Polyhedral Results for A Class of Cardinality Constrained Submodular Minimization Problems

Transcription

1 Polyhedral Results for A Class of Cardinality Constrained Submodular Minimization Problems Shabbir Ahmed and Jiajin Yu Georgia Institute of Technology

2 A Motivating Problem [n]: Set of candidate investment projects c: Cost vector (of size n) b: Budget µ i, σi 2 : Mean and variance of project i return (uncorrelated) Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

3 A Motivating Problem [n]: Set of candidate investment projects c: Cost vector (of size n) b: Budget µ i, σi 2 : Mean and variance of project i return (uncorrelated) Pick projects to optimize mean-variance criterion: Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

4 A Motivating Problem [n]: Set of candidate investment projects c: Cost vector (of size n) b: Budget µ i, σi 2 : Mean and variance of project i return (uncorrelated) Pick projects to optimize mean-variance criterion: min µ i x i + λ σi 2 xi 2 : c x b, x {0, 1} n i i Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

5 A Motivating Problem [n]: Set of candidate investment projects c: Cost vector (of size n) b: Budget µ i, σi 2 : Mean and variance of project i return (uncorrelated) Pick projects to optimize mean-variance criterion: min µ i x i + λ σi 2 xi 2 : c x b, x {0, 1} n i i Mean risk combinatorial optimization [Shen et al 03, Atamturk 08, Nikolova 10, Baumann et al 13] Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

6 Concave Cost Combinatorial Optimization min {d x + f (a x) : x X {0, 1} n} where f is a univariate concave function Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

7 Concave Cost Combinatorial Optimization min {d x + f (a x) : x X {0, 1} n} where f is a univariate concave function min {d x + w : w f (a x), x X {0, 1} n} Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

8 Concave Cost Combinatorial Optimization min {d x + f (a x) : x X {0, 1} n} where f is a univariate concave function min {d x + w : w f (a x), x X {0, 1} n} Goal: Polyhedral description/relaxation of conv(p X) where { } P := (w, x) R {0, 1} n : w f (a x) Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

9 Concave Cost Combinatorial Optimization min {d x + f (a x) : x X {0, 1} n} where f is a univariate concave function min {d x + w : w f (a x), x X {0, 1} n} Goal: Polyhedral description/relaxation of conv(p X) where { } P := (w, x) R {0, 1} n : w f (a x) Simple case: Cardinality constraint X = {x [0, 1] n : e x K } Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

10 Submodular Functions Definition A function F : {0, 1} n R is submodular if F (x + e i ) F(x) F(y + e i ) F(y) x y, s.t. x i = y i = 0 If f is concave and a R n + then F(x) := f (a x) is submodular. F (x1,x2) = p 9x1 + 16x2 {0,0} {1,0} {0,1} {1,1} Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

11 Submodular Functions Definition A function F : {0, 1} n R is submodular if F (x + e i ) F(x) F(y + e i ) F(y) x y, s.t. x i = y i = 0 If f is concave and a R n + then F(x) := f (a x) is submodular. F (x1,x2) = p 9x1 + 16x2 {0,0} {1,0} {0,1} {1,1} Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

12 Submodular Functions Definition A function F : {0, 1} n R is submodular if F (x + e i ) F(x) F(y + e i ) F(y) x y, s.t. x i = y i = 0 If f is concave and a R n + then F(x) := f (a x) is submodular. F (x1,x2) = p 9x1 + 16x2 {0,0} {1,0} {0,1} {1,1} Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

13 Submodular Polyhedra P := {(w, x) : w f (a x)} is the epigraph of a submodular function Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

14 Submodular Polyhedra P := {(w, x) : w f (a x)} is the epigraph of a submodular function Separation/Optimization over conv(p) is easy [Edmonds 71, Lovasz 82]: Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

15 Submodular Polyhedra P := {(w, x) : w f (a x)} is the epigraph of a submodular function Separation/Optimization over conv(p) is easy [Edmonds 71, Lovasz 82]: Given (w, x): Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

16 Submodular Polyhedra P := {(w, x) : w f (a x)} is the epigraph of a submodular function Separation/Optimization over conv(p) is easy [Edmonds 71, Lovasz 82]: Given (w, x): 1 Sort components of x so that x (1) x (2) x (n). Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

