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1 Operaons Research Leers 39 (2011) Conens lss avalable a ScVerse ScenceDrec Operaons Research Leers journal homepage: Rank of Handelman herarchy for Max-Cu Myoung-Ju Park, Sung-Pl Hong Deparmen of Indusral Engneerng, Seoul Naonal Unversy, San 56-1 Shlm-Dong, Kwanahk Gu, Seoul, , Republc of Korea a r c l e n f o a b s r a c Arcle hsory: Receved 10 February 2011 Acceped 28 June 2011 Avalable onlne 28 July 2011 Keywords: Polynomal opmzaon Rank of herarchcal relaxaon Handelman herarchy Max-Cu RLT We consder a herarchcal relaxaon, called Handelman herarchy, for a class of polynomal opmzaon problems. We prove ha he rank of Handelman herarchy, f appled o a sandard quadrac formulaon of Max-Cu, s exacly he same as he number of nodes of he underlyng graph. Also we gve an error bound for Handelman herarchy, n erms of s level, appled o he Max-Cu formulaon Elsever B.V. All rghs reserved. 1. Inroducon Consder he followng problem where p(x), g 1 (x),..., g m (x) R[x] are real polynomals n n varables. p mn mn p(x) s.. x K : x R n g 1 (x) 0,..., g m (x) 0. The polynomal opmzaon problem (1) s NP-hard. Indeed, f p(x) s quadrac and g (x) s are lnear, problem (1) conans he NP-hard Max-Cu problem; see relaon (8) n Secon 3. Problem (1) can be rewren as sup ρ s.. p(x) ρ 0 over K,.e., p mn sup ρ s.. p(x) ρ P (K), (2) where P (K) denoes he se of real polynomals ha are nonnegave on K. However, he nracably of (1) mples ha he problem of esng membershp of an arbrary polynomal n P (K) s already an nracable problem. Several relaxaon deas for (2) have been proposed n he leraure. They replace P (K) wh s subses whose membershp problems are racable. They rely on varous characerzaons, known as Posvsellensaz, of polynomals ha are posve over he semalgebrac se K [8,15,14]. In hs paper, we consder a class of polynomal opmzaon problems n whch K s resrced o be a compac and sold polyhedral se. In hs case, as observed by [7], a Posvsellensaz by Handelman [8] provdes a herarchcal relaxaon of (1) ha converges o an exac opmum. In Secon 2, we revew Handelman Correspondng auhor. E-mal address: pmj0684@snu.ac.kr (M.-J. Park). (1) herarchy and s polynomaly of he relaxaon of a fxed level. Secon 3 s devoed o esablshng he rank of Handelman herarchy for Max-Cu. In Secon 4, we derve an upper bound on he error of Handelman herarchy wh respec o he exac opmum value of Max-Cu. Secon 5 observes he dualy beween Handelman herarchy and RLT and s mplcaons on he rank of RLT wh respec o Max-Cu. 2. Handelman herarchy Our noaon s manly adoped from [12]. Gven a se of polynomals g 1 (x),..., g m (x) R[x] defnng K and β N m, we denoe g β : g β 1 1 gβ m m. For noaonal convenence, for any g, we also denoe g 0 : 1. We assume he se N m s graded lexcographcally ordered and he degree of α N m denoed by α s he sum of enres,.e., α m α 1. Le N m be he se of m- dmensonal vecors whose degree s a mos. I s well known ha N m m+. We use R N m m+ o denoe he se of -dmensonal real vecors y : (y α : α N m ). Gven a polynomal f (x) R[x], deg(f (x)) denoes he degree of f (x). Also we adop a lle abusve bu convenen noaon ha f R Nn denoes he vecor whose α-h enry, f α, s he coeffcen of x α n he polynomal f (x). We wll use vec(f (x)) nsead of f especally when f (x) s gven n a specfc form. Gven a se of polynomals S, vec(s) : {vec(f (x)) f (x) S}. Consder a se of polynomals H(g) : c β g β c β R +, β N m known as he preprme generaed by g 1 (x),..., g m (x). I s easy o see ha each polynomal n H(g) s nonnegave on K and hence /$ see fron maer 2011 Elsever B.V. All rghs reserved. do: /j.orl

