Extension complexity bounds for polygons Numerical factorizations - PowerPoint PPT Presentation

Extension complexity bounds for polygons Numerical factorizations and conjectures François Glineur Université catholique de Louvain (UCLouvain) Center for Operations Research and Econometrics and Information and Communication Technologies, Electronics and Applied Mathematics Institute joint work with Nicolas Gillis (UMons), Arnaud Vandaele (UMons) and Julien Dewez (UCLouvain) Limitations of convex programming: lower bounds on extended formulations and factorization ranks Dagstuhl , February 20 2015 1

Main message Nonnegative factorizations of slack matrices computed numerically can be useful to prove upper bounds and infer exact values for the extension complexity of (small) polytopes 2

Outline Numerical factorization of nonnegative matrices Extension complexity of the correlation polytope Cartesian products and extension complexity Worst-case extension complexity of polygons An explicit factorization for regular n -gons Lower bounds on extension complexity Geometric lower bound on extension complexity Computable bounds for regular n -gons Positive semidefinite rank This work/talk has mostly a computational focus 3

Link between extension and nonnegative factorization Given a polytope P and its slack matrix S Extension complexity of P = rank + ( S ) [Yannanakis, 1991]

Link between extension and nonnegative factorization Given a polytope P and its slack matrix S Extension complexity of P = rank + ( S ) [Yannanakis, 1991] ≡ extension S = UV T S ≥ 0 P U ∈ R m × r with r facets + V ∈ R n × r + 4

Link between extension and nonnegative factorization Given a polytope P and its slack matrix S Extension complexity of P = rank + ( S ) [Yannanakis, 1991] ≡ extension S = UV T S ≥ 0 P U ∈ R m × r with r facets + V ∈ R n × r + Moreover any slack matrix factorization provides an explicit extended formulation (with some redundant equalities) P = { x | Ax ≤ b } = { x | Ax + Uy = b and y ≥ 0 } i.e. columns of U generate the cone of slack vectors 4

Link between extension and nonnegative factorization Given a polytope P and its slack matrix S Extension complexity of P = rank + ( S ) [Yannanakis, 1991] ≡ extension S = UV T S ≥ 0 P U ∈ R m × r with r facets + V ∈ R n × r + Moreover any slack matrix factorization provides an explicit extended formulation (with some redundant equalities) P = { x | Ax ≤ b } = { x | Ax + Uy = b and y ≥ 0 } i.e. columns of U generate the cone of slack vectors Our goal: compute (bounds on) the nonnegative rank of (small) matrices and deduce the extension complexity of (small) polytopes 4

Example for the hexagon  1 0 1 0 0   0 0 0 1 2 1  1 0 0 0 1   1 2 1 0 0 0     0 0 0 1 2     0 0 1 1 0 0     0 1 0 0 1     0 1 0 0 1 0     0 1 1 0 0 5 facets   1 0 0 0 0 1 0 0 2 1 0 = π  0 1 2 2 1 0  0 0 1 2 2 1     1 0 0 1 2 2 6 facets     2 1 0 0 1 2     2 2 1 0 0 1   1 2 2 1 0 0 5

Outline Numerical factorization of nonnegative matrices Extension complexity of the correlation polytope Cartesian products and extension complexity Worst-case extension complexity of polygons An explicit factorization for regular n -gons Lower bounds on extension complexity Geometric lower bound on extension complexity Computable bounds for regular n -gons Positive semidefinite rank 6

Numerical factorization of nonnegative matrices (cf. Nicolas’ talk yesterday) Simulated annealing-based heuristic factorization code (improved multi-start with accelerated coordinate descent) Seems to perform well in practice (e.g. for a few dozen rows and columns) Matlab code (user-friendly) available at http://sites.google.com/site/exactnmf/ Also maintain a list of benchmark matrices and best known lower and upper bounds on rank + 7

Benchmarked matrices m n rank ( X ) rank + ( X ) Code 6 6 3 5 LEDM6 Linear EDM’s 8 8 3 6 LEDM8 X ( i , j ) = ( i − j ) 2 , 12 12 3 7 LEDM12 for 1 ≤ i ≤ m , 1 ≤ j ≤ n 16 16 3 8 LEDM16 32 32 3 10 ∗ LEDM32 Slack Matrix of the Hexagon 6 6 3 5 6-G Slack Matrix of the Heptagon 7 7 3 6 7-G Slack Matrix of the Octagon 8 8 3 6 8-G Slack Matrix of the Nonagon 9 9 3 7 9-G Slack Matrix of the Hexadecagon 16 16 3 8 12-G Slack Matrix of the 32-gon 32 32 3 10 32-G Slack Matrix of the dodecahedron 20 12 4 9 20-D Slack Matrix of the 24-cell 24 24 5 12 ∗ 24-C UDISJ ( n = 4) 16 16 9 9 UDISJ4 UDISJ ( n = 5) 32 32 18 18 UDISJ5 UDISJ ( n = 6) 64 64 27 27 UDISJ6 Randomly generated: X = WH 50 50 10 10 RND1 density = 0.1 50 50 10 10 RND3 density = 0.3 8

