( 1 4 L(G) ffl Y ) Even though the resulting bound of 2= 0:63661 Delta Delta Delta is weaker than 0:87856, the analysis only assumes that L(G) 0 and not the stronger requirement that the weights are nonnegative. Furthermore, it has wider applicability than the term by term analysis (see [52]) Letting arcsin(Y ) arcsin(y ij ) we can write 2 E[w(ffi(S) 1 2 P (i;j)2E w ij arccos(v T i v j ) 1 4 P (i;j)2E w ij ( Gamma 2 arcsin(v T i v j ) 1 4 L(G) ffl arcsin(Y ) Therefore, to derive Nesterov s result, we need to prove that L(G) ffl (arcsin(Y ) Gamma Y ) ....

Y. Nesterov. Quality of semidefinite relaxation for nonconvex quadratic optimization. Manuscript, 1997.

....discuss ongoing research activities. 2.1 Progress in Combinatorial and Global Optimization Recently, there were several theoretical results on approximating combinatorial and quadratic optimization problems by using semidefinite programming. see, e.g. Goemans and Williamson [9] and Nesterov [13]) The positive semi definite relaxation was first proposed by Lov asz and Shrijver [12] also see recent papers by Fujie and Kojima [7] Helmberg and Rendl [10] Karisch, Rendl, and Clausen [11] and Polijak, Rendl and Wolkowicz [15] Consider the quadratic programming (QP) problem with ....

....or ffl maximizer, ffl 2 [0; 1] for (QP) is defined as an x 2 F such that q Gamma q(x) q Gamma q ffl: Yinyu Ye: DMI 9522507 3 Goemans and Williamson [9] proved an approximation result for the max cut problem (all coefficients in Q are nonnegative) where ffl = 1 Gamma 0:878. Nesterov [13] generalized their result to general Q and proved a result of ffl = 4=7. However, approximation algorithms for (QP) were open. Recently, we [W5] established an ffl = 4=7 result to solving (QP) i.e. we developed a fast (polynomial algorithm) to deliver a solution for (QP) that is guaranteed to ....

Yu. E. Nesterov, "Quality of semidefinite relaxation for nonconvex quadratic optimization, " CORE Discussion Paper, #9719, Belgium, March 1997. Yinyu Ye: DMI-9522507 8

....greater than 0:878 of optimality. More recently, provably good approximation algorithms using a positive semidefinite relaxation have been found for MAX 3 SAT, MAX 4 SAT, MAX k CUT[35] MAX 3 CSP, scheduling, and minimum bandwidth, graph bisection, bound constrained quadratic programming[40] [31], graph coloring[22] problems. Much like the vertex cover formulation of Kleinberg and Goemans [23] the SDP relaxation of the MSS problem will assign each vertex an integer value of Gamma1 or 1. One of the two sets will be a stable set. Given a graph G with n Gamma 1 vertices, add an ....

Yu. E. Nesterov. Quality of semidefinite relaxation for nonconvex quadratic optimization. In CORE Discussion Paper, #9719, Belgium, March 1997.

....Recently, there were several results on approximating this quadratic maximum. Bellare and Rogaway [3] established several negative result on approximating this problem; Goemans and Williamson [8] also see Frieze and Jerrum [4] proved an approximation result for the Maxcut problem; Nesterov [11] generalized their result to approximating the Maxcut problem with a more general objective matrix; Ye [18] extended their result to solving the continuous (QP) with simple bound constraints; Ye [19] and Nesterov [12] proved constant approximation quality for problem (QP) such that A i is diagonal ....

Yu. E. Nesterov, Quality of semidefinite relaxation for nonconvex quadratic optimization, CORE Discussion Paper #9719, Belgium, March 1997.

....This fact seems to be corroborated by the recent theoretical work of A. Forsgren [11] Let us consider the Boolean quadratic maximization problem (BQM) max x T Qx x 2 f Gamma1; 1g n ; where Q = Gammacc T and c = n Gamma 1; Gamma1; Gamma1] A primal semidefinite relaxation [28] for (BQM) is (SDP ) max hQ; Xi ; X 2 S n d(X) 1 n ; X 0 ; where d(X) is the diagonal of X. The dual of this problem is then (SDP ) min 1 T n u ; u 2 R n D(u) Gamma Q 0 ; where D(u) is the matrix with u 1 ; u n on its diagonal. It is easy to check that the three ....

Yu. Nesterov. Quality of semidefinite relaxation for nonconvex quadratic optimization. CORE Discussion, Paper # 9719, 1997.

....1 Introduction Semidefinite programming, SDP, has become a very intense area of research in recent years; and, one main reason for this is its success in finding bounds for the Max Cut problem, MC. The current bounds have proven to be very tight both theoretically and in numerical tests, see e.g. [6, 11, 8, 7]. In this paper we present a strengthened SDP relaxation for MC, i.e. an SDP program that provides a strengthened bound for MC relative to the current well known SDP bound. One approach to deriving the SDP relaxation is through the Lagrangian dual, see e.g. 13, 12] i.e. one forms the Lagrangian ....

