The structure of the infinite models in integer programming

Transcription

1 The structure of the infinite models in integer programming January 10, 2017

2 Not Again! Well, Aussois is partly responsible... Some years ago:

3 Not Again! Well, Aussois is partly responsible... Some years ago: Tuesday, January 8, 2008 Chair: Laurence Wolsey 17:15-18:00 Jean-Philippe Richard Group Relaxations for Integer Programming 18:00-18:30 Santanu Dey Facets of High-Dimensional Infinite Group Problem

4 Not Again! Well, Aussois is partly responsible... Some years ago: Tuesday, January 8, 2008 Chair: Laurence Wolsey 17:15-18:00 Jean-Philippe Richard Group Relaxations for Integer Programming 18:00-18:30 Santanu Dey Facets of High-Dimensional Infinite Group Problem Jean-Philippe P. Richard, Santanu S. Dey The Group-Theoretic Approach in Mixed Integer Programming: Theory, Computation and Perspectives, Fifty years of Integer Programming : From early years to the state-of-the-art (M. Juenger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.)), December 2009 (Springer).

5 Not Again! Well, Aussois is partly responsible... Some years ago: Tuesday, January 8, 2008 Chair: Laurence Wolsey 17:15-18:00 Jean-Philippe Richard Group Relaxations for Integer Programming 18:00-18:30 Santanu Dey Facets of High-Dimensional Infinite Group Problem Jean-Philippe P. Richard, Santanu S. Dey The Group-Theoretic Approach in Mixed Integer Programming: Theory, Computation and Perspectives, Fifty years of Integer Programming : From early years to the state-of-the-art (M. Juenger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.)), December 2009 (Springer). a lot of research since then...

6 MIPs in tableau form b R n \Z n. x B + r R rs(r)+ p Ppy(p) = b, x B Z n,s(r) R +,y(p) Z +

7 MIPs in tableau form b R n \Z n. x B + r R rs(r)+ p Ppy(p) = b, x B Z n,s(r) R +,y(p) Z + rs(r)+ p Ppy(p) b +Z n, s(r) R +,y(p) Z + r R

8 MIPs in tableau form b R n \Z n. x B + r R rs(r)+ p Ppy(p) = b, x B Z n,s(r) R +,y(p) Z + rs(r)+ p Ppy(p) b +Z n, s(r) R +,y(p) Z + r R BFS: s(r) = y(p) = 0, x B = b. Want x B Z n

9 The mixed-integer model Mixed-integer infinite group relaxation b R n \Z n, s : R n R + and y : R n Z + M b = {s,y R (Rn ) ) + R(Rn + : rs(r)+ py(p) b +Z n }. r R n p R n R (Rn) is the set of finite support functions from R n to R. R (Rn ) +.

10 The mixed-integer model Mixed-integer infinite group relaxation b R n \Z n, s : R n R + and y : R n Z + M b = {s,y R (Rn ) ) + R(Rn + : rs(r)+ py(p) b +Z n }. r R n p R n R (Rn) is the set of finite support functions from R n to R. R (Rn ) +. (ψ,π,α), ψ,π : R n R, α R, { H ψ,π,α := (s,y) R (Rn) R (Rn) : } ψ(r)s(r)+ π(p)y(p) α r R n p R n (ψ,π,α) is a valid tuple (functions) for M b if M b H ψ,π,α. equivalently: conv(m b ). α { 1,0,1}.

11 A face: The pure integer model The pure integer infinite group relaxation y : R n Z +. I b = {y : (0,y) M b } = {y R (Rn ) + : p R n py(p) b +Z n }.

12 A face: The pure integer model The pure integer infinite group relaxation y : R n Z +. I b = {y : (0,y) M b } = {y R (Rn ) + : p R n py(p) b +Z n }. (π,α), π : R n R, α R, { H π,α := y R (Rn) : } π(p)y(p) α p R n (π,α) is a valid tuple for I b if I b H π,α.