2 324 M.-J. Park, S.-P. Hong / Operaons Research Leers 39 (2011) H(g) P (K). Defne sup ρ s.. p(x) ρ H(g). (3) Then (3) s a relaxaon of (2) n he sense ha p mn. However, he membershp of an arbrary real polynomal o H(g) s no ye an easy problem. The Handelman herarchy s obaned by furher subsung H (g) for H(g) [7]: sup ρ s.. p(x) ρ H (g), (4) where H (g) : β N m c β g β deg(g β ), c β R +. (5) I s easy o see ha, for every deg(p(x)), we have p mn. Also, snce H (g) H +1 (g), we have +1. I s well-known from he dualy heorem for lnear programmng ha f p(x) s lnear, K s a polyhedron, and p(x) aans an opmal soluon over K, hen p(x) p mn s a conc combnaon of g (x) s. Thus, n hs case, 1 p mn : 1 sup ρ s.. p(x) ρ H 1 (g). Of course, hs s no rue n general as H(g) s a proper subse of P (K). However, when K s a compac and sold polyhedron, s guaraneed o converge o p mn. Theorem 2.1 (Handelman s Posvsellensaz [8]). Suppose K s a compac polyhedral se whch s sold,.e., has an neror pon. If p(x) s posve on K, hen p(x) H(g). Now we consder he compuaonal complexy of problem (4) for a fxed. Suppose deg(g β ). Then we can wre g β (x) α N n a α,β x α. Defne he marx A whose (α, β)-h elemen s a α,β for α N n and β N m wh deg(g β ). Le z sand for he vecor of coeffcens c β defned n (5) and e 0 R Nn be he vecor whose α-h enry s 1 f α 0, and 0, oherwse. Then problem (4) s equvalen o he followng LP problem whch compues z ha maxmzes ρ: sup ρ s.. ρe 0 + A z p (6) z 0. The number of varables s less m+ han or equal o and he n+ number of consrans s equal o. Thus for a fxed, p han can be obaned by an LP whose dmenson s polynomal n m and n. (However, he dmenson ncreases exponenally n.) We call he rank of he Handelman herarchy wh respec o a problem he mnmum such ha p mn for every nsance of he problem. There has been a sgnfcan leraure on he rank of varous herarches for combnaoral opmzaon problems [13,2,18,6,1,4,5,9]. In he followng secon, we show ha he rank of he Handelman herarchy, f appled o a sandard quadrac formulaon of Max-Cu, s exacly he number of nodes of he underlyng graph. 3. Applcaon o Max-Cu The maxmum cu problem, or Max-Cu, s he problem of fndng a paron (V 1 ; V 2 ) of he nodes V {1,..., n} of a gven graph G (V, E) ha maxmzes he number of edges beween V 1 and V 2. Max-Cu s ofen formulaed as a bnary quadrac program. Le A G {0, 1} n n be he adjacency marx of G: s (, j)-h enry s 1 f j E, and 0, oherwse. Also, defne x {0, 1} n as follows: 1, f V1, x 0, f V 2. Then s easy o see ha nodes and j, f hey belong o dsnc ses and (, j) E, conrbue o he objecve a un whch can be wren as (A G ) j x (1 x j ) + (A G ) j x j (1 x ). Thus he followng 0 1 quadrac program formulaes Max-Cu: max x {0,1} xt A G (e x), (7) n where e R n s he vecor of all ones. Now, n (7), we relax he bnary resrcon x {0, 1} n o membershp n he hypercube leadng o max x [0,1] xt A G (e x). (8) n Snce every dagonal elemen of A G s 0, he objecve x T A G (e x) s lnear wh respec o each varable x. Ths mples ha here s an opmal soluon of (8) n whch each x s eher 0 or 1. Thus (8) s an exac formulaon of Max-Cu. Snce he feasble soluon se s a hypercube, Handelman herarchy appled o (8) wll converge o he opmal value. Now we show ha s rank