17 Submodular Polyhedra P := {(w, x) : w f (a x)} is the epigraph of a submodular function Separation/Optimization over conv(p) is easy [Edmonds 71, Lovasz 82]: Given (w, x): 1 Sort components of x so that x (1) x (2) x (n). 2 Let (i) = f ( j i a (j)) f ( j<i a (j)), i > 0, and 0 = f (0). Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

18 Submodular Polyhedra P := {(w, x) : w f (a x)} is the epigraph of a submodular function Separation/Optimization over conv(p) is easy [Edmonds 71, Lovasz 82]: Given (w, x): 1 Sort components of x so that x (1) x (2) x (n). 2 Let (i) = f ( j i a (j)) f ( j<i a (j)), i > 0, and 0 = f (0). 3 Then (w, x) conv(p) 0 + i (i)x (i) w Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

19 Submodular Polyhedra P := {(w, x) : w f (a x)} is the epigraph of a submodular function Separation/Optimization over conv(p) is easy [Edmonds 71, Lovasz 82]: Given (w, x): 1 Sort components of x so that x (1) x (2) x (n). 2 Let (i) = f ( j i a (j)) f ( j<i a (j)), i > 0, and 0 = f (0). 3 Then (w, x) conv(p) 0 + i (i)x (i) w Submodular inequalities in MINLP: Atamturk et al. 08, 09, 12, Tawarmalani 10, A.+Papageorgiou 13, Bauman et al Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

20 Cardinality Constrained Submodular Polyhedra We are interested in conv(p X) where X is a cardinality constraint Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

21 Cardinality Constrained Submodular Polyhedra We are interested in conv(p X) where X is a cardinality constraint General cardinality constrained submodular minimization min{f(x) : x X} is strongly NP-hard Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

22 Cardinality Constrained Submodular Polyhedra We are interested in conv(p X) where X is a cardinality constraint General cardinality constrained submodular minimization min{f(x) : x X} is strongly NP-hard However concave cardinality constrained minimization min{f (a x) : x X} is easy (Onn 03) Combinatorial separation algorithm Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

23 Cardinality Constrained Submodular Polyhedra We are interested in conv(p X) where X is a cardinality constraint General cardinality constrained submodular minimization min{f(x) : x X} is strongly NP-hard However concave cardinality constrained minimization min{f (a x) : x X} is easy (Onn 03) Combinatorial separation algorithm Goal: Extend the submodular inequalities to incorporate cardinality constraint Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

24 Illustration w p x x2 2 Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

25 Illustration 0 1 x 1 2 x 2 Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

26 Illustration w x x 2 Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

27 Illustration w x x 2 Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

28 Results P = { } (w, x) R {0, 1} n : w f (a x) X = {x [0, 1] n : e x K } Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

29 Results P = { } (w, x) R {0, 1} n : w f (a x) X = {x [0, 1] n : e x K } Unweighted case (identical a i s): Complete description of conv(p X) Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

30 Results P = { } (w, x) R {0, 1} n : w f (a x) X = {x [0, 1] n : e x K } Unweighted case (identical a i s): Complete description of conv(p X) Weighted case (general a i ): Family of facets/valid inequalities by lifting Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

31 Results P = { } (w, x) R {0, 1} n : w f (a x) X = {x [0, 1] n : e x K } Unweighted case (identical a i s): Complete description of conv(p X) Weighted case (general a i ): Family of facets/valid inequalities by lifting Computational results for mean-risk knapsack Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

32 Unweighted Case Assume a i = 1 thus f (a x) = f ( S ) where x i = 1 for all i S Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

33 Unweighted Case Assume a i = 1 thus f (a x) = f ( S ) where x i = 1 for all i S Let S := {S [n] : S K } Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

34 Unweighted Case Assume a i = 1 thus f (a x) = f ( S ) where x i = 1 for all i S Let S := {S [n] : S K } (w, x) conv(p X) ρ 0 + ρ i x i w i [n] Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

35 Unweighted Case Assume a i = 1 thus f (a x) = f ( S ) where x i = 1 for all i S Let S := {S [n] : S K } (w, x) conv(p X) ρ 0 + ρ i x i w i [n] where ρ is an optimal solution to the (dual) LP: max ρ 0 + i [n]ρ i x i s.t. ρ 0 + i S ρ i f ( S ) S S Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