13 Gomory functions n = 1, α = 1, very simple to describe (0 < b < 1). { r ψ(r) = b if r 0 r 1 b if r < 0 π(p) = { f(p) b 1 f(p) 1 b if f(p) b if f(p) > b

14 Gomory functions n = 1, α = 1, very simple to describe (0 < b < 1). { r ψ(r) = b if r 0 r 1 b if r < 0 π(p) = { f(p) b 1 f(p) 1 b if f(p) b if f(p) > b 1 1 b b 1

15 Gomory functions n = 1, α = 1, very simple to describe (0 < b < 1). { r ψ(r) = b if r 0 r 1 b if r < 0 π(p) = { f(p) b 1 f(p) 1 b if f(p) b if f(p) > b 1 1 b b 1 Subadditive if φ(r 1 )+φ(r 2 ) φ(r 1 +r 2 ), r 1,r 2 R n. (ψ, π)

16 Gomory functions n = 1, α = 1, very simple to describe (0 < b < 1). { r ψ(r) = b if r 0 r 1 b if r < 0 π(p) = { f(p) b 1 f(p) 1 b if f(p) b if f(p) > b 1 1 b b 1 Subadditive if φ(r 1 )+φ(r 2 ) φ(r 1 +r 2 ), r 1,r 2 R n. (ψ, π) Positively homogenous if φ(λr) = λφ(r) r R n and λ 0. (ψ)

17 Gomory functions n = 1, α = 1, very simple to describe (0 < b < 1). { r ψ(r) = b if r 0 r 1 b if r < 0 π(p) = { f(p) b 1 f(p) 1 b if f(p) b if f(p) > b 1 1 b b 1 Subadditive if φ(r 1 )+φ(r 2 ) φ(r 1 +r 2 ), r 1,r 2 R n. (ψ, π) Positively homogenous if φ(λr) = λφ(r) r R n and λ 0. (ψ) Sublinear subadditive + positive homogenous. (ψ)

18 Gomory functions n = 1, α = 1, very simple to describe (0 < b < 1). { r ψ(r) = b if r 0 r 1 b if r < 0 π(p) = { f(p) b 1 f(p) 1 b if f(p) b if f(p) > b 1 1 b b 1 Subadditive if φ(r 1 )+φ(r 2 ) φ(r 1 +r 2 ), r 1,r 2 R n. (ψ, π) Positively homogenous if φ(λr) = λφ(r) r R n and λ 0. (ψ) Sublinear subadditive + positive homogenous. (ψ) Periodic φ(r) = φ(r +z), r R n and z Z n.(π)

19 Gomory functions n = 1, α = 1, very simple to describe (0 < b < 1). { r ψ(r) = b if r 0 r 1 b if r < 0 π(p) = { f(p) b 1 f(p) 1 b if f(p) b if f(p) > b 1 1 b b 1 Subadditive if φ(r 1 )+φ(r 2 ) φ(r 1 +r 2 ), r 1,r 2 R n. (ψ, π) Positively homogenous if φ(λr) = λφ(r) r R n and λ 0. (ψ) Sublinear subadditive + positive homogenous. (ψ) Periodic φ(r) = φ(r +z), r R n and z Z n.(π) Symmetry condition φ satisfies φ(r)+φ(b r) = 1. (π)

20 Attractiveness of valid functions: Plug and play 3.6s s 2 +.2y 1.2y 2 =.4+Z s 1,s 2 R +,y 1,y 2 Z +

21 Attractiveness of valid functions: Plug and play 3.6s s 2 +.2y 1.2y 2 =.4+Z s 1,s 2 R +,y 1,y 2 Z +...not so easy as it seems, but attractive nevertheless.

22 Attractiveness of valid functions: Plug and play 3.6s s 2 +.2y 1.2y 2 =.4+Z s 1,s 2 R +,y 1,y 2 Z +...not so easy as it seems, but attractive nevertheless....library of useful functions in IP solvers.

23 What are the important functions? What should we put in our library?

24 What are the important functions? What should we put in our library? Nontrivial: Not valid for R (Rn ) ) + R(Rn +. Minimal: Undominated in R (Rn ) ) + R(Rn +. Extreme: Later. Facet: Later.