s exacly n. To do so, we use he mnmzaon verson of (8). p mn mn x T A G (e x) s.. x Q : {x R n x 0, 1 x 0, 1,..., n}. Le g x, and g n+ 1 x for 1,..., n. An upper bound on he rank can be derved from he followng heorem. Theorem 3.1 (De Klerk and Lauren [7]). Le A R n n, b R n, and p(x) x T Ax + b T x. Consder he problem of mnmzng p(x) over he hypercube Q. Then for any neger k 1, here s max(kn, 2) such ha p(x) p mn + 1 n max{a, 0} H (g). k 1 In oher words, p mn 1 k n 1 max{a, 0} when max (kn, 2) for any neger k 1. Corollary 3.2. Suppose n 2. Then he rank of Handelman herarchy, f appled o (9), s no greaer han n. Proof. The corollary s mmedae from Theorem 3.1 and he fac ha every dagonal elemen of A G s 0. Now we esablsh he lower bound, namely, ha he rank s a leas n. To do so, he followng lemma s useful. Lemma 3.3. If here s an nsance of a problem such ha p(x) p mn H 1 (g) n (4), hen he rank of he Handelman herarchy s no less han. Proof. Suppose here s an nsance such ha p(x) p mn H 1 (g). Snce vec(h 1 (g)) s a closed convex se, here s a hyperplane srcly separang vec(p(x) p mn ) from vec(h 1 (g)). Hence, here s an ϵ > 0 such ha p(x) p mn + ϵ H 1 (g). Also, f p(x) σ H 1 (g), hen p(x) ρ H 1 (g) for every ρ > σ. Therefore, we have 1 pmn ϵ < p mn. Ths mples ha s less han or equal o he rank. From Lemma 3.3, suffces o consruc an nsance for whch p(x)p mn H n1 (g). Consder a Max-Cu whose underlyng graph (9)

3 s he complee graph K n wh n 2k + 1 for some neger k. Is mn-verson opmal value s p mn (k 2 + k). Thus, we need o show ha p n (x) : x T A Kn (e x) + k(k + 1) n 2k x x x j + k(k + 1) H n1 (g). (10) 1 j E Theorem 3.4. p n (x) H n1 (g). Proof. Condon (10) s equvalen o he exsence of a hyperplane a T w 0 srcly separang p n from vec(h n1 (g)),.e. a T p n < 0 and a T g β 0 for every β N 2n n1. Denoe by Υ R Nn n1 he vecor whose α-h enry s 1 f α, and 0, oherwse. We wll show f we se 2k a n Υ 0 + k 2k 1 k 1 2k + + k Υ 1 Υ + + k 0 M.-J. Park, S.-P. Hong / Operaons Research Leers 39 (2011) Υ k, hen a T n w 0 s a hyperplane srcly separang p n from vec (H n1 (g)). Frs, we wll show a T n p n < 0. Suppose k 1. Then, snce a 3 2Υ 0 + Υ 1, we ge a T 3 p < 0. If k 2, we have 2k 2k 1 a T n p n k(k + 1) (2k + 1) 2k k k 1 2k 2 + (2k + 1) 2k k 2 2k(2k 1) 2k 2 k(k + 1) k(k 1) k 2 2k 1 2k 2 (2k + 1) 2k k 1 k 2 2k 2 + (2k + 1) 2k k 2 2k 2k 2 [(2k 1)(k + 1) (2k 1)(2k + 1) k 1 k 2 + (k 1)(2k + 1)] 2k 2k 2 < 0. k 1 k 2 I remans o show ha a T n gβ 0 for every β N 2n n1. Wre g β (x) x α (e x) γ. Noe ha for any real polynomal h(x), h s he sum of coeffcens of monomals n h(x) whose degree s. Hence h vec(h(x 1, x 1,..., x 1 )). Then, for every, g β s 0 f < α. If, on he oher hand, α, we have g β vec(x α (e x) γ ) vec(x α 1 (1 x 1) γ ) α vec((1 x 1) γ ) (1) α γ α. (11) From (11), for each β, g β s deermned only by α and γ. Relyng on he observaon, we can show a T n gβ 0 when β n1. Suppose k+1 α n1. Then g β 0 for 0,..., k and, hence, a T n gβ k k+ 0 k gβ 0. Suppose, on he oher hand, 0 α k and le j k α. Then we ge γ k + j and hence a T n gβ j k + k gβ 0 j k + (1) k α 0 0 j k + k + j (1) j j γ k α j j (k + )! (k + j)! (1)!k! (j )!(k + )! 0 j j (k + j)! j! (1) k!j!!(j )! 