Approximation algorithms for trilinear optimization with nonconvex constraints and its extensions

Transcription

1 Approximation agorithms for triinear optimization with nonconvex constraints and its extensions Yuning Yang Schoo of Mathematics Science and LPMC Nankai University Tianjin , P.R. China Qingzhi Yang Schoo of Mathematics Science and LPMC Nankai University Tianjin , P.R. China Apri 2, 2011 Abstract In this paper, we study triinear optimization probems with nonconvex constraints under some assumptions. We first consider the semidefinite reaxation (SDR) of the origina probem. Then motivated by So [3], we reduce the probem to that of determining the L 2-diameters of certain convex bodies, which can be approximatey soved in deterministic poynomia-time. After the reaxed probem being soved, the feasibe soution of the origina probem with a good approximation ratio can be obtained from the feasibe soution of the reaxed probem by state-of-art agorithms. Last we consider a cass of biquadratic optimization probems, which has a cose reationship with the triinear optimization probems. Key words: triinear optimization, biquadratic optimization, semidefinite reaxation, approximation soution, convex bodies. MSC 74B99, 15A18, 15A69 Corresponding author: Qingzhi Yang. This work was supported by the Nationa Natura Science Foundation of China (Grant No ) and Scientific Research Foundation for the Returned Overseas Chinese Schoars, State Education Ministry. 1

2 1 Introduction The poynomia optimization is a hot topic in the recent years and studied extensivey in the iterature, e.g., the mutiinear and homogeneous optimization over unit spheres [3, 15], quadratic constraints [15] and discrete sets [17]; the biquadratic optimization over unit spheres [5, 20, 21] and quadratic constraints [20, 6]; and other genera forms [12, 16]. In this paper, we study the triinear optimization of the form (T LP ) maximize Bxyu subject to x S R m, y T R n, u 2 = 1, u R where S R m and T R n are certain bounded nonconvex sets, respectivey. B = (b ijk ) R m n is a 3-th order tensor. (T LP ) generaizes the triinear optimization probems over unit spheres studied by [3], and over discrete sets studied by He et a [17]. We consider two cases of the constraints S and T in this paper: 1. S = {x x A i x 1, i = 0,..., p, } and T = {y y B j y 1, j = 1,..., q, }, where A i and B j are symmetric positive semidefinite matrices, i = 1,..., p, j = 1,..., q whie A 0 may be indefinite. 2. S = {x x 2 F 1 } and T = {y y 2 F 2 }, where F 1 and F 2 are some cosed convex subsets; The second case incudes some we-known probems, e.g., when F 1 and F 2 are the standard simpexes, it hods that m i=1 x2 i = 1 and n j=1 y2 j = 1. Then the probem is the triinear optimization with unit sphere constraints, which is consider by He et a [15] and So [3]; when F 1 = {e} and F 2 = {e} where e denotes the a-one vector, i.e., x { 1, 1} m and y { 1, 1} n, or, F 1 is the singeton point set and F 2 is the standard simpex, they are the mixed-variabes forms studied by He et a [17]. Thus we give a unified framework to treat these probems. The first case is reated to the biquadratic form studied by Zhang et a [20] under some assumptions, and we wi discuss it in the ast section. We wi show that the bounds obtained there can be improved by our approach. To sove the probem, we first consider the semidefinite reaxation of the origina probem. Then motivated by So [3], we reduce the reaxed probem to that of determining the L 2 -diameters of certain convex bodies, which can be approximatey soved in poynomia-time by using powerfu resuts from the agorithmic theory of convex bodies [24, 1], with an approximation ratio of Ω( ). We say a poynomia-time approximation agorithm appied to a probem P to find the maxima vaue P max has an approximation ratio α (0, 1], if can find a ower bound p, such that { P max p αp max if P max 0 αp P max p if P max < 0. It is worth pointing out that in [3], So used this approach to sove the mutiinear optimization probem over unit spheres, who got the best approximation ratio known to date. After the reaxed probem being soved, the feasibe soution of the origina probem with a good approximation ratio can be obtained from the feasibe soution of the reaxed probem by state-of-art agorithms. 2

3 The rest of this paper is organized as foows. In Section 2, we review the approach using by So [3] and introduce some concerned definitions, which makes the content more cear and sef-contained. In section 3 and Section 4, we discuss the two cases above. The reationship between the triinear form and the biquadratic form is studied in Section 5. We first add a comment on the notation that is used in the seque. Q denotes the rationa number fied. Vectors are written as owercase etters (x, y,...), matrices correspond to itaic capitas (A, B,...), and tensors are written as caigraphic capitas (A, B, ). The spaces of n-dimensiona rea vectors are denoted by R n, and the spaces of n n rea (symmetric) matrices are denoted by R n n (S n n ). For two rea matrices A and B with same dimension, A B stands for the usua matrix inner product, i.e., A B = tr(a B), where tr( ) denotes the trace of a matrix. In addition, A F denotes the Frobenius norm of A, i.e., A F = (A A) 1/2. The symbo. denotes the entry-wise production of two vectors with same dimension foowing that of MATLAB. For a vector x R n, x 2 = x. x. The symbo A B means that (A B) is in the positive semidefinite cone. v( ) denotes the maxima vaue of a probem. 2 A powerfu approach to sove the triinear optimization probem over unit spheres Let us review the approach proposed by So [3] to sove the triinear optimization probem over unit spheres. Briefy speaking, the probem is reduced to that of maximizing a certain norm over the sphere, which, by standard duaity arguments, is equivaent to determining the L 2 -diameter of a certain convex body. Let us reca the probem: (T LP ) maximize Bxyu subject to x 2 = y 2 = u 2 = 1, x R m, y R n, u R. Note that for any given u R, this probem is nothing but finding the argest singuar vaue of matrix B(u), where B(u) R m n is the matrix such that B(u) ij = B ijk u k. (2.1) k=1 Let B(u) 2 be the argest singuar vaue of B(u), then the maxima vaue of the origina probem is equa to max u 1 B(u) 2. As such, the author first showed that the function u u B B(u) 2 defines a norm on R. Then he caimed that the unit ba of the norm B given by B B = {u R u B 1} is a 0-centered, centray symmetric convex body. Moreover, this body is we-bounded [24], i.e. a rationa number R 1 is given expicity such that B B B 2(R 1 ) and a rationa number R 2 is given expicity such that B 2(R 2 ) B B, where B 2(R) denotes the -dimensiona Eucidean ba centered at the origin with radius R > 0. Note that if a convex body K satisfying the above conditions (n-dimensiona, a 0 -centered, we-bounded with radiuses R 1 and R 2 known expicity), then it can be denoted by a quintupe (K; n, R 1, R 2, a 0 ), see Definition of [24]. 3

