Relaxations of multilinear convex envelopes: dual is better than primal

Transcription

1 of multilinear convex envelopes: dual is better than primal 1 LIX, École Polytechnique, Palaiseau, France June 7th, th International Symposium on Experimental Algorithms (SEA) Bordeaux (France) of multilinear convex envelopes: dual is better than p

2 1 Definitions Multilinear terms Applications of multilinear convex envelopes: dual is better than p

3 1 Definitions Multilinear terms Applications 2 of multilinear convex envelopes: dual is better than p

4 1 Definitions Multilinear terms Applications 2 3 of multilinear convex envelopes: dual is better than p

5 1 Definitions Multilinear terms Applications of multilinear convex envelopes: dual is better than p

6 Where are we? Definitions Multilinear terms Applications 1 Definitions Multilinear terms Applications of multilinear convex envelopes: dual is better than p

7 Convex sets Definitions Multilinear terms Applications Let S R n be non-empty of multilinear convex envelopes: dual is better than p

8 Convex sets Definitions Multilinear terms Applications Let S R n be non-empty Any convex set containing S is a convex relaxation of S of multilinear convex envelopes: dual is better than p

9 Convex sets Definitions Multilinear terms Applications Let S R n be non-empty Any convex set containing S is a convex relaxation of S The convex hull convs of S is the intersection of all convex relaxations of S relaxation hull of multilinear convex envelopes: dual is better than p

10 Convex functions Definitions Multilinear terms Applications Let C R n be convex and compact of multilinear convex envelopes: dual is better than p

11 Convex functions Definitions Multilinear terms Applications Let C R n be convex and compact Let f : C R be lower semicontinuous of multilinear convex envelopes: dual is better than p

12 Convex functions Definitions Multilinear terms Applications Let C R n be convex and compact Let f : C R be lower semicontinuous Any convex function underestimating f is a convex relaxation of f of multilinear convex envelopes: dual is better than p

13 Convex functions Definitions Multilinear terms Applications Let C R n be convex and compact Let f : C R be lower semicontinuous Any convex function underestimating f is a convex relaxation of f The convex envelope convf of f is the pointwise supremum of all convex underestimators of f of multilinear convex envelopes: dual is better than p

14 Definitions Definitions Multilinear terms Applications x = (x 1,...,x k ) some decision variables of multilinear convex envelopes: dual is better than p

15 Definitions Definitions Multilinear terms Applications x = (x 1,...,x k ) some decision variables Assume finite variable bounds x L x x U of multilinear convex envelopes: dual is better than p

16 Definitions Definitions Multilinear terms Applications x = (x 1,...,x k ) some decision variables Assume finite variable bounds x L x x U The function w(x) = x 1 x 2 x k is a multilinear term of multilinear convex envelopes: dual is better than p

17 Definitions Definitions Multilinear terms Applications x = (x 1,...,x k ) some decision variables Assume finite variable bounds x L x x U The function w(x) = x 1 x 2 x k is a multilinear term If k 1 vars are fixed, w is a linear function of 1 var of multilinear convex envelopes: dual is better than p

18 Definitions Definitions Multilinear terms Applications x = (x 1,...,x k ) some decision variables Assume finite variable bounds x L x x U The function w(x) = x 1 x 2 x k is a multilinear term If k 1 vars are fixed, w is a linear function of 1 var Smallest nontrivial case: w(x) = x 1 x 2 (bilinear term) of multilinear convex envelopes: dual is better than p

19 Some applications Definitions Multilinear terms Applications Process synthesis in chemical engineering (e.g., Haverly s pooling problems [Haverly; ACM SIGMAP Bull., 1978]) of multilinear convex envelopes: dual is better than p

20 Some applications Definitions Multilinear terms Applications Process synthesis in chemical engineering (e.g., Haverly s pooling problems [Haverly; ACM SIGMAP Bull., 1978]) Molecular Distance Geometry Problem (MDGP) [Liberti et al.; ITOR, JOGO, ITOR, JOGO, OPTL, 2011] of multilinear convex envelopes: dual is better than p