36 Unweighted Case Assume a i = 1 thus f (a x) = f ( S ) where x i = 1 for all i S Let S := {S [n] : S K } (w, x) conv(p X) ρ 0 + ρ i x i w i [n] where ρ is an optimal solution to the (dual) LP: max ρ 0 + i [n]ρ i x i s.t. ρ 0 + i S ρ i f ( S ) S S The primal LP: min s.t. f ( S )P(S) S S P(S) = x i S S:i S P(S) = 1, P(S) 0 S S i [n] S S Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

37 Unweighted Case: Construction Given x X, construct a feasible solution to the dual based on order and values of x i Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

38 Unweighted Case: Construction Given x X, construct a feasible solution to the dual based on order and values of x i Gives a valid inequality Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

39 Unweighted Case: Construction Given x X, construct a feasible solution to the dual based on order and values of x i Gives a valid inequality Construct a feasible solution to the primal with equal objective value Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

40 Unweighted Case: Construction Given x X, construct a feasible solution to the dual based on order and values of x i Gives a valid inequality Construct a feasible solution to the primal with equal objective value Proves optimality Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

41 Unweighted Case: Dual Solution Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

42 Unweighted Case: Dual Solution Assume 1 =: x 0 x 1 x 2 x n (by reindexing) Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

43 Unweighted Case: Dual Solution Assume 1 =: x 0 x 1 x 2 x n (by reindexing) Let z i := Kx i K 1 j=i x j for i = 0, 1,..., K Note that z i z i+1 for all i = 0, 1,..., K Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

44 Unweighted Case: Dual Solution Assume 1 =: x 0 x 1 x 2 x n (by reindexing) Let z i := Kx i K 1 j=i x j for i = 0, 1,..., K Note that z i z i+1 for all i = 0, 1,..., K Let y := n j=k x j Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

45 Unweighted Case: Dual Solution Assume 1 =: x 0 x 1 x 2 x n (by reindexing) Let z i := Kx i K 1 j=i x j for i = 0, 1,..., K Note that z i z i+1 for all i = 0, 1,..., K Let y := n j=k x j Let i 0 := argmax{0 i K 1 : z i y z i+1 } Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

46 Unweighted Case: Dual Solution Assume 1 =: x 0 x 1 x 2 x n (by reindexing) Let z i := Kx i K 1 j=i x j for i = 0, 1,..., K Note that z i z i+1 for all i = 0, 1,..., K Let y := n j=k x j Let i 0 := argmax{0 i K 1 : z i y z i+1 } The dual solution: ρ i = f (0) i = 0 f (i) f (i 1) 1 i i 0 i 0 < i n f (K ) f (i 0 ) K i 0 Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

47 Unweighted Case: Dual Solution Assume 1 =: x 0 x 1 x 2 x n (by reindexing) Let z i := Kx i K 1 j=i x j for i = 0, 1,..., K Note that z i z i+1 for all i = 0, 1,..., K Let y := n j=k x j Let i 0 := argmax{0 i K 1 : z i y z i+1 } The dual solution: ρ i = f (0) i = 0 f (i) f (i 1) 1 i i 0 i 0 < i n f (K ) f (i 0 ) K i 0 Note that ρ i = i for i i 0 Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

48 Unweighted Case: Validity Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

49 Unweighted Case: Validity Consider any i 0 {0,..., K 1} Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

50 Unweighted Case: Validity Consider any i 0 {0,..., K 1} Consider arbitrary S S Let S 1 = S {1,..., i 0 } and S 2 = S \ S 1 Let i 1 = S 1 and i 2 = S 2 Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

51 Unweighted Case: Validity Consider any i 0 {0,..., K 1} Consider arbitrary S S Let S 1 = S {1,..., i 0 } and S 2 = S \ S 1 Let i 1 = S 1 and i 2 = S 2 Need to show: ρ 0 + i S ρ i f (S) = f (i 1 + i 2 ) ( ) Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

52 Unweighted Case: Validity Consider any i 0 {0,..., K 1} Consider arbitrary S S Let S 1 = S {1,..., i 0 } and S 2 = S \ S 1 Let i 1 = S 1 and i 2 = S 2 Need to show: ρ 0 + i S ρ i f (S) = f (i 1 + i 2 ) ( ) The lhs of ( ): 0 + i S 1 i + i S 2 ρ i f (i 1 ) + i 2 f (K ) f (i 0) K i 0 by construction and validity of the usual submodular inequality corresponding to S 1 Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