4 Here we use (B B ;, R 1, R 2, 0) to denote B B. The we-boundness of B B pays an important roe in the whoe method. We wi mention it ater. Now consider the poar of B B, where the poar of a convex body K R is defined as K = {y R y x 1 for a x K}. It is known that if (K; n, R, r, 0) is a centray symmetric convex body, then so is (K ; n, 1/r, 1/R, 0), see e.g. [24] p By this property, we can use (B B ;, 1/R 2, 1/R 1, 0) to represent B B. The purpose of considering the poar of B B is that finding the maxima vaue of (T LP ) is equivaent to cacuating the L 2 -diameter of B B : v(t LP ) = max u 1 u B = max (max u v) u 1 v BB = max v B ( max u 1 u v) = max v B v 2 = 1 2 diam 2(B B) where diam 2 (K) represents the L 2 -diameter of a convex body K, i.e., diam 2 (K) = sup x,y K x y 2, where the ast equaity comes from the centray symmetry of BB. Thus the probem reduces to cacuate diam 2 (BB ). By the NP-hardness of (T LP ), we can not hope to cacuate it in poynomia-time. However, there are agorithms to approximate it in poynomia-time, see e.g. [1]. The foowing theorem points out that for a convex body K R n, diam 2 (K) can be og n approximated within a ratio Ω( n ) in poynomia-time if certain probems associated with K can be soved in poynomia-time: Theorem 2.1 (Theorem 2 of [3]) Given an integer n 1, one can construct in deterministic poynomia-time a centray symmetric poytope P R n such that 1. B n 2 (1) P B n 2 (O( n og n )) 2. for any we-bounded centray symmetric convex body K in R n, one has og n Ω( n ) diam 2(K) diam P (K) diam 2 (K), where diam P (K) is the diameter of K with respect to the poytopa norm P induced by P (i.e., for any x R n, one has x P = min{λ 0 x λp }, and P is the unit ba of the induced norm). Moreover, if K is equipped with a weak membership orace, then for any given rationa number ɛ > 0, the quantity diam P (K) can be computed to an accuracy of ɛ in deterministic orace-poynomia-time, and a vector x K(ɛ) is deivered with x P (1/2)diam P (K) ɛ. In this theorem, some definitions shoud be emphasized: An Orace can be regarded as a device that soves certain probems, ([1], p.26.) or simpy speaking, it is a subroutine (back box) that can be used by any agorithm ([24]). A Weak 4

5 Membership Orace soves the Weak Membership Probem (WMEM) ([1], p.51) for a body K R n : given a vector y Q n and a rationa number ɛ > 0, either (i) assert that y K(ɛ), or (ii) assert that y K( ɛ), where K(ɛ) and K( ɛ) are the outer ba and inner ba of K given respectivey by K(ɛ) = K + B n 2 (ɛ), K( ɛ) = {x R n x + B n 2 (ɛ) K}. Weak refers to the fact that we have to aow for a rounding error, since ony finite precision is avaiabe. An agorithm has orace-poynomia-time compexity if its runtime is poynomia in both the input size and the number of cas to the orace [3]. An important consequence is that any orace-poynomia-time agorithm can be turned into a genuiney poynomia agorithm for any situation in which the orace s action can be carried out in poynomia-time [1]. Thus Theorem 2.1 impies that if the WMEM of BB can be computed in poynomiatime, then diam P (BB ) can be approximated to arbitrary accuracy in poynomia-time. From [1], it is known that if we want to approximate diam P (BB ) to arbitrary accuracy in poynomia-time, a Weak Separation Orace associated with BB which soves the Weak Separation Probem (WSEP) shoud be given. One shoud note that if the WSEP associated with BB can be soved in poynomia-time, then the WMEM of B B can aso be soved in poynomia-time, but not vice versa. However, by the we-boundness of BB, these two probems are the same, in the sense of poynomia-time-compexity, see e.g., [24]. Actuay, by the we-boundness of K, and the reationship between K and its poar K, we ony need to check whether the WMEM associated with B B can be soved in poynomia-time. To see this, we need two other probems associated with a body K: Weak Optimization Probem (WOPT) and Weak Vaidity Probem (WVAL). The WOPT is the strongest one in these probems, since other probems can be soved in orace-poynomia-time if a body K is given by a weak optimization orace, but not vice versa. Nevertheess, the foowing emma shows the equivaence of WMEM and WOPT in the sense of poynomia-time-compexity, if the body is we-bounded: Lemma 2.1 (Coroary of [24]) There exists an orace-poynomia-time agorithm that soves the weak optimization probem for every centered body (K; n, R, r, a 0 ) given by a weak membership orace. These emma shows the equivaence of WOPT, WMEM and WVAL. The foowing emma shows the reationship between WVAL of K and WMEM of K Lemma 2.2 (Lemma of [24]) There exists an orace-poynomia-time agorithm that soves the weak membership probem for K, where K is a 0-centered convex body given by a weak vaidity orace. 5