21 Some applications Definitions Multilinear terms Applications Process synthesis in chemical engineering (e.g., Haverly s pooling problems [Haverly; ACM SIGMAP Bull., 1978]) Molecular Distance Geometry Problem (MDGP) [Liberti et al.; ITOR, JOGO, ITOR, JOGO, OPTL, 2011] Multilinear Least-Squares (MLLS) [Paatero; JCGS, 1999] of multilinear convex envelopes: dual is better than p

22 Haverly s pooling problem Definitions Multilinear terms Applications Inputs x 11 3% Sulphur $ 6 y 11 Blend 1 Outputs 2.5% Sulphur $ 9 Demands 100 Pool x 21 1% Sulphur $ 16 y 12 y 21 2% Sulphur 1.5% Sulphur x 12 Blend $ 10 y $ Find oil routing minimizing costs and satisfying mass+sulphur balance, quantity and quality demands of multilinear convex envelopes: dual is better than p

23 Haverly s pooling problem Definitions Multilinear terms Applications Inputs x 11 3% Sulphur $ 6 y 11 Blend 1 Outputs 2.5% Sulphur $ 9 Demands 100 Pool x 21 1% Sulphur $ 16 y 12 y 21 2% Sulphur 1.5% Sulphur x 12 Blend $ 10 y $ Find oil routing minimizing costs and satisfying mass+sulphur balance, quantity and quality demands Decision variables: input quantities x, routed quantities y, percentage p of sulphur in pool of multilinear convex envelopes: dual is better than p

24 Haverly s pooling problem Definitions Multilinear terms Applications Inputs x 11 3% Sulphur $ 6 y 11 Blend 1 Outputs 2.5% Sulphur $ 9 Demands 100 Pool x 21 1% Sulphur $ 16 y 12 y 21 x 12 2% Sulphur $ 10 y 22 Blend 2 1.5% Sulphur Find oil routing minimizing costs and satisfying mass+sulphur balance, quantity and quality demands $ Decision variables: input quantities x, routed quantities y, percentage p of sulphur in pool Sulphur balance: 3x 11 +x 21 = p(y 11 +y 12 ) Quality demands (blend 1): py 11 +2y (y 11 +y 21 ) of multilinear convex envelopes: dual is better than p

25 Molecular Distance Geometry Definitions Multilinear terms Applications Known set of atoms V, determine 3D structure of multilinear convex envelopes: dual is better than p

26 Molecular Distance Geometry Definitions Multilinear terms Applications Known set of atoms V, determine 3D structure Some inter-atomic distances d ij known (NMR) of multilinear convex envelopes: dual is better than p

27 Molecular Distance Geometry Definitions Multilinear terms Applications Known set of atoms V, determine 3D structure Some inter-atomic distances d ij known (NMR) Find atomic positions x i R 3 preserving given distances given weighted graph G = (V,E,d), find embedding in R 3 of multilinear convex envelopes: dual is better than p

28 Molecular Distance Geometry Definitions Multilinear terms Applications Known set of atoms V, determine 3D structure Some inter-atomic distances d ij known (NMR) Find atomic positions x i R 3 preserving given distances given weighted graph G = (V,E,d), find embedding in R 3 of multilinear convex envelopes: dual is better than p

29 Molecular Distance Geometry Definitions Multilinear terms Applications Known set of atoms V, determine 3D structure Some inter-atomic distances d ij known (NMR) Find atomic positions x i R 3 preserving given distances given weighted graph G = (V,E,d), find embedding in R 3 Continuous quartic formulation: min ( x i x j 2 d 2 x ij )2 {i,j} E involves quadrilinear terms of multilinear convex envelopes: dual is better than p

