Mixed Volume Computation, A Revisit

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1 Mixed Voume Computation, A Revisit Tsung-Lin Lee, Tien-Yien Li October 31, 2007 Abstract The superiority of the dynamic enumeration of a mixed ces suggested by T Mizutani et a for the mixed voume computation over the existing methods was reported in [15] In this artice, we embed the idea of dynamic enumeration into our origina agorithm for the mixed voume computation presented in [7], and a new agorithm is deveoped by empoying both prima and dua simpex agorithms Iustrated by the numerica resuts, our new agorithm improves the speed of the code in [15] by a substantia margin 1 Introduction For a system of poynomias P (x) = (p 1 (x),, p n (x)) with x = (x 1,, x n ), write p j (x) = X a S j c j,a x a, j = 1,, n, where a = (a 1,, a n ) N n, c j,a C = C\{0} and x a = x a 1 1 xan n Here S j, a finite subset of N n, is caed the support of p j (x), and S = (S 1,, S n ) is caed the support of P (x) Let Q j = conv(s j ) for j = 1,, n For positive numbers λ 1,, λ n, the n-dimensiona voume of the Minkovski sum λ 1 Q λ n Q n {λ 1 q λ n q n q j Q j, j = 1,, n} is a homogeneous poynomia of degree n in the variabes λ 1,, λ n The coefficient of λ 1 λ n in this poynomia is defined to be the mixed voume of S = (S 1,, S n ), denoted by M(S) In most occasions, we aso ca M(S) the mixed voume of P (x) By Bernshtein s theory [1], the mixed voume M(S) of S = (S 1,, S n ) of the poynomia system P (x) = (p 1 (x),, p n (x)) provides an upper bound for the number of isoated zeros Department of Mathematics, Michigan State University, East Lansing, MI 48824, emai: eetsung@msuedu Department of Mathematics, Michigan State University, East Lansing, MI 48824, emai: i@mathmsuedu Research supported in part by NSF under Grant DMS

2 in (C ) n, counting mutipicities And this bound can be reached if the coefficients of P (x) are generic, or the system is in genera position This root count in (C ) n has been extended to root count in C n [14, 18] They are, in genera, much sharper than the cassica Bézout number and its variants for sparse poynomia systems Based on this combinatoria root count, the poyhedra homotopies are estabished [9] to approximate a the isoated zeros of P (x) by the homotopy continuation method, yieding a drastic improvement over the cassica inear homotopies for sparse poynomia systems When the poyhedra homotopy is empoyed to find a isoated zeros of P (x), the process of ocating a the fine mixed ces in a fine mixed subdivision of the support S = (S 1,, S n ) during the mixed voume computation pays a cruciay important roe [10, 11, 12]: The mixed voume determines the number of soution paths needed to be traced and the fine mixed ces provide starting points of the soution paths Cacuating the fine mixed ces (and thus the mixed voume) of the support S consumes arge part of the computation and therefore dictates the efficiency of the method as we as the scope of its appications In 2005, a software package, MixedVo [8], produced by T Gao, TY Li and M Wu emerged which ed the existing codes for the mixed voume computation by a great margin However, soon after MixedVo was pubished, T Mizutani, A Takeda and M Kojima [15] deveoped a more advanced agorithm which overshadowed MixedVo in speed by a big amount A major ingredient for the efficiency of their agorithm is the nove idea of dynamic enumerations of mixed ces which heps to branch the parent node in the enumeration tree into its chid nodes where the size of feasibe chid nodes is as sma as possibe When ocating mixed ces, one must dea with a arge scae of inear programming (LP) probems by the simpex method Different from the prima simpex method used in MixedVo (as we as the previous works [6, 7, 13] which ed to the deveopment of MixedVo), the new agorithm DEMiCs-095 [15] by T Mizutani et a adopted the dua simpex method to the LP probems It was noted in their artice: In terms of the size of probems, the dua probems are superior to the prima ones and At east, the appication of the dua simpex method is popuar in the fied of optimization to effectivey dea with such a situation Nonetheess, we beieve the prima simpex method for this particuar set of LP probems sti has advantages of its own In particuar, the invovement of the eve sets of the inear functionas, or the hyperpanes, in the prima simpex method as in [6, 7, 8, 13] can hep to utiize the important informations that were generated in the process of pivotings more effectivey On the other hand, we found that the new idea of dynamic enumerations of mixed ces can be embedded in the agorithm in MixedVo with the spirit of the dua simpex method Thus, in this artice, we propose a new agorithm for finding mixed ces (with mixed voume as a by-product) where the prima simpex method and the dua simpex method are both in use We wi maintain the prima simpex method in the main course But when the dynamic enumerations is used to determine a proper order of the supports for each individua mixed ce as suggested in [15] which invoves checking the feasibiities of sets of inequaities, we aways ook into the boundedness of their dua probems The detais wi be eaborated in Section 3 Our method has been impemented successfuy and the preiminary numerica resuts isted in Section 4 are quite impressive Our agorithm eads the agorithm DEMiCs-095 by T Mizutani et a in speed on a the benchmark systems provided in [15] and the speedups range from 13 to 48 2