0 k + j j j (1) j j j 0 k + j (1 1) j 0. j Fnally, we prove f a T n gβ 0 for every β wh β l, hen also holds for every β wh β l 1. To do so, suffces o esablsh he followng for 0,..., k: vec(x α 1 (1 x 1) γ ) vec(x α 1 (1 x 1) γ +1 ) + vec(x α +1 1 (1 x 1 ) γ ). (12) Bu, f < α, (12) holds snce boh sdes are 0. Suppose α. a+1 Relyng on he deny a b b1 a b for every nonnegave negers a, b, we have γ + 1 Rgh-hand-sde of (12) (1) α α γ + (1) α 1 α 1 γ (1) α α Lef-hand-sde of (12). Thus we have compleed he proof. Proposon 3.5. The rank of Handelman herarchy, appled o formulaon (8) of Max-Cu, s equal o he number of nodes of he underlyng graph. We can observe ha when G s bpare, he rank s 2. Proposon E. Proof. Defne p(x) : x T A G (e x) δ x 2 j E x x j, where δ s he number of edges ncden o he node n G. Then E p(x) E δ x x x j j E [(1 x )(1 x j ) + x x j ] H 2 (g). j E Hence 2 E. Now, we wll show ha f λ p(x) H 2 (g), hen λ E. Suppose ha λ p(x) H 2 (g), ha s, λ δ x j E x x j has a decomposon of he form:

4 326 M.-J. Park, S.-P. Hong / Operaons Research Leers 39 (2011) a <j a x + d (1 x ) 2 + <j b (1 x ) + e j x x j f j (1 x )(1 x j ) +,j c x 2 g j x (1 x j ) for some nonnegave scalars a, b, c, d, e j, f j, g j. Lookng a he consan coeffcen we ge λ a 0 + b + d + <j Lookng a he coeffcen of x we ge δ a b 2d j j whch mples b d + j j f j + j f j δ + a + j f j. (13) g j, Summng up over n (14), we oban b d δ + <j f j a +,j g j. (14) g j 2 E, (15) where we use he facs ha δ 2 E and a, g j 0. Usng (13), we oban: 2λ 2a 0 + b + b d f j 2 E. <j 0 2 E by (15) Ths concludes he proof. Snce OPT E when G s bpare, he above proposon shows he rank s no greaer han 2. Bu, he mnmum value of for whch Handelman herarchy s defned s he degree of he objecve funcon. Therefore, follows ha he rank s 2 when G s bpare. 4. Error of Handelman herarchy for Max-Cu The error bound for Handelman herarchy from Theorem 3.1 only apples o he case when s a mulple of n. As he rank of Handelman herarchy s n for (8), we need o explore an error bound for he case when < n. The dea s smlar o he one used n [7]. Frs, we approxmae p(x) wh a lnear combnaon of polynomals from H (g). Second, we show ha p(x) p mn + some consan (dependng on ) ϵ() s a polynomal n H (g) usng he relaon beween p(x) and he approxmaon. Then follows ha ϵ() s an upper bound on he error p mn. In hs paper, o derve an error bound for < n from H (g), we wll choose a se of polynomals ha are dfferen from hose used n [7]. Gven S V s.. S and α {0, 1} S, defne P S,α (x) x α (1 x ) 1α. S Noce ha P S,α (x) H (g). We approxmae f (x), a connuous funcon on [0, 1] n, as a lnear combnaon of P S,α (x) s: B (f (x)) f ( α)p S,α (x), (16) S V, S α {0,1} S where α {0, 1} n s a vecor whose -h enry s α f S, and 0, oherwse. Then, for each, s easy o check ha B s a lnear operaor: B (cf (x) + dg(x)) cb (f (x)) + db (g(x)). Applyng approxmaon (16) o he Max-Cu objecve, f (x) δ V x + 2 j E x x j, we ge B (f (x)) δ B (x ) B (x x j ). (17) V j E Noe ha B (x ) S V, S α {0,1} S α P S,α (x) S V, S, S α {0,1} S,α 1 S V, S, S x n 1 x. 1 Smlarly, we ge B (x x j ) S V, S α {0,1} S P S,α (x) α α j P S,α (x) S V, S,,j S α {0,1} S,α α j 1 S V, S,,j S x x j n 2 x x j. 2 P S,α (x) Subsung hese wo for hose of (17), we have B (f (x)) n 1 n 2 δ x x x j. (18) 1 2 V j E n n2 Add 2 j E x 1 2 x j o boh sdes of (18) and hen dvde hem by o ge n1 1 n 1 B (f (x))/ 1 j E n n 1 x x j f (x). (19) Agan, add n OPT o boh sdes of (19) o ge f (x) + n n n 1 n 1 OPT OPT/ + B (f (x))/ 1 1 n n 1 x x j j E n 1 P S,α (x)opt/ 1 S V, S α {0,1} S + n 1 P S,α (x)f ( α)/ 1 S V, S α {0,1} S n n 1 x x j j E n 1 (OPT + f ( α))p S,α (x)/ 1 S V, S α {0,1} S n n 1 x x j. j E The second equaly follows from he fac ha S V, S α {0,1} S P S,α (x) n snce α {0,1} S P S,α(x) 1. As P S,α (x), x x j H (g) and he coeffcens are nonnegave, snce f ( α) (T, T) where