Benchmark results MS2 SA RBR Hybrid LEDM6 112/150 (3.1) 100/100 (19.6) 100/100 (1.4) 100/100 (19) LEDM8 107/600 (27.1) 100/100 (60.9) 100/100 (16.7) 148/150 (63.6) LEDM12 0/1000 ( ∼ ) 119/200 (42.9) 107/650 (15.1) 103/150 (36.9) LEDM16 0/1000 ( ∼ ) 100/250 (118) 100/550 (29.1) 121/250 (104) LEDM32 0/1000 ( ∼ ) 14/1000 (2592) 0/1000 ( ∼ ) 28/1000 (1371) 6-G 100/100 (2.1) 100/100 (1.2) 100/100 (1.4) 100/100 (1.5) 7-G 100/100 (2.2) 100/100 (4.2) 100/100 (1.5) 100/100 (4.4) 8-G 129/200 (3.8) 100/100 (15.4) 100/100 (1.5) 100/100 (15.3) 9-G 117/200 (4.6) 100/100 (22.9) 100/100 (1.6) 100/100 (23.2) 16-G 0/1000 ( ∼ ) 102/350 (91.6) 143/150 (1.9) 118/150 (34.2) 32-G 0/1000 ( ∼ ) 31/1000 (1086) 107/250 (6.6) 105/300 (97) 20-D 21/1000 (161) 100/100 (7.8) 129/150 (2.3) 100/100 (5.6) 24-C 0/1000 ( ∼ ) 100/100 (3.1) 119/200 (4.1) 100/100 (4.4) UDISJ4 100/100 (2.4) 100/100 (1.2) 100/100 (1.9) 100/100 (1.9) UDISJ5 102/500 (38) 100/100 (2.8) 100/100 (4.9) 100/100 (5.2) UDISJ6 8/1000 (1594) 100/100 (7.8) 112/450 (66.4) 100/100 (18.5) RND1 100/100 (2.8) 100/100 (1.1) 100/100 (2.2) 100/100 (2.2) RND3 100/100 (2.8) 100/100 (1.1) 100/100 (2.2) 100/100 (2.2) 9

Outline Numerical factorization of nonnegative matrices Extension complexity of the correlation polytope Cartesian products and extension complexity Worst-case extension complexity of polygons An explicit factorization for regular n -gons Lower bounds on extension complexity Geometric lower bound on extension complexity Computable bounds for regular n -gons Positive semidefinite rank 10

Correlation polytope Correlation polytope COR ( n ) is COR ( n ) = conv { xx T such that x ∈ { 0 , 1 } n } xc ( COR ( n )) ≥ 2 Ω( n ) (Fiorini et al., 2012) xc ( COR ( n )) ≥ 1 . 5 n (Kaibel and Weltge, 2013) 11

A submatrix of the slack matrix of COR(n) Define the following 2 n × 2 n nonnegative matrix M n ( a , b ) = ( 1 − | a T b | ) 2 for any a ∈ { 0 , 1 } n , b ∈ { 0 , 1 } n M n appears as a submatrix of the slack matrix of COR(n) M n has rank 1 2 n ( n + 1 ) + 1 Numerical tests suggest that for 3 ≤ n ≤ 6 we have rank + M n = 2 n (1000 attempts for each n ) which would imply xc ( COR ( n )) ≥ 2 n 12

Outline Numerical factorization of nonnegative matrices Extension complexity of the correlation polytope Cartesian products and extension complexity Worst-case extension complexity of polygons An explicit factorization for regular n -gons Lower bounds on extension complexity Geometric lower bound on extension complexity Computable bounds for regular n -gons Positive semidefinite rank 13

Cartesian product of polyhedral cones Let C 1 , C 2 be two polyhedral cones (with f i faces, r i extreme rays) and S 1 ∈ R f 1 × r 1 , S 2 ∈ R f 2 × r 2 their slack matrices + + Their Cartesian product C 1 × C 2 = { ( x , y ) such that x ∈ C 1 , y ∈ C 2 } features f 1 + f 2 faces and v 1 + v 2 extreme rays and its slack matrix S 12 satisfies rank + S 12 = rank + S 1 + rank + S 2 or xc ( C 1 × C 2 ) = xc ( C 1 ) + xc ( C 2 ) Is the same true for polytopes ? 14

Cartesian product for polytopes and nonnegative matrices Let P 1 , P 2 be two polytopes (with f i faces, v i vertices) and S 1 ∈ R f 1 × v 1 , S 2 ∈ R f 2 × v 2 their slack matrices + + We have that P 1 × P 2 features f 1 + f 2 faces and v 1 v 2 vertices and xc ( P 1 × P 2 ) ≤ xc ( P 1 ) + xc ( P 2 ) and its slack matrix S 12 = S 1 ⊙ S 2 (size ( f 1 + f 2 ) × v 1 v 2 ) satisfies rank + S 1 ⊙ S 2 ≤ rank + S 1 + rank + S 2 ( ⊙ can be defined for arbitrary nonnegative matrices S 1 and S 2 ) Does equality always hold ? Intuitively obvious but ... No one knows ! (except when P 2 is a simplex or S 2 identity) (for standard rank we have rank S 1 ⊙ S 2 = rank S 1 + rank S 2 unless 1 ∈ rowspan S 1 and 1 ∈ rowspan S 2 , in which case rank S 1 ⊙ S 2 = rank S 1 + rank S 2 − 1) 15

Numerical tests for Cartesian product Numerical tests for products of two (or three) of the following 1. 5-gon, 6-gon, 7-gon, 8-gon (with respective xc equal to 5,5,6,6) 2. dodecahedron (xc=9) 3. cuboctahedron (xc=8) 4. polar of any of the above 5. generalized slack matrix obtained by relaxing 5-gon (adding a constant δ ≈ 0 . 499 to all entries of its slack matrix) 0 ≤ δ < 1 2 ⇒ rank + ( S 5 + δ E ) = 5 and rank + ( S 5 + 1 2 E ) = 4 All tests strongly suggest that xc ( P 1 × P 2 ) = xc ( P 1 ) + xc ( P 2 ) holds as well as rank + S 1 ⊙ S 2 = rank + S 1 + rank + S 2 (with the same final error observed in all tests for each pair) 16