Lifts of convex sets and cone factorizations
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1 Lifts of convex sets and cone factorizations João Gouveia Universidade de Coimbra 20 Dec CORE - Université Catholique de Louvain with Pablo Parrilo (MIT) and Rekha Thomas (U.Washington)
2 Lifts of Polytopes What is a hard domain to do linear programming in?
3 Lifts of Polytopes What is a hard domain to do linear programming in? First guess: a polytope with many vertices and facets.
4 Lifts of Polytopes What is a hard domain to do linear programming in? First guess: a polytope with many vertices and facets. However, polytopes with many facets can be projections of much simpler polytopes.
5 Lifts of Polytopes What is a hard domain to do linear programming in? First guess: a polytope with many vertices and facets. However, polytopes with many facets can be projections of much simpler polytopes. An example is the Parity Polytope: PP n = conv({x {0, 1} n : x has odd number of 1}).
6 Lifts of Polytopes What is a hard domain to do linear programming in? First guess: a polytope with many vertices and facets. However, polytopes with many facets can be projections of much simpler polytopes. An example is the Parity Polytope: PP n = conv({x {0, 1} n : x has odd number of 1}). For every even set A {1,..., n}, x i x i A 1 i A i A is a facet, so we have at least 2 n 1 facets.
7 Parity Polytope There is a much shorter description.
8 Parity Polytope There is a much shorter description. PP n is the set of x R n such that there exists for every odd 1 k n a vector z k R n and a real number α k such that
9 Parity Polytope There is a much shorter description. PP n is the set of x R n such that there exists for every odd 1 k n a vector z k R n and a real number α k such that k z k = x;
10 Parity Polytope There is a much shorter description. PP n is the set of x R n such that there exists for every odd 1 k n a vector z k R n and a real number α k such that k z k = x; k α k = 1;
11 Parity Polytope There is a much shorter description. PP n is the set of x R n such that there exists for every odd 1 k n a vector z k R n and a real number α k such that k z k = x; k α k = 1; z k 1 = k α k ;
12 Parity Polytope There is a much shorter description. PP n is the set of x R n such that there exists for every odd 1 k n a vector z k R n and a real number α k such that k z k = x; k α k = 1; z k 1 = k α k ; 0 ( z k ) i α k.
13 Parity Polytope There is a much shorter description. PP n is the set of x R n such that there exists for every odd 1 k n a vector z k R n and a real number α k such that k z k = x; k α k = 1; z k 1 = k α k ; 0 ( z k ) i α k. O(n 2 ) variables and O(n 2 ) constraints.
14 Motivation Polytopes with many facets can be projections of much simpler polytopes.
15 Motivation Polytopes with many facets can be projections of much simpler polytopes. Canonical LP Lift Given a polytope P, a canonical LP lift is a description P = Φ(R k + L) for some affine space L and affine map Φ. We say it is a R k +-lift.
16 Motivation Polytopes with many facets can be projections of much simpler polytopes. Canonical LP Lift Given a polytope P, a canonical LP lift is a description P = Φ(R k + L) for some affine space L and affine map Φ. We say it is a R k +-lift. The smallest k such that P has a R k +-lift is a much better measure of LP-complexity than number of facets and vertices.
17 Two definitions Let P be a polytope with facets defined by h 1 (x) 0,..., h f (x) 0, and vertices p 1,..., p v.
18 Two definitions Let P be a polytope with facets defined by h 1 (x) 0,..., h f (x) 0, and vertices p 1,..., p v. Slack Matrix The slack matrix of P is the matrix S P R v f defined by S P (i, j) = h j (p i ).
19 Two definitions Let P be a polytope with facets defined by h 1 (x) 0,..., h f (x) 0, and vertices p 1,..., p v. Slack Matrix The slack matrix of P is the matrix S P R v f defined by S P (i, j) = h j (p i ). Nonnegative Factorization Given a nonnegative matrix M R n m + we say that it has a k-nonnegative factorization, or a R k +-factorization if there exist matrices A R n k + and B Rk m + such that M = A B.
20 Yannakakis Theorem Theorem (Yannakakis 1991) A polytope P has a R k +-lift if and only if S P has a R k +-factorization.
21 Yannakakis Theorem Theorem (Yannakakis 1991) A polytope P has a R k +-lift if and only if S P has a R k +-factorization. Our questions: Does it work for other types of lifts?
22 Yannakakis Theorem Theorem (Yannakakis 1991) A polytope P has a R k +-lift if and only if S P has a R k +-factorization. Our questions: Does it work for other types of lifts? Does it work for other types of convex sets?