25 What are the important functions? What should we put in our library? Nontrivial: Not valid for R (Rn ) ) + R(Rn +. Minimal: Undominated in R (Rn ) ) + R(Rn +. Extreme: Later. Facet: Later. A (seemingly) technical detour: π 0 Why? There are minimal functions π 0, but they are pathological: Every disc contains some (x,f(x)).

26 What are the important functions? What should we put in our library? Nontrivial: Not valid for R (Rn ) ) + R(Rn +. Minimal: Undominated in R (Rn ) ) + R(Rn +. Extreme: Later. Facet: Later. A (seemingly) technical detour: π 0 Why? There are minimal functions π 0, but they are pathological: Every disc contains some (x,f(x))....but ignorance should not be an excuse... We try to answer this.

27 What are the important functions? Theorem (Yildiz and Cornuéjols, related to Johnson.) Let ψ : R n R, π : R n R be any functions, and α { 1,0,1}. Then (ψ,π,α) is a nontrivial minimal valid tuple for M b if and only if: π is subadditive; π(ǫr) ψ(r) = sup ǫ>0 ǫ = lim π(ǫr) ǫ 0 + ǫ = limsup π(ǫr) ǫ 0 + ǫ every r R n ; sublinear π is Lipschitz continuous with Lipschitz constant L := max r =1 ψ(r); π 0, π(z) = 0 for every z Z n, and α = 1; π satisfies the symmetry condition (and is periodic). for

28 (One of our) Goal(s) Q b = R (Rn ) ) + R(Rn + = R (Rn ) ) + R(Rn + H ψ,π,α (ψ,π,α) valid (ψ,π,α) minimal, nontrivial H ψ,π,α

29 (One of our) Goal(s) Q b = R (Rn ) ) + R(Rn + = R (Rn ) ) + R(Rn + H ψ,π,α (ψ,π,α) valid (ψ,π,α) minimal, nontrivial H ψ,π,α Clearly conv(m b ) Q b. But what is Q b?

30 Norms, closed sets... While in finite dimensions all norms are equivalent to the Euclidean norm, In infinite dimensions this is not so... Norm on R (Rn) R (Rn) (BCCZ): (s,y) := s(0) + r s(r) + y(0) + p y(p) r R n p R n

31 Norms, closed sets... While in finite dimensions all norms are equivalent to the Euclidean norm, In infinite dimensions this is not so... Norm on R (Rn) R (Rn) (BCCZ): (s,y) := s(0) + r s(r) + y(0) + p y(p) r R n p R n Theorem Under the topology induced by (, ), Q b = cl(conv(m b )) = conv(m b )+R (Rn ) + R(Rn ) +.

32 Norms, closed sets... While in finite dimensions all norms are equivalent to the Euclidean norm, In infinite dimensions this is not so... Norm on R (Rn) R (Rn) (BCCZ): (s,y) := s(0) + r s(r) + y(0) + p y(p) r R n p R n Theorem Under the topology induced by (, ), Q b = cl(conv(m b )) = conv(m b )+R (Rn ) + R(Rn ) +. conv(m b ) conv(m b )+R (Rn ) + R(Rn )

33 Liftable functions G b = {y R (Rn) : (0,y) Q b } (Minimal, nontrivial) (π,α) valid for I b liftable if ψ s.t. (ψ,π,α) (Minimal, nontrivial) valid for M b. G b = R (Rn ) + {H π,α : (π,α) minimal nontrivial liftable tuple}.

34 Liftable functions G b = {y R (Rn) : (0,y) Q b } (Minimal, nontrivial) (π,α) valid for I b liftable if ψ s.t. (ψ,π,α) (Minimal, nontrivial) valid for M b. G b = R (Rn ) + {H π,α : (π,α) minimal nontrivial liftable tuple}. Theorem G b = cl(conv(i b )) = conv(i b )+R (Rn ) +.