6 Since B B and BB are 0-centered, centray symmetric and we-bounded, we can deduce the foowing emma Lemma 2.3 If the weak membership probem of B B can carry out its computation in poynomiatime, then so is the weak membership probem of its poar B B. Hence, it suffices to prove the poynomia-time-compexity of WMEM of B B for approximating diam 2 (BB ). For (T LP ), this condition hods. Finay, So showed that how to construct a feasibe soution of (T LP ) that attains the approximation ratio Ω( ). To begin, et ɛ > 0 be such that ɛ < γ 4 max i,j,k b ijk γ 4 v(t LP ), where γ 2 5 is the constant behind the Ω-notation in Theorem 2.1. By such theorem, we can in deterministic-poynomia-time compute a vector u BB (ɛ) such that u P 1 2 diam P (B B) ɛ. Then the author proved the foowing hods u 2 γ v(t LP ) ɛ. (2.2) Let u = u and (x, y) = arg max u Bxyu 2 x 2=1, y 2=1 we have (x, y, u) is feasibe for (T LP ) and Bxyu = u B. Moreover, it foows from the choice of ɛ and (2.2) that u ɛ u 2 u BB. Thus Bxyu = u B = max u v v BB u (u ɛ u) u 2 = u 2 ɛ γ v(t LP ). 2 This competes the whoe summary of the method of [3]. Not ike the SDR-based approach proposed by Luo and Zhang [23], and Ling et a [5], or the SOS method (cf. Lasserre [9, 10]), or other reaxation methods proposed by He et a [16, 15], So provides a totay different approach for the poynomia optimization probems over unit spheres, where the main toos come from the agorithmic theory of convex bodies, with the best-known approximation ratio to date. Looking backward, the foowing are some key points of this powerfu method Condition.1 1. For any fixed u R, the probem (T LP ) defines a norm on R. 2. The unit ba of the norm is 0-centered and we-bounded. 6

7 3. The WMEM associated with this unit ba can be soved in deterministic-poynomia-time. Naturay, we wi ask a question: can we appy this method to more genera constrained probems? E.g., the probem introduced in the beginning of this paper (T LP ) maximize Bxyu subject to x S R m, y T R n, u 2 = 1, u R where S and T may be some nonconvex bounded sets, or even discrete sets. We aways assume the probem under consideration has an optima soution. We sti use (T LP ) to denote this probem. Note that in the approach, we do not need to know the detai of the norm, which impies that if (T LP ) defines a norm on R, then computing v(t LP ) is aso equivaent to cacuating the L 2 -diameter of the unit ba of the norm. Furthermore, if the unit ba is webounded and the WMEM probem associated with the ba can be soved in poynomia-time, then we can appy the approach to (T LP ). The foowing theorem summarizes it Theorem 2.2 If (T LP ) satisfies Condition.1, then there is a deterministic-poynomia-time agorithm that, given an instance of (T LP ), returns a feasibe soution (x, y, u), such that Bxyu Ω( )v(t LP ). In the next section we wi consider two cases that satisfy these conditions. Here we give some other form of the objective function Bxyu for further anaysis. Let B(u) be defined in (2.1). Then Bxyu = B(u) xy = x B(u)y. Denote B R mn the unfoded matrix of B such that B (i 1)n+j,k = B ijk. Then Bxyu is equivaent to (x y) Bu, where denotes the Kronecker products such that x y = (x 1 y 1, x 1 y 2,..., x 1 y n, x 2 y 1,..., x 2 y n,..., x m y n ). We suppose mn in the foowing of this paper. 3 Quadratic constraints In this section we assume that S = {x R m x A i x 1, i = 0,..., p} and T = {y R n y B j y 1}, where A i, B j are a symmetric positive semidefinite matrices, i = 1,..., p, j = 1,..., q, whie A 0 may be indefinite. We assume that there exists u k 0, k = 0,..., p and v 0, = 1,..., q such that p q u i A i 0, v j B j 0. (3.3) i=0 The probem we consider in this section is j=1 (T LP ) maximize Bxyu = (x y) Bu (3.4) subject to x A i x 1, i = 0,..., p, y B j y 1, j = 1,..., q, u 2 = 1, u R 7