53 Unweighted Case: Validity (contd.) Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

54 Unweighted Case: Validity (contd.) Sufficient to show that f (i 1 ) + i 2 f (K ) f (i 0) K i 0 f (i 1 + i 2 ) Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

55 Unweighted Case: Validity (contd.) Sufficient to show that f (i 1 ) + i 2 f (K ) f (i 0) K i 0 f (i 1 + i 2 ) Assume i 2 > 0 otherwise we are done Rearranging f (K ) f (i 0 ) K i 0 f (i 1 + i 2 ) f (i 1 ) i 2 Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

56 Unweighted Case: Validity (contd.) Sufficient to show that f (i 1 ) + i 2 f (K ) f (i 0) K i 0 f (i 1 + i 2 ) Assume i 2 > 0 otherwise we are done Rearranging f (K ) f (i 0 ) K i 0 f (i 1 + i 2 ) f (i 1 ) i 2 The above inequality can be shown using concavity of f (i) and i 1 i 0 K Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

57 Unweighted Case: Primal Solution Need to construct P(S) for all S S := {S [n] : S K } ; {1} {1, 2}. {1, 2,...,i 0 } {1, 2,...,i 0 } [ C {1, 2,...,i 0 } [ C i 0 i n Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

58 Unweighted Case: Primal Solution Need to construct P(S) for all S S := {S [n] : S K } ; {1} {1, 2}. {1, 2,...,i 0 } {1, 2,...,i 0 } [ C {1, 2,...,i 0 } [ C i 0 i n Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

59 Unweighted Case: Primal Solution Need to construct P(S) for all S S := {S [n] : S K } ; {1} {1, 2}. {1, 2,...,i 0 } {1, 2,...,i 0 } [ C {1, 2,...,i 0 } [ C i 0 i n C {i 0 +1,...,n}, C = K i 0 Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

60 Unweighted Case: Primal Solution Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

61 Unweighted Case: Primal Solution P( ) = 1 x 1 P({1,..., i}) = x i x i+1 for i = 1,..., i 0 1 P({1,..., i 0 }) = z i 0 y K i 0 Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

62 Unweighted Case: Primal Solution P( ) = 1 x 1 P({1,..., i}) = x i x i+1 for i = 1,..., i 0 1 P({1,..., i 0 }) = z i 0 y K i 0 Key Lemma: There exists a nonnegative solution to the following linear system C:C C Q(C) = x i, i = i 0 + 1,..., n where C = {C {i 0 + 1,..., n} : C = K i 0 } Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

63 Unweighted Case: Primal Solution P( ) = 1 x 1 P({1,..., i}) = x i x i+1 for i = 1,..., i 0 1 P({1,..., i 0 }) = z i 0 y K i 0 Key Lemma: There exists a nonnegative solution to the following linear system C:C C Q(C) = x i, i = i 0 + 1,..., n where C = {C {i 0 + 1,..., n} : C = K i 0 } P({1,..., i 0 } C) = Q(C) for all C C Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

64 Unweighted Case: Primal Solution P( ) = 1 x 1 P({1,..., i}) = x i x i+1 for i = 1,..., i 0 1 P({1,..., i 0 }) = z i 0 y K i 0 Key Lemma: There exists a nonnegative solution to the following linear system C:C C Q(C) = x i, i = i 0 + 1,..., n where C = {C {i 0 + 1,..., n} : C = K i 0 } P({1,..., i 0 } C) = Q(C) for all C C The constructed primal solution is feasible and has the same objective value as that of the dual problem Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

65 Weighted Case: Exact Lifting Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

66 Weighted Case: Exact Lifting Consider a set S [n] with S = K Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

67 Weighted Case: Exact Lifting Consider a set S [n] with S = K The usual submodular inequality 0 + i S ix i w is a facet of conv(p {x : x i = 0 j S}) Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

68 Weighted Case: Exact Lifting Consider a set S [n] with S = K The usual submodular inequality 0 + i S ix i w is a facet of conv(p {x : x i = 0 j S}) Compute coefficients of variables not in S by (sequence dependent) lifting Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