35 Liftable functions G b = {y R (Rn) : (0,y) Q b } (Minimal, nontrivial) (π,α) valid for I b liftable if ψ s.t. (ψ,π,α) (Minimal, nontrivial) valid for M b. G b = R (Rn ) + {H π,α : (π,α) minimal nontrivial liftable tuple}. Theorem G b = cl(conv(i b )) = conv(i b )+R (Rn ) +. conv(i b ) conv(i b )+R (Rn ) +

36 Canonical faces, Finite faces Ultimately we want valid inequalities for Integer Programs (with rational data). Given R,P R n, let V R,P = { (s,y) R (Rn) R (Rn) : s(r) = 0 r R, y(p) = 0 p P }. A canonical face of conv(m b ) is F = conv(m b ) V R,P. When R, P finite, F is a finite canonical face.

37 Canonical faces, Finite faces Ultimately we want valid inequalities for Integer Programs (with rational data). Given R,P R n, let V R,P = { (s,y) R (Rn) R (Rn) : s(r) = 0 r R, y(p) = 0 p P }. A canonical face of conv(m b ) is F = conv(m b ) V R,P. When R, P finite, F is a finite canonical face. Notice: conv(m b ) = F finite canonical face

38 Canonical faces, Finite faces Ultimately we want valid inequalities for Integer Programs (with rational data). Given R,P R n, let V R,P = { (s,y) R (Rn) R (Rn) : s(r) = 0 r R, y(p) = 0 p P }. A canonical face of conv(m b ) is F = conv(m b ) V R,P. When R, P finite, F is a finite canonical face. Notice: conv(m b ) = F finite canonical face Same for conv(i b ). Finite canonical faces of conv(i b ) are the corner polyhedra.

39 Rational finite faces What happens when R, P Q n? Theorem Let R,P Q n. Then conv(m b ) V R,P = cl(conv(m b )) V R,P. Let P Q n. Then conv(i b ) V P = cl(conv(i b )) V P. Corollary The restrictions of the minimal, nontrivial valid tuples give all the (nontrivial) facets of rational mixed-integer polyhedra. The restrictions of the minimal, nontrivial liftable functions give all the (nontrivial) facets of rational corner polyhedra,

40 Extreme functions and facets π 0 extreme if (π,1) valid and π 1 = π 2 = π for every π 1 0, π 2 0 such that (π 1,1) (π 2,1) valid and π = 1 2 π π 2. π 0 facet if (π,1) valid and π 1 = π for every π 1 0, such that H π1,1 I b H π1,1 I b. π facet π extreme. The converse is not known. Köppe and Zhou: Coincide for the case of continuous piecewise linear functions On the notions of facets, weak facets, and extreme functions of the Gomory-Johnson infinite group problem n = 1, software to test extremality of piecewise linear functions.

41 Discontinuous extreme functions (n = 1) Dey, Richard, Li and Miller: The following function is extreme: n = 1, 0 < b < 1 2, π : π(r) = { r b 0 r b r 1+b b < r < 1 1 b 1

42 More discontinuous functions (n = 1) Letchford, Lodi Strong fractional functions Minimal, dominate fractional functions. Dash, Günlük Extended two-step MIR (mixed integer rounding) functions. limit of sequences of two-step MIR functions, dominate LetcLo. Hildebrand, two-sided discontinuous at the origin with 1 or 2 slopes, extreme Köppe, Zhou: Extreme functions that are continuous but not Lipschitz continuous. see Köppe, Zhou Equivariant perturbation in Gomory and Johnson s infinite group problem. vi. the curious case of two-sided discontinuous functions.

43 Not all extreme functions are needed Summarizing our results: cl(conv(i b )) = conv(i b )+R (Rn ) + = = R (Rn ) + {H π,α : (π,α) minimal nontrivial liftable tuple}. When P Q n, we have that cl(conv(i b )) V P = conv(i b ) V P (π,α) minimal nontrivial liftable tuple: π 0, α = 1, π Lipschitz continuous. ONLY THESE ARE NEEDED

44 Cauchy equation and Hamel bases The Cauchy functional equation in R n : θ(u)+θ(v) = θ(u +v) for all u,v R n. (subadditivity) θ(x) = c T x is obviously a solution to the equation. A Hamel basis B in a basis of R n over the field Q. i.e. a subset of R n s.t. x R n, there exists a unique finite subset {β 1,...,β t } B and λ 1,...,λ t Q such that x = t i=1 λ iβ i. (axiom of choice).