8 For a fixed u R, this mode reduces to the probem of maximizing a homogeneous quadratic form over quadratic constraints, which has been studied intensivey in the iterature [4, 18]. Even in this case, the probem is NP-hard [18], so we can not hope to find a maxima soution in poynomia-time. Instead, we want to use the approach of Section 2 to approximatey sove it. But we can not directy appy the approach on this probem. The reason wi be expained after Proposition 4.8. Instead, we wi consider its semidefinite reaxation. To begin with, et B(u) = 1 2 [ 0 B(u) B(u) 0 be a (m + n) (m + n) symmetric matrix and z = [x, y ], and et C i R (m+n) (m+n), i = 0,..., p + q be [ ] A i 0 if i = 0,..., p 0 0 C i = [ ] 0 0 if i = p + 1,..., p + q. 0 B (i p) Then (3.4) can be formuated as the homogeneous form with quadratic constraints The SDP reaxation is ] maximize z B(u)z (3.5) subject to z C i z 1, i = 0,..., p + q, u 2 = 1, u R. (RT LP ) maximize B(u) Z (3.6) subject to C i Z 1, i = 0,..., p + q, Z 0, u 2 = 1, u R. It is cear that v(rt LP ) v(t LP ). In what foows, we mainy prove that (RT LP ) satisfies Condition.1. To this end, we first show that the maxima vaue of (RT LP ) is bounded away from zero, where the bound is expicity known Proposition 3.1 If B = 0, then v(rt LP ) âˆb max ijk b ijk > 0, where â = 1 max0 i p max 1 k m (A i ) kk, ˆb 1 = max1 i q max 1 k n (B j ) kk. Proof. By (3.3), the denominator of â (resp. ˆb) must be positive, hence â and ˆb are wedefined. Without oss of generaity suppose b 111 = arg max b ijk. Let u = [sign(b 111 ), 0,..., 0], x = [a, 0,..., 0] and y = [b, 0,..., 0]. Then (x, y) S T and we have v(rt LP ) v(t LP ) Bxyu = âˆb max ijk b ijk > 0, as desired. For any fixed u, (RT LP ) is a we-formuated semidefinite programming g(u) := maximize B(u) Z (3.7) subject to C i Z 1, i = 0,..., p + q, Z 0. 8

9 By (3.3), it hods that p u i C i + i=0 q v j C j+p 0. (3.8) j=1 Under this condition, the SDP reaxation satisfies the dua Sater condition. Thus the primadua optima soutions exist and the prima-dua optima objective vaues are attainabe, and (3.7) can be soved by the interior-point method in poynomia-time. Now we caim that g(u) defines a norm on R Proposition 3.2 g(u) defines a norm on R. Proof. Firsty, for any u R, Bxyu is inear on x and y, hence we have max (x,y) S T Bxyu 0 for fixed u, which impies g(u) 0 for any u. Next we prove g(u) = g( u). Let Z = rz i=1 zi (z i ) be such that B(u) Z = g(u) where Z is feasibe for (3.7). For i = 1,..., r z, denote z i = [x i, y i ] where x i R m and y i R n satisfying z i = [x, y ]. Let Z = rz i=1 zi (z i ). Then we see that Z is aso feasibe for (4.11) and B( u) Z = g(u), which shows that g( u) g(u); using a simiar argument, we can aso deduce g(u) g( u). Thus we have g(u) = g( u) for any u R. Secondy, it is cear that g(α 1 u 1 + α 2 u 2 ) α 1 g(u 1 ) + α 2 g(u 2 ) for any u 1, u 2 α 1, α 2 0. Thus g(u) defines a semi-norm on R. R and Thirdy, denote L = {u g(u) = 0}. To compete the proof, we need to show that L = {0}. The equation g(u) = 0 impies that for any feasibe soution of (3.7), the objective vaue is identicay zero, which means that for any feasibe pair (x, y) S T, Bxyu = 0. If Bu 0 and without oss suppose (Bu) 1 0, then et x = [â sign((bu) 1 ), 0,..., 0], y = [ˆb, 0,..., 0] where â and ˆb are defined in Proposition 3.1. We have Bxyu > 0, which deduces a contradiction. Thus g(u) = 0 impies Bu = 0. This means that L is the nuspace of matrix B. As that of [3], armed with Proposition 3.1, we see that any optima soution (Z, u ) of (RT LP ) must satisfy u L where L R is the orthogona compement of L. With the assumption that mn, we can without oss assume that matrix B R mn has fu coumn rank, otherwise we can reduce the dimension of B and obtain a probem that is equivaent to (RT LP ). So Bu = 0 impies u = 0 and hence L = {0}. This competes the proof. Denote the norm by u B. It is cear that v(rt LP ) = max u 2=1 u B = max u 2 1 u B. Consider the unit ba of the norm u B given by B B = {u R u B 1}, which is 0-centered, centray symmetric and convex. As that of Section 2, computing v(rt LP ) is equivaent to cacuating the L 2 -diameter of the unit ba. The foowing concusion shows that B B is a we-bounded convex body, whose outer ba s radius and inner ba s radius are expicit known. Proposition 3.3 There exist rationa numbers 0 < C 1 C 2 <, whose encoding engths are poynomiay bounded by the input size of the probem, such that B 2(C 1 ) B B B 2(C 2 ). Proof. To prove this proposition, we first give a bound on the feasibe set of (3.7). Denote Ω = {Z R (m+n) (m+n) C i Z 1, i = 0,..., p + q, Z 0}. 9

10 Let C = p i=0 u ic i + q j=1 v jc j+p. From (3.8) we know that C 0 and we have Ω Ω := {Z R (m+n) (m+n) C Z p u i + i=0 q v j, Z 0}. By the symmetry of C, there is an orthogona matrix such that C = QΛQ where Λ is diagona and positive definite. Then it is equivaent that Ω = {Z R (m+n) (m+n) Λ Z p u i + i=0 j=1 q v j, Z 0}, which shows that for any Z Ω, we have m+n i=1 λ ic ii p i=0 u i + q j=1 v j, where λ i is an eigenvaue of C (it is aso the i-th diagona entry of Λ). By the positive definiteness of C and the positive semidefiniteness of Z, we have for any index i {1,..., m + n}, m+n λ min Z ii λ min i=1 Z ii Λ Z j=1 p u i + i=0 q v j, where λ min is the smaest eigenvaue of C. Thus we have for any Z Ω, Z ij Z ii Z jj ( p u i + i=0 j=1 q v j )/λ min, which is expicit bounded. Let e k be the k-th basis vector in R, k = 1,...,. Then we have 0 e k B = ( 1 i m,m+1 j m+n 1 i m,m+1 j m+n p u i + i=0 q v j ) j=1 j=1 B(e k ) ij Z ij = B ijk Zii Z jj 1 i m,1 j n where Z is a maximizer of (3.7). For k = 1,...,, denote c k = ( p u i + i=0 q v j ) j=1 1 i m,1 j n 1 i m,m+1 j m+n B ijk /λ min, B ijk /λ min, c max = max k=1 c k. Then e k 1 /c max B B. Let C 1 = (. We have c max) B 2(C 1 ) = {u R u 2 B ijk Z ij 1 ( c max ) } {u R u 1 1 } B B, c max where the first incusion comes from the fact that u 1 u 2 for any u R and the ast incusion comes from the equivaence between {u R u 1 1 c max } and the convex 1 hu c max {±e 1,..., ±e }. Thus we have proved that B B contains a ba with radius C 1, whose encoding ength is poynomia bounded by the input size of (RT LP ). On the other hand, for any u R \ {0}, it foows from Proposition 3.2 that B(u) 0. Suppose B(u) 11 = arg max B(u) ij 0. Let x = [sign(b(u) 11 )â, 0,..., 0], y = [ˆb, 0,..., 0] 10