3 2 The main course Amost a of the existing codes for the mixed voume computation cacuate mixed voumes via cacuating the mixed ces: For a generic ifting ω = (ω 1,, ω n ) on S = (S 1,, S n ) with ω j : S j R for j = 1,, n, write bs j = {ba = (a, ω j (a)) a S j } A coection of pairs {a 1, a 1 } S 1,, {a n, a n} S n bα = (α, 1) R n+1 with α R n such that is caed a mixed ce if there exists ba j, bα = ba j, bα < ba, bα for a S j \ a j, a j, j = 1,, n It is known that the mixed voume of S = (S 1,, S n ) equas the sum of voumes of a such mixed ces That is, X M (S) = det a 1 a 1,, a n a n α On the other hand, those mixed ces pay a criticay important roe in finding isoated zeros of poynomia systems numericay by the poyhedra homotopies [10, 11, 12] To find a the mixed ces for a given generic ifting ω = (ω 1,, ω n ) on S = (S 1,, S n ), we first construct the Reation Tabe T (i, j) for 1 i j which dispay the reationships between eements of Si b and Sj b in the foowing sense: Given eements ba (i) Si b and ba (j) m Sj b where m when i = j, does there exist an bα = (α, 1) R n+1 such that D E ba (i), bα ba (i), bα for a ba (i) Si b (1) and ba (j) m, bα ba (j), bα for a ba (j) b Sj? Denote the entry on Tabe ht (i, j) at i the intersection h of i the row containing ba (i) and the coumn containing ba (j) m by ba (i), ba (j) m and set ba (i), ba (j) m = 1 when the answer of Probem h i (1) is positive and ba (i), ba (j) m = 0 otherwise An efficient agorithm to construct those tabes was presented in [7] To empoy the dynamic enumeration suggested by [15], for k distinct integers {i 1,, i k } {1,, n}, we ca F k := a i1, a i 1,, aik, a i k where aij, a i j Sij for j = 1,, k (2) a eve-k subface of S b = bs1,, Sn b (or simpy eve-k subface if no confusions exist) if there exists bα = (α, 1) R n+1 such that for each j = 1,, k baij, bα = ba i j, bα ba, bα for a S ij \ a ij, a i j 3