23 Yannakakis Theorem Theorem (Yannakakis 1991) A polytope P has a R k +-lift if and only if S P has a R k +-factorization. Our questions: Does it work for other types of lifts? Does it work for other types of convex sets? Can we include symmetry in the result?
24 The Hexagon Consider the regular hexagon.
25 The Hexagon Consider the regular hexagon.
26 The Hexagon Consider the regular hexagon. It has a 6 6 slack matrix S H.
27 The Hexagon Consider the regular hexagon. It has a 6 6 slack matrix S H
28 The Hexagon Consider the regular hexagon. It has a 6 6 slack matrix S H =
29 Hexagon - continued It is the projection of the slice of R 5 + cut out by y 1 + y 2 + y 3 + y 5 = 2, y 3 + y 4 + y 5 = 1.
30 Hexagon - continued It is the projection of the slice of R 5 + cut out by y 1 + y 2 + y 3 + y 5 = 2, y 3 + y 4 + y 5 = 1.
31 Hexagon - continued It is the projection of the slice of R 5 + cut out by y 1 + y 2 + y 3 + y 5 = 2, y 3 + y 4 + y 5 = 1. For irregular hexagons a R 6 + -lift is the only we can have.
32 Generalizing to non-lp We want to generalize this result to other types of lifts.
33 Generalizing to non-lp We want to generalize this result to other types of lifts. K -Lift Given a polytope P, and a closed convex cone K, a K -lift of P is a description P = Φ(K L) for some affine space L and affine map Φ.
34 Generalizing to non-lp We want to generalize this result to other types of lifts. K -Lift Given a polytope P, and a closed convex cone K, a K -lift of P is a description P = Φ(K L) for some affine space L and affine map Φ. Important cases are R n +, PSD n, SOCP n, CP n, CoP n,...
35 Generalizing to non-lp We want to generalize this result to other types of lifts. K -Lift Given a polytope P, and a closed convex cone K, a K -lift of P is a description P = Φ(K L) for some affine space L and affine map Φ. Important cases are R n +, PSD n, SOCP n, CP n, CoP n,... We also need to generalize the nonnegative factorizations.
36 K -factorizations Recall that if K R l is a closed convex cone, K R l is its dual cone, defined by K = {y R l y, x 0, x K }.
37 K -factorizations Recall that if K R l is a closed convex cone, K R l is its dual cone, defined by K = {y R l y, x 0, x K }. K -Factorization Given a nonnegative matrix M R n m + we say that it has a K -factorization if there exist a 1,... a n K and b 1,..., b m K such that M i,j = a i, b j.
38 K -factorizations Recall that if K R l is a closed convex cone, K R l is its dual cone, defined by K = {y R l y, x 0, x K }. K -Factorization Given a nonnegative matrix M R n m + we say that it has a K -factorization if there exist a 1,... a n K and b 1,..., b m K such that M i,j = a i, b j. We can now generalize Yannakakis. Theorem (G-Parrilo-Thomas) A polytope P has a K -lift if and only if S P has a K -factorization.
39 The Square The 0/1 square is the projection onto x and y of the slice of PSD 3 given by 1 x y x x z y z y 0.
40 The Square The 0/1 square is the projection onto x and y of the slice of PSD 3 given by 1 x y x x z y z y 0.
41 The Square The 0/1 square is the projection onto x and y of the slice of PSD 3 given by 1 x y x x z y z y 0. Its slack matrix is given by S P = and should factorize in PSD ,
42 Square - continued is factorized by S P = , , , , , for the rows and , for the columns , , ,
43 Slack Operator To further generalize Yannakakis to other convex sets, we have to introduce a slack operator.
44 Slack Operator To further generalize Yannakakis to other convex sets, we have to introduce a slack operator. Given a convex set C R n, consider its polar set C = {x R n : x, y 1, y C},
45 Slack Operator To further generalize Yannakakis to other convex sets, we have to introduce a slack operator. Given a convex set C R n, consider its polar set C = {x R n : x, y 1, y C}, and define the slack operator S C : ext(c) ext(c ) R + as S C (x, y) = 1 x, y.
46 Slack Operator To further generalize Yannakakis to other convex sets, we have to introduce a slack operator. Given a convex set C R n, consider its polar set C = {x R n : x, y 1, y C}, and define the slack operator S C : ext(c) ext(c ) R + as S C (x, y) = 1 x, y. Note that this generalizes the slack matrix.