11 where â and ˆb are defined in Proposition 3.1. Then we have (x, y) S T and obviousy u B B(u) xy λmin (B = âˆb Bu âˆb B) u 2. mn Since B R mn has fu coumn rank by the proof of Proposition 3.2 and mn, we have λ min (B B) > 0. Hence B B B2(C 2 ), where C 2 = (ab) mn/ λ min (B B). Thus we have proved B B is contained in a ba with radius C 2, whose encoding ength is aso bounded by the input size of (RT LP ). This competes the proof. The ony thing eft is to prove that the weak membership probem associated with B B can be soved in poynomia-time. We caim that this concusion hods Proposition 3.4 The weak membership probem associated with BB can be soved in deterministic poynomia-time. Proof. By the anaysis of Section 2, since B B is we-bounded, we ony need to prove that the weak membership probem associated with B B can be soved in deterministic poynomia-time. From the definition of the WMEM probem, we see that a key point to make the concusion hod is the poynomia-time computationa compexity of (3.7). For a given u Q \{0} and a rationa number ɛ > 0, we can use the interior-point method to compute a Z Ω in poynomia-time, such that u B ω(u) B(u) Z u B ɛ u 2, where Ω is the feasibe soution set of (3.7). Simiar to the proof of Proposition 3 of [3], if ω(u) > 1, then u B > 1, which impies u B B ( ɛ); if ω(u) 1, then u B B (ɛ). This competes the proof. We mention here that for fixed u R, since (T LP ) is NP-hard, it can not be soved in poynomia-time, which, by the proof above, does not meet the requirement of the poynomiatime sovabiity of the WMEM probem. That is why we must consider the reaxed probem of (T LP ). Now we have proved that Condition.1 hods for this case. Then we can in poynomia-time find a soution of (RT LP ) with ratio Ω( ). Foowing Section 2, we wi show that how to construct such a feasibe soution of (RT LP ). Let ɛ be chosen such that ɛ < γ âˆb max 4 b ijk < γ v(rt LP ), ijk 4 where â and ˆb are defined in Proposition 3.2. We can in deterministic poynomia-time compute a vector u BB (ɛ) such that u 2 γ v(rt LP ) ɛ and u ɛu/ u 2 B B. Let û = u u 2, and Z be a maximizer of (3.7) with respect to û. We 11

12 have that (û, Z ) is feasibe for (RT LP ) and B(û) Z = û B = max û v v BB u 2 ɛ γ v(rt LP ). 2 Thus we have in poynomia-time found a feasibe soution of (RT LP ) with approximation ratio Ω( ). The task eft is to extract a rank-one feasibe soution of (T LP ) from Z. Note again that for fixed u, it reduces to the probem studied by He et a [18], who used a randomized procedure to get a rank-one soution from a maximizer of the semidefinite reaxation probem. Since Z is a maximizer of the semidefinite probem (3.7) with respect to û, we can appy their method on our case. Combing their method and the previous anaysis, we have Theorem 3.3 There exists a deterministic poynomia-time agorithm that, given any instance of (RT LP ), returns a feasibe soution pair (û, Z ), such that B(û) Z Ω( ) v(rt LP ); where Z is a maximizer of (3.7) with respect to û. There exists a feasibe soution (x, y, û) of (T LP ) such that the probabiity Bxyû > B(û) Z α is at east (p+q)µe α/2, where α > 2 og[(174(p+q)µ] where µ = min{(p+q), max p+q i=0 rank(c iz )}. Moreover, we can find such a feasibe soution in randomized poynomia-time time. To sum up, we can in poynomia-time find a feasibe soution of (3.4) with approximation ratio Ω( og[175(p+q)u] ). Note that if A 0 is positive semidefinite, then foowing Nemirovski [4], the feasibe soution (x, y, û) of (T LP ) returned by the randomized agorithm satisfies Prob{Bxyû > B(û) Z } > 1 2(p + q)µ e α /2 α where α > 2 og[(2(p + q)µ ] where µ = min{(p + q), max p+q i=0 rank(c i)}. 4 cosed convex set In this section, we consider this case: S = {x R m x 2 F 1 }, T = {y R n y 2 F 2 }, where F 1 and F 2 are some cosed convex subsets. Through out this section, we assume the foowing hod for S and T. 12