.... rounding a solution to the SDP relaxation, they find a ae approximation algorithm, i.e. a solution with value at least ae times the optimal value, where ae = 878: Numerical tests are presented in e.g. 7, 8] Further results on problems with general quadratic objective functions are presented in [11, 17], e.g. Nesterov [11] uses the SDP bound to provide estimates of the optimal value of MC, with arbitrary L = L T ; with constant relative accuracy. 2 Lagrangian Relaxation A quadratic model for MC with a general homogeneous quadratic objective function is MC = max v T Qv s.t. v 2 i ....

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Y. E. NESTEROV. Quality of semidefinite relaxation for nonconvex quadratic optimization. Technical report, CORE, Universite Catholique de Louvain, Belgium, 1997.

....the following Boolean QP problem which arises in combinatorial optimization, namely q max = max Phi x T Qx : x i 2 f Gamma1; 1g (8i) Psi ; 7) which has as Shor relaxation 1 q = maxfTr (QX) diag(X) e; X 2 Sg ; 8) with e the all one vector. For this general relaxation Nesterov [20] recently proved that q Gamma q q max Gamma q min 4 Gamma Gamma q Gamma q Delta where (q min ; q max ) is the range of feasible objective values in (7) and (q; q) is the range of feasible values in the relaxation problem (8) Moreover, a random feasible solution x ....

Yu. Nesterov. Quality of semidefinite relaxation for nonconvex quadratic optimization. CORE Discussion paper 9719, Belgium, March 1997.

....Management Sciences The University of Iowa Iowa City, Iowa 52242, U.S.A. March 31, 1997 Abstract We consider the problem of approximating the global maximum of a quadratic program (QP) with n variables subject to bound constraints. Based on the results of Goemans and Williamson [4] and Nesterov [6], we show that a 4=7 approximate solution can be obtained in polynomial time. Key words. Quadratic programming, global maximizer, approximation algorithm This author is supported in part by NSF grant DMI 9522507. 1 Introduction Consider the quadratic programming (QP) problem q(Q) Maximize ....

....ffl: Note that according to this definition any feasible solution x is a 1 maximizer. Recently, there were several significant results on approximating specific quadratic problems. Goemans and Williamson [4] proved an approximation result for the Maxcut problem where ffl 1 Gamma 0:878. Nesterov [6] generalized their result to approximating a boolean QP problem Maximize q(x) x T Qx Subject to jx j j = 1; j = 1; n: where ffl 4=7. Some negative results were given by Bellare and Rogaway [1] There are also several approximation algorithms developed for approximating (QP) when the ....

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Yu. E. Nesterov, "Quality of semidefinite relaxation for nonconvex quadratic optimization," CORE Discussion Paper, #9719, Belgium, March 1997.

....q ffl: Recently, there were several significant results on approximating specific quadratic problems. Goemans and Williamson [7] also see Frieze and Jerrum [3] proved an approximation result for the Maxcut problem where ffl 1 Gamma 0:878 when all arc weights are nonnegative. Nesterov [9] generalized their result to approximating a boolean QP problem Maximize q(x) x T Qx Subject to jx j j = 1; j = 1; n; where ffl 4=7. Ye [14] extended the 4=7 result to solving the continuous QP problem Maximize q(x) x T Qx Subject to jx j j 1; j = 1; n: Some negative ....

....In what follows, we let x = x(Q) X = X(Q) Since X is positive semi definite, there is a factorization matrix V = v 1 ; v n ) 2 n Thetan , i.e. v j is the jth column of V , such that X = V T V . The algorithm (Goemans and Williamson [7] Nesterov [9], and Ye [14] generates a random vector u uniformly distributed on the n dimensional unit ball and then assigns x = Doe( V T u) 3) where D = diag(k v 1 k; k v n k) diag( p x 11 ; p x nn ) and for any x 2 n , oe(x) is the vector whose components are ....

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Yu. E. Nesterov, Quality of semidefinite relaxation for nonconvex quadratic optimization, CORE Discussion Paper #9719, Belgium, March 1997.

....Williamson [8] provided a 0.878 approximation algorithm for the max cut and s Gamma t max cut problems. Frieze and Jerrum [6] generalized the Goemans and Williamson approach to the k max cut problem and provided a 0.65 approximation algorithm for the maximum graph bisection problem. Nesterov [12] provided a 2 approximation algorithm for the boolean QP problem Maximize q(x) x 0 Qx subject to x 2 j = 1; j = 1; n; where Q is positive semidefinite. Ye [17] further extended the 2 result to solving the general QP problem Maximize q(x) x 0 Qx subject to n X j=1 a ....