47 Generalized Yannakakis for convex sets We can then define a K -factorization of S C as a pair of maps A : ext(c) K B : ext(c ) K such that for all x, y. A(x), B(y) = S C (x, y)
48 Generalized Yannakakis for convex sets We can then define a K -factorization of S C as a pair of maps A : ext(c) K B : ext(c ) K such that for all x, y. A(x), B(y) = S C (x, y) Theorem (G-Parrilo-Thomas) A convex set C has a K -lift if and only if S C has a K -factorization.
49 The Disk The unit disk D is the projection onto x and y of the slice of PSD 2 given by [ ] 1 + x y 0. y 1 x
50 The Disk The unit disk D is the projection onto x and y of the slice of PSD 2 given by [ ] 1 + x y 0. y 1 x D = D, there must be A : S 1 PSD 2 and B : S 1 PSD 2 such that A(x), B(y) = 1 x, y
![The Disk The unit disk D is the projection onto x and y of the slice of PSD 2 given by [ ] 1 + x y 0.](/docs-images/84/89523380/images/51-0.jpg "y 1 x D = D, there must be A : S 1 PSD 2 and B : S 1 PSD 2 such that A(x), B(y) = 1 x, y [ 1 + x y A(x, y) = y 1 x ] [ 1 x")
51 The Disk The unit disk D is the projection onto x and y of the slice of PSD 2 given by [ ] 1 + x y 0. y 1 x D = D, there must be A : S 1 PSD 2 and B : S 1 PSD 2 such that A(x), B(y) = 1 x, y [ 1 + x y A(x, y) = y 1 x ] [ 1 x y, B(x, y) = y 1 + x ].
52 Cone ranks of polytopes Recall - rank + (M) is the smallest k such that M has an R k +-factorization. rank + (P) := rank + (S P )
53 Cone ranks of polytopes Recall - rank + (M) is the smallest k such that M has an R k +-factorization. rank + (P) := rank + (S P ) Given K = {K 1, K 2, }, (e.g. R k +, PSD k, CP k, CoP k,... ) rank K (M) is the smallest i such that M has a K i -factorization.
54 Cone ranks of polytopes Recall - rank + (M) is the smallest k such that M has an R k +-factorization. rank + (P) := rank + (S P ) Given K = {K 1, K 2, }, (e.g. R k +, PSD k, CP k, CoP k,... ) rank K (M) is the smallest i such that M has a K i -factorization. Again rank K (P) := rank K (S P ).
55 Cone ranks of polytopes Recall - rank + (M) is the smallest k such that M has an R k +-factorization. rank + (P) := rank + (S P ) Given K = {K 1, K 2, }, (e.g. R k +, PSD k, CP k, CoP k,... ) rank K (M) is the smallest i such that M has a K i -factorization. Again rank K (P) := rank K (S P ). We are specially interested in rank psd (M).
56 Bounds for matrices For M R p q +. rank(m) rank + (M) min{p, q}.
57 Bounds for matrices For M R p q +. rank(m) rank + (M) min{p, q}. rank(m) ( rank psd (M)+1) 2.
58 Bounds for matrices For M R p q +. rank(m) rank + (M) min{p, q}. rank(m) ( rank psd (M)+1). rank psd (M) rank + (M). 2
59 Bounds for matrices For M R p q +. rank(m) rank + (M) min{p, q}. rank(m) ( rank psd (M)+1). rank psd (M) rank + (M). 2 Proposition If M R n n + is zero on the diagonal and positive everywhere else then rank + (M) k, where k is the smallest integer such that n ( k k/2 ).
60 Bounds for matrices - 2 Proposition If M R p q has rank k, then the matrix M obtained by squaring each entry of M has psd-rank at most k.
61 Bounds for matrices - 2 Proposition If M R p q has rank k, then the matrix M obtained by squaring each entry of M has psd-rank at most k. Consider the matrix A R n n defined by a i,j = (i j) 2. rank(a) = 3;
62 Bounds for matrices - 2 Proposition If M R p q has rank k, then the matrix M obtained by squaring each entry of M has psd-rank at most k. Consider the matrix A R n n defined by a i,j = (i j) 2. rank(a) = 3; rank psd (A) = 2;
63 Bounds for matrices - 2 Proposition If M R p q has rank k, then the matrix M obtained by squaring each entry of M has psd-rank at most k. Consider the matrix A R n n defined by a i,j = (i j) 2. rank(a) = 3; rank psd (A) = 2; rank + (A) log 2 (n) grows with n.
64 Bounds for matrices - 2 Proposition If M R p q has rank k, then the matrix M obtained by squaring each entry of M has psd-rank at most k. Consider the matrix A R n n defined by a i,j = (i j) 2. rank(a) = 3; rank psd (A) = 2; rank + (A) log 2 (n) grows with n. rank + can be arbitrarily larger than rank and rank psd.