13 Assumption There exist R 1 (m, n) > 0 and R 2 (m, n) > 0 such that F 1 B m 2 (R 1 ) and F 2 B n 2 (R 2 ), where R 1 and R 2 are poynomias with variabes m, n; 2. there exists a positive number β, for any given subindex pair (i, j), there exists (x, y) S T, such that x i β, y j β; 3. F 1 and F 2 are fu dimensiona, i.e., there exists at east a pair (x, y) S T such that x i 0 and y j 0, i = 1,..., m, j = 1,..., n; 4. R 1, R 2 and β are known expicity. These assumptions, athough seems restrictive, they do not excude some common cases. E.g. F 1 = {e}. In this case, S is exacty the binary set S = {1, 1} m ; F 1 = {y R m + m i=1 y i = 1}. In this case, S is exacty the unit sphere which reduces to that of [3]. It is easy to see these two case satisfy Assumption 4.1. The probem can be written as the foowing triinear form (T LP ) maximize Bxyu = B(u) xy = (x y) Bu (4.9) subject to x 2 F 1 R m, y 2 F 2 R n, u 2 = 1, u R Even for a fixed u R, this probem is sti NP-hard [19]. As the previous section, the WMEM associated with this probem can not be soved in poynomia-time. For this reason, we sti consider its semidefinite reaxation. To begin with, et B(u) = [ 0 B(u) 0 0 ] R (m+n) (m+n) and z = [x, y ] R m+n, F := F 1 F 2. (T LP ) can be rewritten as Its SDR is maximize B(u) zz (4.10) subject to z 2 F, u 2 = 1 (RT LP ) maximize B(u) Z subject to diag(z) F, Z 0, u 2 = 1 where diag(z) R m+n denotes the diagona entries of Z. Note that for any fixed u, (RT LP ) is a we-formuated convex optimization probem. Obviousy we have v(rt LP ) v(t LP ). Moreover, it hods that Proposition 4.5 v(rt LP ) v(t LP ) β 2 max ijk b ijk > 0, where β is given in Assumption 4.1. Proof. Without oss of generaity suppose b 111 = arg max ijk b ijk 0. Let u = [sign(b 111 ), 0,..., 0] R. 13

14 By Assumption 4.1 (2), there exists a pair (ˆx, ŷ) S T such that ˆx 1, ŷ 1 β. Let x = ˆx. [1, σ x ] and y = ŷ. [1, σ y ] where (σ x, σ y ) { 1, 1} (m 1) { 1, 1} (n 1). Traversing a the eements of { 1, 1} (m 1) { 1, 1} (n 1), we have (σ x,σ y) { 1,1} (m 1) { 1,1} (n 1) [1, σ x ] [1, σ y ] = [2 m+n 2, 0,..., 0] R m n, and so Bxyu = 2 m+n 2 b 111 x 1 y 1 2 m+n 2 β 2 b 111, x,y which means that there is at east a pair (x, y) S T satisfying Bxyu β 2 b 111 = β 2 max i,j,k b ijk. It foows that v(rt LP ) v(t LP ) β 2 max i,j,k b ijk > 0. Now, Denote g(u) := maximize B(u) Z (4.11) subject to diag(z) F, Z 0 To ensure the WMEM of the associated convex body can be soved in poynomia-time, we sha make an assumption Assumption 4.2 For any fixed u, (4.11) can be soved in poynomia-time by the interior-point method or the eipsoid method. Note that in many cases, this semidefinite probem can be soved by the interior-method or the eipsoid method in poynomia-time. We prove the foowing, which shows that (4.11) satisfies Condition.1 (1) Proposition 4.6 g(u) defines a norm on R. Proof. As that of Proposition 3.2, it is straightforward to show that g(u) 0 for any u R and g(α 1 u 1 + α 2 u 2 ) α 1 g(u 1 ) + α 2 g(u 2 ) for any u i R and α i 0, i = 1, 2. The ony nontrivia task is to show that g(u) = 0 ony if u = 0. Denote L = {u g(u) = 0}. As the previous section, we need to show that L = {0}. The equation g(u) = 0 aso means that for such u and for any (x, y) S T, Bxyu = 0. Under Assumption 4.1 (3), there exists a pair (ˆx, ŷ) such that a the components are not zero. Without oss of generaity suppose ˆx > 0, ŷ > 0. This impies that for a x = ˆx. σ x and y = ŷ. σ y where (σ x, σ y ) { 1, 1} m { 1, 1} n, the components of a these vectors are not zero. Since there are 2 m 1 ineary independent vectors in { 1, 1} m and 2 n 1 ineary independent vectors in { 1, 1} n, we have totay 2 m+n 2 ineary independent vectors of x y R mn. Reca that (x y) Bu 0 for a these x y, we have Bu 0, which means that L is the nuspace of matrix B. As that of Proposition 3.2, we can without oss of generaity suppose B has fu coumn rank, which deduces u = 0. Thus g(u) defines a norm on R, as needed. Denote the norm by u B. Consider the unit ba of the norm u B given by B B = {u R u B 1}, 14