5 M.-J. Park, S.-P. Hong / Operaons Research Leers 39 (2011) T { V α 1}, we oban ha f (x) + n OPT H (g). Thus we ge he error bound n OPT, for all n. (20) Noce ha he error bound (20) s conssen wh he rank n proved n Secon 3. Ths bound, however, s meanngful only when n/2. Indeed, he Handelman bound s equal o E by Proposon 3.6 whle OPT of Max-Cu s a leas E /2. Hence he Handelman herarchy always provdes a bound a mos 2OPT. 5. Handelman herarchy and RLT In hs secon, we show ha he Handelman herarchy s dual o he Reformulaon Lnearzaon Technque(RLT). RLT s orgnally proposed by Sheral and Adams [16] for he compuaon of he convex hull P I of he 0 1 lace pons from a polyhedron P {x {0, 1} n g l (x) 0, l 1,..., m}. I compues a se of vald nequales of P I by usng he bound nequales, 0 x 1. Gven a par of subses I, J {1, 2,..., n}, defne f (I, J)(x) x (1 x j ). I j J Mulply each consran by f (I, J)(x) o ge he followng se of vald nequales: G I,J : {g l (x) f (I, J)(x) 0 l 1,..., m}. For each fxed, RLT consrucs he vald nequales G I,J for all pars I and J such ha I J and I J. Then, lnearze each vald nequaly frs by replacng x 2 wh x, and hen by subsung y α for each monomal x α. Denoe by P he projecon of he nersecon of he se {y : y 1} and he polyhedron defned by he lnearzed nequales o he orgnal space. Then, can be shown ha P 1 P 2 P n1 P n P I. RLT uses he relaxaon of he problem n whch P I s relaxed o P for a fxed n. Charkar e al. [3] explored he compuaonal effcency of RLT, when appled o he IP-formulaon of Max-Cu based on he rangle nequales among edge varables. They showed ha for every ϵ > 0 here exss γ > 0 such ha he negraly gap of he above relaxaon for O(n γ ) s a leas 2 ϵ. Sheral and Tuncblek [17] exended RLT o he polynomal opmzaon problems n an analogous manner, usng he varable bounds 0 l x u <. Namely, consruc f (I, J)(x) (x l ) (u j x j ), I, J s.. I J. I j J Then lnearze each vald nequaly by subsung y α for each monomal x α. Now we consder he followng varaon RLT of RLT. Namely, for our semalgebrac se K {x R n g (x) 0, 1,..., m}, we use he produc of he powers of g s. For a fxed we consder he vald nequales n g β (x) : g (x) β 0, β N m s.. deg(g β ) 1 and lnearze hem by subsung y α for x α. Then RLT for (1) becomes he followng LP mn p T y s.. y 0 1 A T y 0, (21) where A s he same marx as n (6). I s easy o see ha (21) s dual o (6). Thus for each, Handelman herarchy s dual o RLT. Remark 5.1. Afer he submsson of he frs verson of hs paper, we found ha he dualy beween wo herarches had already been observed n [10]. From he dualy beween wo herarches, we have he followng corollary o Proposon 3.5. Corollary 5.2. The rank of RLT, appled o he quadrac formulaon (8) of Max-Cu s also equal o he number of nodes of he underlyng graph. Remark 5.3. The dualy beween he wo herarches mmedaely ransfers any error bound of one herarchy o he oher one for polynomal opmzaon on compac and sold polyhedra. See, e.g., Theorem 1.4 from [7]. Remark 5.4. The dualy also provdes a nonconsrucve proof of Theorem 3.4. For Max-Cu, he orgnal RLT by Sheral and Adams [16] s a leas as powerful as RLT. Indeed, he orgnal RLT explos he addonal condons x 2 x for 1,..., n ha are represened by he lnear consrans