65 Bounds for polytopes - LP Proposition An R k +-lift of P induces an embedding from the lattice of faces of P, L(P), to the boolean lattice 2 [k]. In particular:
66 Bounds for polytopes - LP Proposition An R k +-lift of P induces an embedding from the lattice of faces of P, L(P), to the boolean lattice 2 [k]. In particular: If p is the size of the largest antichain in L(P), then rank + (P) k where k is the smallest integer such that p ( k k/2 ).
67 Bounds for polytopes - LP Proposition An R k +-lift of P induces an embedding from the lattice of faces of P, L(P), to the boolean lattice 2 [k]. In particular: If p is the size of the largest antichain in L(P), then rank + (P) k where k is the smallest integer such that p ( k k/2 ). [Goemans] If n P is the number of faces of P, rank + (P) log 2 (n P ).
68 Bounds for polytopes - LP Proposition An R k +-lift of P induces an embedding from the lattice of faces of P, L(P), to the boolean lattice 2 [k]. In particular: If p is the size of the largest antichain in L(P), then rank + (P) k where k is the smallest integer such that p ( k k/2 ). [Goemans] If n P is the number of faces of P, rank + (P) log 2 (n P ). P = 3-cube: rank + (P) 6.
69 Bounds for polytopes - LP Proposition An R k +-lift of P induces an embedding from the lattice of faces of P, L(P), to the boolean lattice 2 [k]. In particular: If p is the size of the largest antichain in L(P), then rank + (P) k where k is the smallest integer such that p ( k k/2 ). [Goemans] If n P is the number of faces of P, rank + (P) log 2 (n P ). P = 3-cube: rank + (P) 6. n P = 28 rank + (P) log 2 (28)
70 Bounds for polytopes - LP Proposition An R k +-lift of P induces an embedding from the lattice of faces of P, L(P), to the boolean lattice 2 [k]. In particular: If p is the size of the largest antichain in L(P), then rank + (P) k where k is the smallest integer such that p ( k k/2 ). [Goemans] If n P is the number of faces of P, rank + (P) log 2 (n P ). P = 3-cube: rank + (P) 6. n P = 28 rank + (P) log 2 (28) n edges = 12, ( 5 2) = 10, ( 6 3) = 20, hence rank+ (P) 6.
71 Bounds for polytopes - SDP Theorem If a polytope P in R n has rank psd = k than it has at most k O(k 2 n) facets.
72 Bounds for polytopes - SDP Theorem If a polytope P in R n has rank psd = k than it has at most k O(k 2 n) facets. For P n = n-gon, rank + (P n ) and rank psd (P n ) grow to infinity as n grows, despite rank(s Pn ) = 3. Open questions:
73 Bounds for polytopes - SDP Theorem If a polytope P in R n has rank psd = k than it has at most k O(k 2 n) facets. For P n = n-gon, rank + (P n ) and rank psd (P n ) grow to infinity as n grows, despite rank(s Pn ) = 3. Open questions: Can we find a separation between rank psd and rank + for polytopes?
74 Bounds for polytopes - SDP Theorem If a polytope P in R n has rank psd = k than it has at most k O(k 2 n) facets. For P n = n-gon, rank + (P n ) and rank psd (P n ) grow to infinity as n grows, despite rank(s Pn ) = 3. Open questions: Can we find a separation between rank psd and rank + for polytopes? Recently, [Fiorini-Massar-Pokutta-Tiwary-de Wolf] proved rank + (TSP) grows exponentially. What about rank psd?
75 Symmetric Lifts In the LP case there has been much interest in symmetric lifts. [Kaibel-Pashkovich-Theis] Symmetric lifts Let P be a polytope and P = Φ(K L) a lift of P. We say the lift is symmetric if there exists a group homomorphism sending g Aut(P) to ψ g Aut(K ) such that ψ g (L) = L and Φ ψ g = g. Symmetric lift preserves symmetries of the lifted objects.
76 Symmetric Lifts In the LP case there has been much interest in symmetric lifts. [Kaibel-Pashkovich-Theis] Symmetric lifts Let P be a polytope and P = Φ(K L) a lift of P. We say the lift is symmetric if there exists a group homomorphism sending g Aut(P) to ψ g Aut(K ) such that ψ g (L) = L and Φ ψ g = g. Symmetric lift preserves symmetries of the lifted objects. Common lift-and-project methods are symmetric (w.r.t. permutation of variables): LS, SA, Las...
77 Example:The square Recall the lift of the 0/1 square [ 1 x y x x z y z y ] 0.
78 Example:The square Recall the lift of the 0/1 square [ 1 x y x x z y z y ] 0. Aut(g) = g(x, y) = (y, x), h(x, y) = (1 x, y).