15 We wi prove that B B is a we- which is aso 0-centered, centray symmetric and convex. bounded Proposition 4.7 There exist rationa numbers 0 < C 1 C 2 <, whose encoding engths are poynomiay bounded by the input size of the probem, such that B 2(C 1 ) B B B 2(C 2 ). Proof. Foowing the previous section, et e k be the k-th basis vector in R, k = 1,..., r. Then we have 0 e k B = B(e k ) ij Zij = B ijk Zij 1 i m,m+1 j m+n 1 i m,m+1 j m+n R 1 R 2 1 i m,1 j n B ijk Zii Z jj B ijk, 1 i m,m+1 j m+n where Z is a maximizer of (4.11) with respect to u = e k, and the ast inequaity comes from Assumption 4.1 (1). Denote c k = R 1 R 2 i,j B ijk, c max = max k=1 c 1 k and C 1 = (, we c max) have B2(C 1 ) = {u R 1 u 2 ( c max ) } {u R u 1 1 } B B. c max Thus we have proved that B B contains a ba with radius C 1, whose encoding ength is poynomia bounded by the input size of (T LP ). On the other hand, for any u R \ {0}, we have Bu R mn \ {0} by Proposition 4.6, which means that B(u) R m n \{0}. Without oss of generaity suppose B(u) 11 = arg max B(u) ij = arg max k=1 B ijku k. By Assumption 4.1 (2), there exists a pair (ˆx, ŷ) S T such that ˆx 1, ŷ 1 β > 0. Let x = ˆx. [1, σ x ] and y = ŷ. [1, σ y ] where (σ x, σ y ) { 1, 1} (m 1) { 1, 1} (n 1). As the proof of Proposition 4.5, there exists at east a pair (x, y) S T such that B(u) xy = (x y) Bu β 2 max i,j B(u) ij = β 2 Bu. Thus we have u B B(u) xy β 2 Bu β 2 λmin (B B) u 2. mn Since B R mn has fu coumn rank by Proposition 4.6 and mn, we have λ min (B B) > 0. Hence B B B2(C 2 ), where C 2 = mn β 2. Thus we have proved B λ min(b B is contained in a B) ba with radius C 2, whose encoding ength is aso bounded by the input size of (T LP ). We caim that Condition.1 (3) hods Proposition 4.8 The weak membership probem associated with BB can be soved in deterministicpoynomia-time. Proof. We ony need to prove that the weak membership probem associated with B B can be soved in deterministic poynomia-time. By Assumption 4.2, (4.11) can be sove in poynomiatime. For a given u Q \ {0} and a rationa number ɛ > 0 (without oss of generaity suppose ɛ < u 2 ), if (4.11) can be soved by the eipsoid method [25], then we can in deterministic poynomia-time compute a matrix Z R (m+n) (m+n), such that ɛ d(z, K) max{2 B(u) F, 1} u 2 15

16 and ω(u) B(u) Z u B ɛ u B ɛ, max{2 B(u) F, 1} u 2 u 2 where K := {Z R (m+n) (m+n) diag(z) F, Z 0} is the feasibe soution set of (4.11), where d(z, K) denotes the Eucidean distance between Z and K. If ω(u) 1, it foows from [3] that u B B (ɛ); if ω(u) > 1, by assumption, there exists a Z K such that Z Z F ɛ max{2 B(u) F,1} u 2. Then and so u + ɛ which impies u B B ( ɛ). u B B(u) Z = ω(u) + B(u) (Z Z) ɛ 1 B(u) F max{2 B(u) F, 1} u 2 1 ɛ 2 u 2 u u 2 B = (1 + ɛ u 2 ) u B (1 + ɛ u 2 )(1 ɛ 2 u 2 ) > 1, If (4.11) can be sove by the interior-point method in deterministic poynomia-time, then it foows from Proposition 4.8 that the concusion hods. This competes the proof. Now we have proved that Condition.1 hods for this case. Hence we can in poynomia-time find a soution of (RT LP ) with ratio Ω( ). Let ɛ be chosen such that ɛ < γ β 2 max 4 b ijk < γ v(rt LP ). ijk 4 Then we can in deterministic poynomia-time compute a vector u BB (ɛ) such that u 2 γ v(rt LP ) ɛ. Let û = u u 2, and Z be a maximizer of (4.11) with respect to û, we have that (û, Z ) is feasibe for (RT LP ) and B(û) Z = max û v v BB u 2 ɛ γ v(rt LP ). 2 Thus we have found a feasibe soution (û, Z ) of (RT LP ) with approximation ratio Ω( ) in poynomia-time. Then we must extract a z R m+n from Z such that z = [x, y ] is a feasibe soution of (4.9). Note that when u is given and F = {e}, (4.10) reduces to a quadratic integer programming, where Aon and Naor [14] provided a randomized poynomiatime agorithm to get a binary vector from Z, with approximation ratio ρ = 2 n(1+ 2) π > Actuay, their approach can be generaized to fit our case. Motivated by Zhang [19], et d = diag(z ) R m+n and X = (D ) + Z (D ) + + D, (4.12) 16

17 where d is the root of diagona entries of Z, D is the diagona matrix representation of d, (D ) + stands for the pseudo-inverse of D, i.e., it is a diagona and { (D ) + ii = (d i ) 1, d i > 0, 0, d i = 0, and D denotes a binary diagona matrix where D ii = 1 if Z ii = 0 and D ii = 0 otherwise. Then we have (d ) 2 F, diag(x ) = e, X 0 and Z = D X D where e denotes the a-one vector. Thus we can denote X = V V where V = [v 1,..., v m+n ] where v i are a unit vector, i = 1,..., m + n. To continue, we reca a emma given in [14] Lemma 4.4 (Lemma 4.2 of [14]) For any set {u i 1 i n} {v j 1 j m} of unit vectors in a Hibert space H, and for c = sinh 1 (1) = n(1 + 2), there is a set {u i 1 i n} {v j 1 j m} of unit vectors in a Hibert space H, such that if z is chosen randomy and uniformy in the unit sphere of H then for a 1 i n, 1 j m. π 2 E([sign(z u i)] [sign(z v j)]) = cu i v j (4.13) Based on this emma, given X R (m+n) (m+n) such that X = V V, we can find v 1,..., v m+n in another space in poynomia-time satisfying Lemma 4.4 as that of [14]. Let ξ be chosen randomy and uniformy in the unit sphere of the same space as v i and et x R m+n where x i = sign(ξ v i ), we have for i j E(x ix j) = 2c π v i v j = 2c π X ij. By noticing that the diagona entries of B(u) are zero, we get E(B(u) D x (x ) D ) = 2 n(1 + 2) π B(u) D X D = 2 n(1 + 2) B(u) Z. π Let [x, y ] = z = D x. Then (x, y) is a feasibe soution of (T LP ). Summarizing the whoe anaysis of this section, we get the foowing concusion Theorem 4.4 Suppose Assumption 4.1 and 4.2 hod. Then there is a deterministic poynomiatime agorithm that, given an instance of (RT LP ), returns a feasibe soution (û, Z ), such that B(û) Z Ω( )v(rt LP ); (4.14) there is a randomized poynomia-time agorithm that, given the feasibe soution (û, Z ) of (RT LP ) produced by the agorithm above, returns a feasibe soution (x, y, û) of (T LP ), such that E(Bxyû) = ρb(û) Z ρω( )v(t LP ), where ρ = 2 n(1+ 2) π > To sum up, we can in poynomia-time find a feasibe soution of (T LP ) with approximation ratio Ω( ). 17