30 Multilinear Least Squares Definitions Multilinear terms Applications Decision variables x 1,...,x n Sampled data d 1,...,d m Theoretical model: i m d i = l L i where L l {1,...,n} for all l j J l x j Minimize error Q p = d ( l L i j J l x j i m) p, where p N { } With 1 or norms, get multilinear terms of multilinear convex envelopes: dual is better than p

31 Where are we? 1 Definitions Multilinear terms Applications of multilinear convex envelopes: dual is better than p

32 Relaxing problems having multilinear terms Two ways to relax multilinear terms are presented and compared: of multilinear convex envelopes: dual is better than p

33 Relaxing problems having multilinear terms Two ways to relax multilinear terms are presented and compared: of multilinear convex envelopes: dual is better than p

34 Relaxing problems having multilinear terms Two ways to relax multilinear terms are presented and compared: of multilinear convex envelopes: dual is better than p

35 For the general case, convex envelopes for multilinear terms are available explicitly in function of x L,x U for k = 2,3 and partly k = 4 of multilinear convex envelopes: dual is better than p

36 For the general case, convex envelopes for multilinear terms are available explicitly in function of x L,x U for k = 2,3 and partly k = 4 They consist of sets of constraints to be adjoined to the Mathematical Programming formulation of multilinear convex envelopes: dual is better than p

37 For the general case, convex envelopes for multilinear terms are available explicitly in function of x L,x U for k = 2,3 and partly k = 4 They consist of sets of constraints to be adjoined to the Mathematical Programming formulation No further variables are needed of multilinear convex envelopes: dual is better than p

38 Bilinear terms: McCormick s inequalities Let W = {(w,x 1,x 2 ) w = x 1 x 2 (x 1,x 2 ) = [x L,x U ]}, then conv(w) is given by: w x L 1x 2 +x L 2x 1 x L 1x L 2 w x U 1 x 2 +x U 2 x 1 x U 1 x U 2 w x L 1x 2 +x U 2 x 1 x L 1x U 2 w x U 1 x 2 +x L 2x 1 x U 1 x L 2 Stated [McCormick; MP, 1976], proved [Al-Khayyal, Falk; MOR, 1983] of multilinear convex envelopes: dual is better than p

39 McCormick s envelopes Lower envelopes Upper envelopes Both of multilinear convex envelopes: dual is better than p

40 Special case: Fortet s linearization If x 1 and x 2 are binary variables, the McCormick s inequalities lead to the Fortet s inequalities [Fortet; RFRO, 1960]: w 0 w x 2 +x 1 1 w x 1 w x 2 The resulting reformulation is an exact linearization as shown in [Liberti; RAIRO-RO, 2009] of multilinear convex envelopes: dual is better than p

41 Trilinear case It is not as easy as bilinear convex relaxation: of multilinear convex envelopes: dual is better than p

42 Trilinear case It is not as easy as bilinear convex relaxation: the number of constraints is greater than 4 of multilinear convex envelopes: dual is better than p

43 Trilinear case It is not as easy as bilinear convex relaxation: the number of constraints is greater than 4 there are several cases, depending on sign of bounds of the variables: x L i xu i 0 [Meyer, Floudas; 2003]; mixed case [Meyer, Floudas; JOGO, 2004] of multilinear convex envelopes: dual is better than p

44 Trilinear case It is not as easy as bilinear convex relaxation: the number of constraints is greater than 4 there are several cases, depending on sign of bounds of the variables: x L i xu i 0 [Meyer, Floudas; 2003]; mixed case [Meyer, Floudas; JOGO, 2004] there are further conditions to check of multilinear convex envelopes: dual is better than p

45 Example (1): x U 1, x U 2, x U 3 0 Permute variables x 1, x 2 and x 3 such that: x U 1 x L 2x L 3 +x L 1x U 2 x U 3 x L 1x U 2 x L 3 +x U 1 x L 2x U 3 x U 1 x L 2x L 3 +x L 1x U 2 x U 3 x U 1 x U 2 x L 3 +x L 1x L 2x U 3 of multilinear convex envelopes: dual is better than p