79 Example:The square Recall the lift of the 0/1 square [ 1 x y x x z y z y ] 0. Aut(g) = g(x, y) = (y, x), h(x, y) = (1 x, y). φ h (A) = φ g (A) = [ 1 1 x y 1 x 1 x y z y y z y [ 1 y x y y z x z x ] = ] = P 23 AP 23, [ ] [ A ], generate a homomorphism so the lift is symmetric.
80 Example 2 : Regular n-gons Proposition For p prime the smallest k for which there exists a symmetric R k +-lift of the p-gon is p.
81 Example 2 : Regular n-gons Proposition For p prime the smallest k for which there exists a symmetric R k +-lift of the p-gon is p. Note that we know that there are actually O(log(n)) dimensional lifts of these polytopes [Ben-Tal, Nemirovski].
82 Example 2 : Regular n-gons Proposition For p prime the smallest k for which there exists a symmetric R k +-lift of the p-gon is p. Note that we know that there are actually O(log(n)) dimensional lifts of these polytopes [Ben-Tal, Nemirovski]. Open (small) problem: prove that the smallest symmetric lift of an n-gon is to R n +.
83 Symmetric Yannakakis K -Factorization Given a polytope P and its slack matrix S P R n m + and its K -factorization given by a 1,... a n K, b 1,..., b m K, we say that it is symmetric if there is an homomorphism φ : Aut(P) Aut(K ) such that if g send the i-th vertex to the j-th vertex, φ(a i ) = a j.
84 Symmetric Yannakakis K -Factorization Given a polytope P and its slack matrix S P R n m + and its K -factorization given by a 1,... a n K, b 1,..., b m K, we say that it is symmetric if there is an homomorphism φ : Aut(P) Aut(K ) such that if g send the i-th vertex to the j-th vertex, φ(a i ) = a j. Theorem (G-Parrilo-Thomas) A convex set C has a symmetric K -lift if and only if S C has a symmetric K -factorization.
85 Matchings Matchings Given the complete graph K n = ([n], E n ) a matching is a collection M of edges such that there s one and only one edge incident to each vertex.
86 Matchings Matchings Given the complete graph K n = ([n], E n ) a matching is a collection M of edges such that there s one and only one edge incident to each vertex.
87 Matchings Matchings Given the complete graph K n = ([n], E n ) a matching is a collection M of edges such that there s one and only one edge incident to each vertex.
88 Matchings Matchings Given the complete graph K n = ([n], E n ) a matching is a collection M of edges such that there s one and only one edge incident to each vertex. χ M {0, 1} En is the indicator vector of M. For this example χ M = (0, 0, 0, 0, 1, 1).
89 The matching Polytope MaxMatch Given a complete graph K 2n with edge weights ω : E n R, find the matching with maximum weight.
90 The matching Polytope MaxMatch Given a complete graph K 2n with edge weights ω : E n R, find the matching with maximum weight. This has a geometrical version. MaxMatch Maximize ω, x over the polytope conv({χ M : M is a matching}).
91 The matching Polytope MaxMatch Given a complete graph K 2n with edge weights ω : E n R, find the matching with maximum weight. This has a geometrical version. MaxMatch Maximize ω, x over the polytope conv({χ M : M is a matching}). This polytope is the Matching Polytope, denoted PMatch 2n.
92 Symmetric lifts of matching polytope Yannakakis Although the max-matching problem is polynomial time solvable, there is no polynomial size linear symmetric lift for the matching polytope.
93 Symmetric lifts of matching polytope Yannakakis Although the max-matching problem is polynomial time solvable, there is no polynomial size linear symmetric lift for the matching polytope. What about non-symmetric?
94 Symmetric lifts of matching polytope Yannakakis Although the max-matching problem is polynomial time solvable, there is no polynomial size linear symmetric lift for the matching polytope. What about non-symmetric? With other versions of the matching polytope, Kaibel, Pashkovich and Theis show that symmetry does matter, but the general question is still open.
95 Symmetry in SDP Further thoughts:
96 Symmetry in SDP Further thoughts: SDP lift-and-project algorithms don t work polynomially for matchings [Tuncel].
97 Symmetry in SDP Further thoughts: SDP lift-and-project algorithms don t work polynomially for matchings [Tuncel]. Do all polynomial sized symmetric SDP lifts fail?
98 Symmetry in SDP Further thoughts: SDP lift-and-project algorithms don t work polynomially for matchings [Tuncel]. Do all polynomial sized symmetric SDP lifts fail? What about non-symmetric?
99 The end Thank You
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