18 We shoud point out that, if both F 1 and F 2 are the singeton point set {e}, or one of them is {e} whie another is the standard simpex, then (T LP ) reduces to the triinear mix-variabes forms studied by He et a [17]. If min{m, n}, then we improve their ratio from Ω( 1 ) to Ω( ). 5 The biquadratic case In this section, we mainy discuss a specia cass of biquadratic form (BQP ) maximize f(x, y) := Axxyy subject to x S, y T, where Axxyy = m i,j=1 n k,=1 A ijkx i x j y k y, A R m m n n is a 4-th order partiay symmetric tensor, i.e. a ijk = a jik = a ijk. The biquadratic optimization has many appications in quantum physics, see e.g., [2, 8, 7, 11, 13]. Some specia forms of (5.15) are studied in [5, 20, 6]. Now we expore the structure of A. Note that A can be unfoded into a matrix A R mn mn, with A (i 1)n+k,(j 1)n+ = A ijk. By such unfoding, A is a symmetric matrix. Denote its eigenvaue decomposition by A = r s=1 λ sη s (η s ) where r is the rank of A. In what foows, we make the foowing assumption Assumption 5.3 The unfoded matrix A is positive semidefinite. Note that a partiay symmetric tensor A is positive semidefinite if its unfoded matrix is positive semidefinite, but not vice versa, see, e.g., [21]. If the unfoded matrix of a partiay symmetric tensor A is positive semidefinite, then we ca A strongy positive semidefinite. Under this assumption, for s = 1,..., r, et b s = λ s η s. Then f(x, y) = Axxyy = r s=1 ((bs ) (x y)) 2. Denote matrix B = [b 1, b 2,..., b r ] R mn r. It is cear that the foowing proposition hods for B Proposition 5.9 B has fu coumn rank, i.e. rank(b) = r. From inear agebra, we immediatey have another concusion Proposition 5.10 For any u R r, Bu = 0 if and ony if u = 0. Denote 3-th order tensor B R m n r where B ijk = B (i 1)n+j,k. Then B is the foded tensor of matrix B. Now consider the foowing triinear form (T LP ) maximize g(x, y, u) := Bxyu = (x y) Bu subject to x S, y T, u 2 = 1 Actuay (T LP ) is equivaent to (BQP ). we state the proposition here, which is aso mentioned in He et a [15] 18

19 Proposition 5.11 v(bqp ) = v 2 (T LP ). Moreover, for any feasibe soution (x, y, u) of (T LP ), it hods that Axxyy (Bxyu) 2. Proof. Suppose (x, y, u) is a feasibe soution of (T LP ). For such given (x, y), u = Bxy/ Bxy 2 maximizes g(x, y, u) where (Bxy) R r with (Bxy) k = i,j=1 B ijkx i y j. Thus Bxy 2 = (Bxy) u = g(x, y, u) g(x, y, u). On the other hand, by the construction of B, we have Bxy 2 2 = r ((b s ) (x y)) 2 = Axxyy. s=1 Thus for any feasibe soution (x, y, u) of (T LP ), (x, y) is a feasibe soution pair of (BQP ) satisfying Axxyy (Bxyu) 2. Next suppose (x, y ) is an optima soution of (BQP ). By etting u = Bx y / Bx y 2, we have v(bqp ) = f(x, y ) = g 2 (x, y, u) v 2 (T LP ). Combining above, we have the concusion hods. By the anaysis above, when A is strongy positive semidefinite, we can reate the biquadratic optimization probem (BQP ) with the triinear optimization probem mainy studied in this paper. It is cear that the foowing theorem hods Theorem 5.5 Suppose A is strongy positive semidefinite. 1. If S = {x x A i x 1, i = 0,..., p, } and T = {y y B j y 1, j = 1,..., q, }, then there is a randomized poynomia-time agorithm that returns a feasibe soution (x, y) of (BQP ) og r with approximation ratio Ω( og 2 [175(p+q)u] r ); 2. if S = {x x 2 F 1 } and T = {y y 2 F 2 }, then there is a randomized poynomiatime agorithm that returns a feasibe soution (x, y) of (BQP ) with approximation ratio Ω( og r r ), where r is the rank of the unfoded matrix A R mn mn. Note that for case 1, Zhang et a [20] got an approximation ratio under some different assumptions ( ) 1 Ω (p + q) 2 (1 + 2 og(100p 2. )) og(100q) max{m, n}(max{m, n} 1) Thus in our case, we get a better approximation ratio than that. References [1] A. Brieden, P. Gritzmann, R. Kannan, V. Kee. L. Lovász and M. Simonovits, Deterministic and randomized poynomia-time approximation of radii, Mathematika 48 (2001)

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