46 Example (1): x U 1, x U 2, x U 3 0 Permute variables x 1, x 2 and x 3 such that: Lower envelope: x U 1 x L 2x L 3 +x L 1x U 2 x U 3 x L 1x U 2 x L 3 +x U 1 x L 2x U 3 x U 1 x L 2x L 3 +x L 1x U 2 x U 3 x U 1 x U 2 x L 3 +x L 1x L 2x U 3 w x L 2x L 3x 1 +x L 1x L 3x 2 +x L 1x L 2x 3 2x L 1x L 2x L 3 w x U 2 x U 3 x 1 +x U 1 x U 3 x 2 +x U 1 x U 2 x 3 2x U 1 x U 2 x U 3 w x L 2 xu 3 x 1 +x L 1 xu 3 x 2 +x U 1 xl 2 x 3 x L 1 xl 2 xu 3 xu 1 xl 2 xu 3 w x U 2 x L 3x 1 +x U 1 x L 3x 2 +x L 1x U 2 x 3 x U 1 x U 2 x L 3 x L 1x U 2 x L 3 w c 1 x 1 +x U 1 x L 3x 2 +x U 1 x L 2x 3 +x L 1x U 2 x U 3 c 1 x L 1 x U 1 x U 2 x L 3 x U 1 x L 2x U 3 w c 2 x 1 +x L 1x U 3 x 2 +x L 1x U 2 x 3 +x U 1 x L 2x L 3 c 2 x U 1 x L 1x L 2x U 3 x L 1x U 2 x L 3, where c 1 = xu 1 xu 2 xl 3 xl 1 xu 2 xu 3 xu 1 xl 2 xl 3 +xu 1 xl 2 xu 3 x U 1 xl 1 and c 2 = xl 1 xl 2 xu 3 xu 1 xl 2 xl 3 xl 1 xu 2 xu 3 +xl 1 xu 2 xl 3 x L 1 xu 1 of multilinear convex envelopes: dual is better than p

47 Example (2): x U 1, x U 2, x U 3 0 Upper envelope: w x L 2x L 3x 1 +x U 1 x L 3x 2 +x U 1 x U 2 x 3 x U 1 x U 2 x L 3 x U 1 x L 2x L 3 w x U 2 x L 3x 1 +x L 1x L 3x 2 +x U 1 x U 2 x 3 x U 1 x U 2 x L 3 x L 1x U 2 x L 3 w x L 2x L 3x 1 +x U 1 x U 3 x 2 +x U 1 x L 2x 3 x U 1 x L 2x U 3 x U 1 x L 2x L 3 w x U 2 x U 3 x 1 +x L 1x L 3x 2 +x L 1x U 2 x 3 x L 1x U 2 x U 3 x L 1x U 2 x L 3 w x L 2x U 3 x 1 +x U 1 x U 3 x 2 +x L 1x L 2x 3 x U 1 x L 2x U 3 x L 1x L 2x U 3 w x U 2 x U 3 x 1 +x L 1x U 3 x 2 +x L 1x L 2x 3 x L 1x U 2 x U 3 x L 1x L 2x U 3. of multilinear convex envelopes: dual is better than p

48 Quadrilinear terms The convex envelope is not known explicitly for quadrilinear terms Combine bilinear and trilinear envelope [Cafieri, Lee, Liberti; JOGO, 2011] Convex envelope for some cases presented in [Balram; M.Sc. Thesis, 2019] (e.g., when x L 1, xl 2, xl 3, xl 4 0, then 44 constraints are generated) of multilinear convex envelopes: dual is better than p

49 Tighter relaxations by associativity Write w = x 1 x 2 x 3 x 4 as: 1 (x 1 x 2 )x 3 x 4 (tri(bi,1,1)) 2 (x 1 x 2 x 3 )x 4 (bi(tri,1)) 3 (x 1 x 2 )(x 3 x 4 ) (bi(bi(1,1),bi(1,1))) 4 ((x 1 x 2 )x 3 )x 4 (bi(bi(bi(1,1),1),1)) of multilinear convex envelopes: dual is better than p