4 Ŝ i Ŝ i 2 3 s i 1 s i 1 [ 1, â(i) 2 ] [â(i) 1, â(i) 3 ] [â(i) 1, â(i) s i 1 ] [â(i) 1, â(i) s i ] 2 [ 2, â(i) 3 ] [â(i) 2, â(i) s i 1 ] [â(i) 2, â(i) s i ] s i 1 [ s i 1, â(i) s i ] Tabe T(i, i) Ŝ j Ŝ i â (j) 1 â (j) 2 â (j) 3 â (j) s j 1 [ 1, â(j) 1 ] [â(i) 1, â(j) 2 ] [â(i) 1, â(j) 3 ] [â(i) 1, â(j) s j ] 2 [ 2, â(j) 1 ] [â(i) 2, â(j) 2 ] [â(i) 2, â(j) 3 ] [â(i) 2, â(j) s j ] s i [ s i, â(j) 1 ] [â(i) s i, â(j) 2 ] [â(i) s i, â(j) 3 ] [â(i) s i, â(j) s j ] Tabe T(i, j) If a eve-k subface F k = a i1, a i 1,, aik, a i k is suppemented with aik+1, a i k+1 S ik+1 for certain i k+1 {1, 2,, n} \ {i 1,, i k } so that F k+1 := F k a ik+1, a i k+1 becomes a eve-(k + 1) subface, we ca F k+1 an extension of F k, and we say F k is extendabe in such situations For finding mixed ces, we pick an appropriate b Si1 with i 1 {1,, n} as our point of departure From Tabe T (i 1, i 1 ), those pairs a i1, a i 1 Si1 with ba i1, ba i 1 = 1 are the ony possibe eve-1 subsurfaces For a fixed pair a i1, a i 1 among them, we search among {S j : j {1,, n} \ {i 1 }} for the support S i2 having minima amount of suitabe points where ony pairs among those points can possiby extend the eve-1 subface a i1, a i 1 The main strategy for finding such a support suggested in [15] is the remova of those points, as many as possibe, in each support S j where j i 1, which have no chances to be part of pairs that can extend a i1, a i 1 and seect the support with minima remaining points as Si2 We wi give the detais of this procedure in the next section Suppose the seected S i2 contains the remaining points b 1,, b We foow by finding a the pairs among them that can extend a i1, a i 1 as in [7], and the procedure is outined beow: Let M := a a S i1 \ a i1, a i 1 and [ba, ba i1 ] = ba, ba i 1 = 1 in Tabe T (i1, i 1 ) i = 1,,, consider the One-Point test on b i : For each Minimize bbi, bα α 0 ba i1, bα = ba i 1, bα ba, bα a M (3) α 0 bk b, bα k = 1,, 4

5 in the variabes bα = (α, 1) R n+1 and α 0 R When the optima vaue is zero, the point b i wi be retained for further considerations Otherwise b i woud pay no roe in any pairs that can extend a i1, a i 1, and therefore it can be removed An important feature here is that one never needs to sove a these Linear Programming (LP) probems when the simpex method is used, because the informations generated by the simpex pivoting aready provide answers to some of the other LP probems Moreover, as shown in [7], feasibe points for the constraints of those LP probems are aways avaiabe when those reation tabes were estabished Let b j1,, b jµ be the remaining points in S i2 after a the resuts of One-Point tests are attained Next the Two-Point test wi be appied on the pairs among these points; that is, for m in {1,, µ}, consider the LP probem Minimize bbj + b bjm, bα 2α 0 ba i1, bα = ba i 1, bα ba, bα a M (4) α 0 b bjk, bα k = 1,, µ Ceary, ony zero optima vaue of this LP probem grants the permission to the pair {b j, b jm } for extending a i1, a i 1 to a eve-2 subface of b S = bs1,, b Sn Again, one never needs to sove these LP probems for a pairs, because most of the pairs {b j, b jm } having zero optima vaue for the corresponding LP probem in (4) are reveaed when constraints invoving both bb j and b bjm are active in certain pivoting stages during One-Point tests in (3) were performed or when Two-Point tests were appied to other pairs In summary, appying One-Point tests foowed by Two-Point tests on {b 1,, b } S i2 resut in a set of pairs in S i2 in which each pair can extend a i1, a i 1 Si1 to a eve-2 subface of b S = bs1,, b Sn Of course, if this set is empty, ie, there exists no pairs in Si2 wi that can extend a i1, a i 1, then we must stop here and focus our attention on extending other eve-1 subfaces in S i1 Extending eve-2 subface ( a i1, a i 1, ai2, a i 2 ) with certain ai2, a i 2 Si2 may foow the simiar procedure as described above and this process can be continued Finay, mixed ces wi be induced by pairs a in, a i n in Sin when we show ( a i1, a i 1,, ain 1, a i n 1 ) with aij, a i j Sij for j = 1,, n 1 is extendabe 3 Removing non-essentia points in the supports For integer k with 1 k < n 1, et F k = a i1, a i 1,, aik, a i k be a eve-k subface of S b = bs1,, Sn b where {a, a } S j {1,, n} \Q, et V := {b 1,, b r } S j where for each µ = 1,, r, bbµ, ba = bbµ, ba = 1 Q for Q := {i 1,, i k } {1,, n} For a fixed in the reation tabe T (, j) We wish to remove, as many as possibe, those points in V which woud be absent in any pairs in V that can extend the eve-k subface F k 5