69 Weighted Case: Exact Lifting Consider a set S [n] with S = K The usual submodular inequality 0 + i S ix i w is a facet of conv(p {x : x i = 0 j S}) Compute coefficients of variables not in S by (sequence dependent) lifting Lifting problem is a concave minimization problem over a cardinality system.. solvable in O(n 3 ) [Atamturk and Naraynan 09, Onn 03] Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

70 Weighted Case: Exact Lifting Consider a set S [n] with S = K The usual submodular inequality 0 + i S ix i w is a facet of conv(p {x : x i = 0 j S}) Compute coefficients of variables not in S by (sequence dependent) lifting Lifting problem is a concave minimization problem over a cardinality system.. solvable in O(n 3 ) [Atamturk and Naraynan 09, Onn 03] Resulting (facet-defining) lifted inequality (LI): 0 + i S i x i + i S λ i x i w Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

71 Weighted Case: Approximate Lifting Sort x 1 x 2 x n Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

72 Weighted Case: Approximate Lifting Sort x 1 x 2 x n Let T i = argmax{a(t ) : T [i 1], T = K 1} for all i > K Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

73 Weighted Case: Approximate Lifting Sort x 1 x 2 x n Let T i = argmax{a(t ) : T [i 1], T = K 1} for all i > K Let γ i = f (a(t i {i})) f (a(t i )) for all i = K + 1,..., n Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

74 Weighted Case: Approximate Lifting Sort x 1 x 2 x n Let T i = argmax{a(t ) : T [i 1], T = K 1} for all i > K Let γ i = f (a(t i {i})) f (a(t i )) for all i = K + 1,..., n The approximate lifted inequality: 0 + is valid for conv(p X) K i x i + i=1 n i=k +1 γ i x i w Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

75 Weighted Case: Approximate Lifting Sort x 1 x 2 x n Let T i = argmax{a(t ) : T [i 1], T = K 1} for all i > K Let γ i = f (a(t i {i})) f (a(t i )) for all i = K + 1,..., n The approximate lifted inequality: 0 + is valid for conv(p X) K i x i + i=1 Proof: 0 γ i λ i for all i > K n i=k +1 γ i x i w Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

76 Weighted Case: Remarks on Approximate Lifting The approximate lifting inequality is a version of the unweighted inequality corresponding to i 0 = K 1 i γ i for all i > K with the inequality being strict when f is strictly monotone (e.g. square root) Can be computed in O(n log n) time Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

77 Computations: Instances Mean-risk Knapsack: min µ i x i + λ σi 2 x i : c x b, x {0, 1} n i i 1,080 unweighted and 1,080 weighted instances randomly generated (n = 50, 80, 100) Knapsack to Cardinality: Let c 1 c n and pick K such that c c K b and c c K +1 > b Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

78 Computations: Experiments Compare branch-and-cut time and nodes for Mixed Integer SOCP formulation (Replace x i by xi 2 ) [SCOP] SOCP + 5 rounds of usual submodular inequalities [Edmonds] SOCP + 5 rounds of exact separated inequalities in the unweighted case, or SOCP + 5 rounds of approximate lifted inequality in the weighted case [Lifted] Implemented in Python + Gurobi 5.6 Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

79 Computations: Performance profiles (Time) SOCP Edmond Lifted SOCP Edmond Lifted Unweighted Weighted Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

80 Computations: Averages SOCP Edmonds Lifted Unweighted Time: Nodes: 221, ,444 75,811 Weighted Time: Nodes: 59,779 34,366 27,928 Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

81 Concluding Remarks P = { (w, x) R {0, 1} n : w f (a x) } X = {x [0, 1] n : e x K } Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

82 Concluding Remarks P = { (w, x) R {0, 1} n : w f (a x) } X = {x [0, 1] n : e x K } Summary: Unweighted case (identical a i s): Complete description of conv(p X) Weighted case (general a i ): Family of facets/valid inequalities by lifting Computational results for mean-risk knapsack Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23

83 Concluding Remarks P = { (w, x) R {0, 1} n : w f (a x) } X = {x [0, 1] n : e x K } Summary: Unweighted case (identical a i s): Complete description of conv(p X) Weighted case (general a i ): Family of facets/valid inequalities by lifting Computational results for mean-risk knapsack Some open issues: Handle correlations x Σx by introducing new variables for cross terms Mixed integer setting Submodular relaxations of other classes of MINLP Ahmed and Yu Constrained Submodular Polyhedra MINLP / 23