50 Tighter relaxations by associativity Write w = x 1 x 2 x 3 x 4 as: 1 (x 1 x 2 )x 3 x 4 (tri(bi,1,1)) 2 (x 1 x 2 x 3 )x 4 (bi(tri,1)) 3 (x 1 x 2 )(x 3 x 4 ) (bi(bi(1,1),bi(1,1))) 4 ((x 1 x 2 )x 3 )x 4 (bi(bi(bi(1,1),1),1)) Apply bilinear/trilinear envelopes, get different relaxations for w: which one is tightest? of multilinear convex envelopes: dual is better than p

51 Tighter relaxations by associativity Write w = x 1 x 2 x 3 x 4 as: 1 (x 1 x 2 )x 3 x 4 (tri(bi,1,1)) 2 (x 1 x 2 x 3 )x 4 (bi(tri,1)) 3 (x 1 x 2 )(x 3 x 4 ) (bi(bi(1,1),bi(1,1))) 4 ((x 1 x 2 )x 3 )x 4 (bi(bi(bi(1,1),1),1)) Apply bilinear/trilinear envelopes, get different relaxations for w: which one is tightest? Theorem Choose smallest number of compositions ((1)-(2) are better than (3)-(4)) of multilinear convex envelopes: dual is better than p

52 Tighter relaxations by associativity Write w = x 1 x 2 x 3 x 4 as: 1 (x 1 x 2 )x 3 x 4 (tri(bi,1,1)) 2 (x 1 x 2 x 3 )x 4 (bi(tri,1)) 3 (x 1 x 2 )(x 3 x 4 ) (bi(bi(1,1),bi(1,1))) 4 ((x 1 x 2 )x 3 )x 4 (bi(bi(bi(1,1),1),1)) Apply bilinear/trilinear envelopes, get different relaxations for w: which one is tightest? Theorem Choose smallest number of compositions ((1)-(2) are better than (3)-(4)) Some empirical indications on choosing (1) or (2) depending on bounds of multilinear convex envelopes: dual is better than p

53 Beyond quadrilinear terms envelopes for multilinear terms larger than quadrilinear: not known explicitly of multilinear convex envelopes: dual is better than p

54 Beyond quadrilinear terms envelopes for multilinear terms larger than quadrilinear: not known explicitly software as PORTA can compute the convex hull of a given set of points in R n of multilinear convex envelopes: dual is better than p

55 Beyond quadrilinear terms envelopes for multilinear terms larger than quadrilinear: not known explicitly software as PORTA can compute the convex hull of a given set of points in R n Balram s thesis reports a similar procedure to compute the convex hull (but less refined) of multilinear convex envelopes: dual is better than p

56 : preliminaries Consider the 2 k point set P: { (x L 1,...,xL k 1,xL k ), (x L 1,...,x L k 1,xU k ), (x L 1,...,xU k 1,xL k ), (x L 1,...,x U k 1,xU k ),..., (x U 1,...,xU k 1,xL k ), (x U 1,...,x U k 1,xU k ) } (i.e., all combinations of lower/upper bounds) Let w(x) = i k x i: lift P to (x,w) space, get P W R k+1 x P ( x,w( x)) P W of multilinear convex envelopes: dual is better than p

57 Dual representation of a point set Convex hull of P = {p 1,...,p m } R n is given by x R n : λ R m x = λ i p i i = 1 i m (λ i 0) i m i mλ of multilinear convex envelopes: dual is better than p

58 Dual representation of a point set Convex hull of P = {p 1,...,p m } R n is given by x R n : λ R m x = λ i p i i = 1 i m (λ i 0) i m i mλ x is a convex combination of points in P of multilinear convex envelopes: dual is better than p