6 For a particuar point b v V, consider the set of constraints: ba, bα = ba, bα Q ba, bα a S \ a, a bbv, bα b b, bα b V \ {b v } (5) Apparenty, when the set of equaities and inequaities in (5) is infeasibe, then there is no b µ in V such that {b v, b µ } can extend F k to become a eve-(k + 1) subface, and therefore b v can be safey removed from V More expicity, equaities and inequaities in (5) yied, a a, α = ω (a ) ω (a ) Q a a, α ω (a) ω (a ) a S \ a, a b v b, α ω j (b) ω j (b v ) b V \ {b v } As in [15], the feasibiity check of (I) can be formuated via an LP probem, (P) Max r, α subject to where r R n is some fixed vector, aong with its dua probem (D) in the variabes where and Φ(x) = X Q Min Φ(x) (I) subject to Ψ(x) = r x a 0 a S \ a, a, Q x b 0 ( b V \ {b v } ) < x a < ( Q ) x = (x a : a S \ {a }, Q; x b : b V \ {b v }) X a S \{a } Ψ(x) = X Q (ω (a) ω (a )) x a + X X a S \{a } b V \{b v} (a a) x a + X b V \{b v} (ω j (b) ω j (b v )) x b (b v b) x b (I) Any vector r R n may be chosen for the cost vector in (P), therefore one may choose r so that (D) becomes feasibe From the duaity theorem, (I) is infeasibe if and ony if (D) is unbounded To determine the boundedness of the LP probem in (D), we first reca that when F k = ai1, a i 1,, aik, a i k with {a, a } S for a Q is decaired to be a eve-k subface of α R n, b S = bs1,, b Sn, the foowing set of constraints must be satisfied for certain a a, α = ω (a ) ω (a ), Q (6) a a, α ω (a) ω (a ), a S \ a, a 6

7 Resuting from impementing the One-Point test or the Two-Point test by soving LP probems in (3) or (4) by the simpex agorithm, there are exacty n equaities in (6), and the inverse for the matrix determined by this n n inear equations is aways avaiabe in the context Let those equaities be a a, α = ω (a ) ω (a ), Q = {i 1,, i k } ajk+h ea k+h, α = ω jk+h (ea k+h) ω jk+h (a j k+h ) where for h = 1,, n k, j k+h Q and ea k+h S jk+h \ a jk+h, a j k+h In matrix form, we have D α = E (7) where D = a i1 a i 1,, a ik a i k, a jk+1 ea k+1,, a jn ea n E = ω i1 (a i 1 ) ω i1 (a i1 ) ω ik (a i k ) ω ik (a ik ) ω jk+1 (ea k+1 ) ω jk+1 (a jk+1 ) ω jn (ea n ) ω jn (a jn ) 3 7 5, D 1 is avaiabe Actuay, the coumns of matrix D pro- and, as mentioned above, vide a basis when we dea with the LP probem in (D) Under this basis, a criteria for the unboundedness of the LP probem is, if the cost r b = ω j (b) ω j (b v ) E D 1 (b v b) < 0 and the ξ th entry of D 1 (b v b) is non-positive for a ξ k + 1, then this LP probem is unbounded However, by (7), r b = ω j (b) ω j (b v ) α D D 1 (b v b) = ω j (b) + b, α (ω j (b v ) + b v, α ) = b b, bα b bv, bα So, ony the sign of entries of D 1 (b v b) for those b V for which b b, bα < b bv, bα needed to be checked to determine the possibe unboundedness of the LP probem in (D) Summarizing the above, yieds the foowing agorithm Agorithm : Removing non-essentia points in the supports Input: A eve-k subface F k := a i1, a i 1,, aik, a i k aong with α and D in (7) Output: i k+1 for which the support S ik+1 has fewest points where ony pairs among 7