45 Cauchy equation and Hamel bases II For β B, let c(β) R be a real number. Define θ as:...θ solves the Cauchy equation. θ(x) = t i=1 λ ic(β i ). Theorem Let B a Hamel basis of R n. Then every solution to the Cauchy equation is of this form.

46 The affine hull of conv(i b ) Theorem (Basu, Hildebrand Köppe) The affine hull of conv(i b ) is described by the equations p Rn θ(p)y(p) = θ(b) for all solutions θ : R n R of the Cauchy equation such that θ(p) = 0 for every p Q n. Extreme functions (without π 0) do not exist... aff(cl(conv(i b ))) = R (Rn ) aff(conv(m b )) = aff(cl(conv(m b ))) = R (Rn) R (Rn).

47 Every valid function is nonnegative. Theorem For every valid tuple (π,α) for I b, there exists a unique solution of the Cauchy equation θ : R n R such that θ(p) = 0 for every p Q n and the valid tuple (π,α ) = (π +θ,α+θ(b)) satisfies π 0. Nonnegative valid functions form a compact, convex set. Its extreme points are the extreme functions and suffice to describe this set. (...but not all of them are necessary)

48 Finite faces and recession cones 2y1.2y 2 +(1 2)y 3.4+Z y 1,y 2,y 3 Z + y 1 = y 3, (1,0,1)

49 Finite faces and recession cones 2y1.2y 2 +(1 2)y 3.4+Z y 1,y 2,y 3 Z + y 1 = y 3, (1,0,1) L the linear space parallel to aff(conv(i b )) Theorem For every P R n finite: the face C P = conv(i b ) V P is a rational polyhedron in R P ; every extreme ray of C P is spanned by some r Z P + such that p P pr(p) Zn ; rec(c P ) = (L V P ) R P +.

50 Finite faces and recession cones Theorem There are finite canonical faces of conv(m b ) that are not closed. All the finite canonical faces of conv(i b ) are rational polyhedra.

51 More work? n = 1: EVERY EXTREME FUNCTION π 0 IS NICE. Gomory Johnson: Every extreme function is piecewise linear. NO Basu, Conforti, Cornuejols, Zambelli.

52 More work? n = 1: EVERY EXTREME FUNCTION π 0 IS NICE. Gomory Johnson: Every extreme function is piecewise linear. NO Basu, Conforti, Cornuejols, Zambelli. Dey and Richard Aussois 2008 Construct extreme functions that are piecewise linear and have > 4 slopes. YES Hildebrand (2013) 6 Köppe and Zhou (2015) 28. Computer search. BCDP (2015) For every k there exists an extreme function that is piecewise linear with k slopes. The pointwise limit of this sequence is extreme with slopes.

53 More work? n = 1: EVERY EXTREME FUNCTION π 0 IS NICE. Gomory Johnson: Every extreme function is piecewise linear. NO Basu, Conforti, Cornuejols, Zambelli. Dey and Richard Aussois 2008 Construct extreme functions that are piecewise linear and have > 4 slopes. YES Hildebrand (2013) 6 Köppe and Zhou (2015) 28. Computer search. BCDP (2015) For every k there exists an extreme function that is piecewise linear with k slopes. The pointwise limit of this sequence is extreme with slopes. Is every bad function (discontinuous, non piecewise linear, slopes) the pointwise limit of a sequence of good functions?

54 Maybe nice functions suffice... Is every facet of conv(i b ) V P, P finite (rational) the restriction of a piecewise linear function?

55 Maybe nice functions suffice... Is every facet of conv(i b ) V P, P finite (rational) the restriction of a piecewise linear function? conv(i b ) = cl(conv(i b )) aff(conv(i b ))?

56 THANK YOU FOR YOUR ATTENTION

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