y α+e y α+2e for all α N n 2 and 1,..., n where e R n s he un vecor whose -h enry s 1. Besdes hese lnear consrans, RLT and he orgnal RLT are dencal. Hence he rank of he orgnal RLT s a lower bound on he rank of RLT. Lauren [11] showed ha he rank of he orgnal RLT for Max-Cu s n. Combned wh he dualy, hs mples ha he rank of Handelman herarchy for Max-Cu s a leas n. Acknowledgmens The auhors are graeful o wo referees for helpful commens n mprovng he presenaon of he paper. Especally, we are n deb o Monque Lauren for he proof of 2 E n Proposon 3.6 and Remark 5.4. Ths research was suppored by Basc Scence Research Program hrough he Naonal Research Foundaon of Korea (NRF) funded by he Mnsry of Educaon, Scence and Technology ( ). References [1] S. Arora, B. Bollobás, L. Lovász, Provng negraly gaps whou knowng he lnear program, n: FOCS 02: Proceedngs of he 43rd IEEE Symposum on Foundaons of Compuer Scence, 2002, pp [2] E. Balas, S. Cera, G. Cornuéjols, A lf-and-projec cung plane algorhm for mxed 0 1 programs, Mahemacal Programmng 58 (3) (1993) [3] M. Charkar, K. Makarychev, Y. Makarychev, Inegraly gaps for Sheral Adams relaxaons, n: STOC 09: Proceedngs of he 41s Annual ACM Symposum on Theory of Compung, ACM, New York, 2009, pp [4] K.K.H. Cheung, On Lovász Schrjver lf-and-projec procedures on he Danzg Fulkerson Johnson relaxaon of he TSP, SIAM Journal on Opmzaon 16 (2) (2005) [5] K.K.H. Cheung, Compuaon of he Lasserre ranks of some polyopes, Mahemacs of Operaons Research 32 (1) (2007) [6] W. Cook, S. Dash, On he marx-cu rank of polyhedra, Mahemacs of Operaons Research 26 (1) (2001) [7] E. De Klerk, M. Lauren, Error bounds for some semdefne programmng approaches o polynomal mnmzaon on he hypercube, SIAM Journal on Opmzaon 20 (6) (2010) [8] D. Handelman, Represenng polynomals by posve lnear funcons on compac convex polyhedra, Pacfc Journal of Mahemacs 132 (1) (1988) [9] S.-P. Hong, L. Tunçel, Unfcaon of lower-bound analyses of he lf-andprojec rank of combnaoral opmzaon polyhedra, Dscree Appled Mahemacs 156 (1) (2008) [10] J.B. Lasserre, Semdefne programmng vs. LP relaxaons for polynomal programmng, Mahemacs of Operaons Research 27 (2) (2002) [11] M. Lauren, A Comparson of he Sheral Admas, Lovász Schrjver and Lasserre relaxaons for 0 1 programmng, Mahemacs of Operaons Research 28 (3) (2003) [12] M. Lauren, Sums of squares, momen marces and opmzaon over polynomals, n: Emergng Applcaons of Algebrac Geomery, n: M. Punar, S. Sullvan (Eds.), IMA Volumes n Mahemacs and s Applcaons, vol. 149, Sprnger, 2009, pp [13] L. Lovász, Schrjver, Cones of marces and se-funcons and 0 1 opmzaon, SIAM Journal on Dscree Mahemacs 1 (2) (1991) [14] M. Punar, Posve polynomals on compac sem-algebrac ses, Indana Unversy Mahemacs Journal 42 (3) (1993)

6 328 M.-J. Park, S.-P. Hong / Operaons Research Leers 39 (2011) [15] K. Schmüdgen, The K -momen problem for compac sem-algebrac ses, Mahemasche Annalen 289 (2) (1991) [16] H.D. Sheral, W.P. Adams, A herarchy of relaxaons beween he connuous and convex hull represenaons for zero-one programmngs, SIAM Journal on Dscree Mahemacs 3 (3) (1990) [17] H.D. Sheral, C.H. Tuncblek, A global opmzaon algorhm for polynomal programmng problems usng a reformulaon lnearzaon echnque, Journal of Global Opmzaon 2 (1) (1992) [18] T. Sephen, L. Tunçel, On a represenaon of he machng polyope va semdefne lfngs, Mahemacs of Operaons Research 24 (1) (1999) 1 7.