8 which can possiby extend F k for a j {1,, n} \Q where Q := {i 1,, i k } do M j for a b µ S j end for do if for a Q, end if end for M j M j {b µ } bbµ, ba = bbµ, ba = 1 in the Reation Tabe T (j, ), then if M j doesn t contain at east 2 eements, then output F k is not extendibe for a b v M j do for a b M j \{b v } do if b b, bα < b bv, bα, then end if end for (A) end for for a ξ {k + 1,, n} do if the ξ th entry of D 1 (b v b) is positive, then end if end for go to (A) M j M j \ {b v } N j the number of eements in M j if N j < 2, then output F k is not extendibe return i k+1 where N ik+1 = min{n j : j {1,, n} \Q} 4 Numerica Resuts Our agorithm has been successfuy impemented in FORTRAN-90 and its Matab interface version is avaiabe at eetsung/softwarehtm To compare our new code MixedVo-20 with existing codes MixedVo and DEMiCs- 095 for the mixed voume computation, we wi mainy concentrate on the benchmark systems 8

9 speed-up system size(n) mixed voume MixedVo-20 MixedVo ratio Cycic-n , m 108m ,704, m 193hr ,795, hr 171hr ,243, hr Noon-n 19 1,162,261, m 783hr ,486,784, hr Eco-n 18 65, m 212hr , hr 145hr , hr 623hr , hr Chandra-n , m 782hr ,048, m Katsura-n 13 8, m 348hr , m 197hr , hr 108hr , hr Gaukwa-n 6 371, s 111m ,390, m 461m ,338, hr 335hr 105 Tabe 1: 16GHz Itanium2 processor, 2G RAM isted in [15], such as, Cycic-n [2, 5], Noon-n [17], Economic-n [16], Chandra-n [4], Katsura-n [3], and Gaukwa-n [19] A the computations were carried out on a 16GHz Itanium2 CPU with 2G RAM Tabe 1 compares our new code MixedVo-20 with MixedVo As expected, it iustrates a very high eve of speed-ups The superiority of DEMiCs-095 in speed had been demonstrated in [15] In Tabe 2, we compare our code with DEMiCs-095 As indicated, our agorithm is uniformy faster than DEMiCs-095 in cpu time on a the systems and the speedups range from 137 to 483 9

10 speed-up system size(n) mixed voume MixedVo-20 DEMiCs-095 ratio Cycic-n , m 331m ,704, m 295m ,795, hr 406hr ,243, hr 378hr 158 Noon-n 19 1,162,261, m 706m ,486,784, hr 269hr ,460,353, hr 946hr ,381,059, hr 258hr ,143,178, hr 744hr 341 Eco-n 18 65, m 523m , hr 331hr , hr 120hr , hr 402hr 143 Chandra-n , m 763m ,048, m 337hr ,097, hr 863hr ,194, hr 278hr ,388, hr 752hr 406 Katsura-n 13 8, m 110m , m 602m , hr 514hr , hr 232hr 147 Gaukwa-n 6 371, s 332s ,390, m 209m ,338, hr 131hr 409 Tabe 2: 16GHz Itanium2 processor, 2G RAM References [1] D N Bernshtein (1975), The number of roots of a system of equations, Functiona Anaysis and App 9(3), Transated from Funktsiona Ana i Priozhen, 9(3), 1-4 [2] G Björk and R Fröberg (1991), A faster way to count the soutions of inhomogeneous systems of agebraic equations, J Symboic Comput 12(3),

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