59 Dual representation of a point set Convex hull of P = {p 1,...,p m } R n is given by x R n : λ R m x = λ i p i i = 1 i m (λ i 0) i m i mλ x is a convex combination of points in P Can express points in P W in function of x,w,x L,x U and of added (dual) variables λ for any k of multilinear convex envelopes: dual is better than p

60 Dual representation of a point set Convex hull of P = {p 1,...,p m } R n is given by x R n : λ R m x = λ i p i i = 1 i m (λ i 0) i m i mλ x is a convex combination of points in P Can express points in P W in function of x,w,x L,x U and of added (dual) variables λ for any k Automatically get explicit convex envelopes for multilinear terms of multilinear convex envelopes: dual is better than p

61 Dual envelopes of multilinear terms We compute the i-th point p i P in O(2 k ) as follows: i 2 k d i = ( ) i 1 2 k j mod 2 j k j k b j (0) = x L j b j (1) = x U j i.e. for all i 2 k, we have p i = (x L/U? j j k) = (b j (d ij ) j k) of multilinear convex envelopes: dual is better than p

62 Dual envelopes of multilinear terms We compute the i-th point p i P in O(2 k ) as follows: i 2 k d i = j k ( ) i 1 2 k j mod 2 j k b j (0) = x L j b j (1) = x U j i.e. for all i 2 k, we have p i = (x L/U? j j k) = (b j (d ij ) j k) We add 2 k new variables λ 0 and k +1 new constraints: j k x j = i 2 k λ i b j (d ij ) w = i 2 k λ i = 1 λ i i 2 k j k b j (d ij ) of multilinear convex envelopes: dual is better than p

63 Dual envelopes of multilinear terms We compute the i-th point p i P in O(2 k ) as follows: i 2 k d i = j k ( ) i 1 2 k j mod 2 j k b j (0) = x L j b j (1) = x U j i.e. for all i 2 k, we have p i = (x L/U? j j k) = (b j (d ij ) j k) We add 2 k new variables λ 0 and k +1 new constraints: j k x j = i 2 k λ i b j (d ij ) w = i 2 k λ i = 1 λ i i 2 k j k b j (d ij ) The projection of this feasible region on (x,w) is conv(w) of multilinear convex envelopes: dual is better than p

64 Example: bilinear term Using a matrix representation, we have: x L 1 x L 2 [ λ1 λ 2 λ 3 λ 4 ] x L 1 x U 2 x U 1 x L = [ ] x 1 x 2 2 x U 1 x U 2 of multilinear convex envelopes: dual is better than p

65 Example: bilinear term Using a matrix representation, we have: x L 1 x L 2 [ λ1 λ 2 λ 3 λ 4 ] x L 1 x U 2 x U 1 x L = [ ] x 1 x 2 2 x U 1 x U 2 x L 1 [ xl 2 λ1 λ 2 λ 3 λ 4 ] x L 1 xu 2 x U 1 xl = w 2 x U 1 xu 2 of multilinear convex envelopes: dual is better than p

66 Example: bilinear term Using a matrix representation, we have: x L 1 x L 2 [ λ1 λ 2 λ 3 λ 4 ] x L 1 x U 2 x U 1 x L = [ ] x 1 x 2 2 x U 1 x U 2 x L 1 [ xl 2 λ1 λ 2 λ 3 λ 4 ] x L 1 xu 2 x U 1 xl = w 2 x U 1 xu 2 x 1 = λ 1 x L 1 +λ 2 x L 1 +λ 3 x U 1 +λ 4 x U 1 x 2 = λ 1 x L 2 +λ 2x U 2 +λ 3x L 2 +λ 4x U 2 w = λ 1 x L 1 xl 2 +λ 2x L 1 xu 2 +λ 3x U 1 xl 2 +λ 4x U 1 xu 2 λ i = 1 i 4 of multilinear convex envelopes: dual is better than p