....f(t) and X 2 n Thetan , let f [X] 2 n Thetan be the matrix with the components f(x ij ) Lemma 2 Let X 0 and d(X) e. Then X Gamma 2 2 ( n X j=1 x jj )I arcsin[X] X; where I is the identity matrix in n Thetan . Proof. The right side inequality is proved by Nesterov [12]. We now prove the left. Note that X 0 and jx ij j 1 for all i; j = 1; n, we have [X] 0 0 for all t = 1; 2; Thus arcsin[X] X 1 2 Delta 3 [X] 3 1 Delta 3 2 Delta 4 Delta 5 [X] 5 : X 1 2 Delta 3 n X j=1 x 3 jj I 1 Delta 3 2 Delta 4 Delta ....

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Yu. E. Nesterov, "Quality of semidefinite relaxation for nonconvex quadratic optimization," CORE Discussion Paper, #9719, Belgium, March 1997.

....y i y j ) then the resulting problem is NP hard to approximate within 11=12 ffl = 0:91666 Delta Delta Delta, while the random hyperplane technique still gives the same guarantee of ff 0:87856. The analysis of the random hyperplane technique can be generalized following an idea of Nesterov [21] for more general Boolean quadratic programs. First observe that (9) can be rewritten as E[w(ffi(S) 1 2 L(G) ffl arcsin(Y ) where arcsin(Y ) arcsin(y ij ) Suppose now that we restrict our attention to weight functions for which L(G) 2 K for a certain cone K. Then a bound of ff would ....

Y. Nesterov. Quality of semidefinite relaxation for nonconvex quadratic optimization. CORE Discussion Paper 9719, Louvain-La-Neuve, Belgium, 1997.

....[7] in the maximum cut problem, one is given a graph G = V; E) with non negative weights w ij for i; j 2 V , and wishes to find a set S ae V so as to maximize P i2S;j = 2S w ij . As time permits, I will also discuss how this technique has been extended to quadratic programming by Nesterov [13]. A survey of applications of semidefinite programming to problems in combinatorial optimization can be found in Goemans [5] ....

Y. Nesterov. Quality of semidefinite relaxation for nonconvex quadratic optimization. CORE Discussion Paper #9719, 1997.

....University of Iowa, Iowa City, Iowa 52242, U.S.A. 1 Introduction Recently, there were several theoretical results on the effectiveness of approximating combinatorial and nonconvex quadratic optimization problems by using semidefinite programming (see, e.g. Goemans and Williamson [11] Nesterov [24], and Ye [33] These results raise the hope that some hard optimization problems could be tackled by solving large scale semidefinite relaxation programs. The positive semidefinite relaxation was early considered by Lov asz [18] and Shor [29] and the field had received further contributions by ....

Yu. E. Nesterov, "Quality of semidefinite relaxation for nonconvex quadratic optimization," CORE Discussion Paper, #9719, Belgium, March 1997.

....relative accuracy. In the second part we consider some natural extensions of the result. 1 Introduction Starting from the seminal paper by Goemans and Williamson [1] we can see an increasing interest to the global quadratic optimization problems over some simple feasible sets. As it was shown in [2], the random hyperplane technique, proposed in [1] for Max Cut problem, can be applied also to get a constant relative accuracy bound for the problem of maximizing a general quadratic form over a Boolean box. Later on, in the papers [4, 5, 3] it was shown that the feasible set of the problems can ....

Yu.Nesterov, "Quality of semidefinite relaxation for nonconvex quadratic optimization, " CORE Discussion Paper #9719, CORE, March 1997.

....implications of the results for the discussion around the question P = NP. # CORE, Catholic University of Louvain, 34 voie du Roman Pays, 1348 Louvain la Neuve, Belgium; e mail: nesterov core.ucl.ac. be 1 1 Introduction Starting from the pioneering paper [2] there were obtained several results [4, 7, 5], which show that a solution of an indefinite quadratic maximization problem with some linear constraints on the squared variables can be approximated with a constant relative accuracy. In this paper we present some improvements and extensions of the results [5] In Section 2 we consider a problem ....

Yu.Nesterov, "Quality of semidefinite relaxation for nonconvex quadratic optimization, " CORE Discussion Paper #9719, CORE 1997.

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Y. Nesterov. Quality of semidefinite relaxation for nonconvex quadratic optimization. CORE Discussion Paper 9719, Louvain-La-Neuve, Belgium, 1997.

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Yu. E. Nesterov, "Quality of semidefinite relaxation for nonconvex quadratic optimization," CORE Discussion Paper, #9719, Belgium, March 1997.

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Y. Nesterov, "Quality of Semidefinite Relaxation for Nonconvex Quadratic Optimization," Technical Report, CORE, Universite Catholique de Louvain, Belgium, 1997.

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Y.E. NESTEROV. Quality of semidefinite relaxation for nonconvex quadratic optimization. Technical report, CORE, Universite Catholique de Louvain, Belgium, 1997.

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