67 Where are we? 1 Definitions Multilinear terms Applications of multilinear convex envelopes: dual is better than p

68 Comparison between relaxations Primal envelopes Dual envelopes Original variables only 2 k added variables O(2 k ) added constraints k +1 added constraints of multilinear convex envelopes: dual is better than p

69 Comparison between relaxations Primal envelopes Dual envelopes Original variables only 2 k added variables O(2 k ) added constraints k +1 added constraints k Primal envelopes Dual envelopes 2 [McCormick] [Meyer, Floudas] [Balram] [Balram] Primal envelopes constraints: apparent growth like k2 k of multilinear convex envelopes: dual is better than p

70 Comparison between relaxations Primal envelopes Dual envelopes Original variables only 2 k added variables O(2 k ) added constraints k +1 added constraints k Primal envelopes Dual envelopes 2 [McCormick] [Meyer, Floudas] [Balram] [Balram] Primal envelopes constraints: apparent growth like k2 k O(k2 k ) vs. 2 k might yield CPU improvements of multilinear convex envelopes: dual is better than p

71 Experimental set-up Generate random multilinear NLPs P linear, bilinear, trilinear terms nonseparable Generate primal convex LP relaxation R P Generate dual convex LP relaxation Λ P Solve R P,Λ P using CPLEX, compare CPU times To get a feel about how R P,Λ P might perform in BB, add integrality constraints on primal variables, get MILP relaxations R P,Λ P Solve R P,Λ P using CPLEX, compare CPU times of multilinear convex envelopes: dual is better than p

72 Instance set 2500 random instances # variables n {10,20} n = 10: # bilinear terms β {0,10,13,17,21,25,29,33} # trilinear terms τ {0,10,22,34,46,58,71,83} n = 20: β {0,20,38,57,76,95,114,133} τ {0,20,144,268,393,517,642,766} 16 instances for each parameter combination yielding multilinear NLPs (and then MINLPs after imposing integrality on some variables) Variable bounds chosen at random, magnitude of multilinear convex envelopes: dual is better than p

73 LP relaxation test, n = 10 CPU time averages over each 16-instance block with given (n,β,τ) primal relaxation dual relaxation 0.02 CPU time [s] instance set of multilinear convex envelopes: dual is better than p

74 LP relaxation test, n = 20 CPU time averages over each 16-instance block with given (n,β,τ) primal relaxation dual relaxation CPU time [s] instance set of multilinear convex envelopes: dual is better than p

75 MILP relaxation test, n = 10 CPU time averages over each 16-instance block with given (n,β,τ) 2.5 primal relaxation dual relaxation 2 CPU time [s] instance set of multilinear convex envelopes: dual is better than p

76 MILP relaxation test, n = 20 CPU time averages over each 16-instance block with given (n,β,τ) primal relaxation dual relaxation CPU time [s] instance set of multilinear convex envelopes: dual is better than p

77 Where are we? 1 Definitions Multilinear terms Applications of multilinear convex envelopes: dual is better than p

78 Conclusion Considerations outperforms the primal when the number of variables increases of multilinear convex envelopes: dual is better than p

79 Conclusion Considerations outperforms the primal when the number of variables increases is better by far than primal for MILP of multilinear convex envelopes: dual is better than p

80 Conclusion Considerations outperforms the primal when the number of variables increases is better by far than primal for MILP is more stable (empirically CPU time increases proportionally to instance size) than primal for MILP of multilinear convex envelopes: dual is better than p

81 Conclusion Considerations outperforms the primal when the number of variables increases is better by far than primal for MILP is more stable (empirically CPU time increases proportionally to instance size) than primal for MILP These are preliminary results of multilinear convex envelopes: dual is better than p

82 Conclusion Considerations outperforms the primal when the number of variables increases is better by far than primal for MILP is more stable (empirically CPU time increases proportionally to instance size) than primal for MILP These are preliminary results Using dual relaxation within spatial Branch-and-Bound could improve computational times of multilinear convex envelopes: dual is better than p