Solving semidefinite programs for packing problems Frank Vallentin - PowerPoint PPT Presentation

Solving semidefinite programs for packing problems Frank Vallentin University of Cologne, Germany Maria Dostert (Cologne), Cristob´ al Guzm´ an (Santiago de Chile), David de Laat (CWI), Fernando Oliveira (Sa˜ o Paulo) partially supported by

1. Introduction

Densest packings different spheres tetrahedra equal spheres M&Ms Applications: information theory, materials science

Rich history Hilbert’s 18th problem Arrange most densely equal solids of a given form, e.g. spheres, regular tetrahedra Extremely difficult solved by Hales (1998, 2014) n = 3 : Viazovska et al. (2016) n = 8 , 24 : n = 4 , 5 , . . . : best upper bounds by de Laat, Oliveira, Vallentin (2014) Conjecture (Torquato, Jiao, 2009): “Kepler’s conjecture for the 21st century” wide open, maximum density between 0 . 85 and 1 − 10 − 26 Densest packings of centrally symmetric Platonic, Archimedean solids, and of l p -unit balls are given by the corresponding lattice packings (Chen, Engel, Glotzer (2010), Gravel, Elser, Kallus (2011))

2. Methods

Independent sets in Cayley graphs 3 Cayley ( Z / 5 Z , { 1 , 4 } ) Cayley ( G, Σ ) ⇒ x − y ∈ Σ x ∼ y ⇐ 4 2 Abelian group Σ ⊆ G, Σ = − Σ undirected graph on G 0 1 I ✓ G independent: 8 x, y 2 I, x 6 = y, x 6⇠ y find indep. sets in Cayley ( G, Σ ) which are as “large” as possible G finite: α ( G ) = max {| I | : I independent } K ⊆ R n centrally symmetric convex body G infinite: G = R n , Σ = K � α (Cayley( R n , Σ )) = maximal density of packing of translates of 1 2 K

Complete SDP proof system polynomial optimization formulation 3 Cayley ( Z / 5 Z , { 1 , 4 } ) α ( G ) = max P i ∈ V x i 4 2 x i ≥ 0 x 2 i − x i = 0 for i ∈ V x i x j = 0 if ij ∈ E 0 1 apply Lasserre’s hierarchy for polynomial optimization

First step of Lasserre’s hierarchy 3 α ( G ) = max P i ∈ V x i Cayley ( Z / 5 Z , { 1 , 4 } ) x i ≥ 0 x 2 i − x i = 0 for i ∈ V 4 2 x i x j = 0 if ij ∈ E 0 1 ≤ max y 0 + y 1 + y 2 + y 3 + y 4 y ∅ = 1 , y 0 , y 1 , y 2 , y 3 , y 4 ≥ 0   y ∅ y 0 y 1 y 2 y 3 y 4 0 0 y 0 y 0 y 02 y 03     0 0 y 1 y 1 y 13 y 14   ⌫ 0   0 0 y 2 y 02 y 2 y 24     0 0 y 3 y 03 y 13 y 3   0 0 y 4 y 14 y 24 y 4 ? linearize x i x j to y ij ✓ ◆ y ∅ y i ) 1 · y i � y 2 ) y 2 ⌫ 0 = i = i � y i  0 ? y i y i first step = ϑ 0 ( G ) ?

t -th step of Lasserre’s hierarchy n X o y { x } : y 2 R I 2 t las t ( G ) = max ≥ 0 , y ∅ = 1 , M t ( y ) ⌫ 0 , x ∈ V I 2 t = independent sets with ≤ 2 t elements ( if J ∪ J 0 ∈ I 2 t , y J [ J 0 ( M t ( y )) J,J 0 = moment matrix 0 otherwise. ∅ 1 2 3 12 13 23   ∅ y ∅ y 1 y 2 y 3 y 12 y 13 y 23 1 y 1 y 1 y 12 y 13 y 12 y 13 y 123     2 y 2 y 12 y 2 y 23 y 12 y 123 y 23     3 y 3 y 13 y 23 y 3 y 123 y 13 y 23     12 y 12 y 12 y 12 y 123 y 12 y 123 y 123     13 y 13 y 13 y 123 y 13 y 123 y 13 y 123   y 23 y 123 y 23 y 23 y 123 y 123 y 23 23

Properties of Lasserre’s hierarchy the t -th step las t ( G ) is a semidefinite program ? ? SDP proof system is complete: (Laurent, 2003) ϑ 0 ( G ) = las 1 ( G ) ≥ las 2 ( G ) ≥ . . . ≥ las α ( G ) ( G ) = α ( G ) every intermediate step gives rigorous upper bound can be generalized to infinite graphs ?

Complete SDP proof system for infinite graphs (de Laat, Vallentin, 2015) need topological assumptions Graph G = ( V, E ) is a topological packing graph if V is a Hausdorff topological space ? every finite clique is contained in a clique which is open ? n X o y { x } : y 2 R I 2 t las t ( G ) = max ≥ 0 , y ∅ = 1 , M t ( y ) ⌫ 0 , x ∈ V n o λ ( I =1 ) : λ ∈ M ( I 2 t ) � 0 , λ ( { ∅ } ) = 1 , A ⇤ t λ ∈ M ( I t × I t ) ⌫ 0 las t ( G ) = sup . Borel measure SDP proof system complete if G compact top. packing graph

3. Explicit Computations

Lasserre’s first step / Lov´ asz ϑ α ( G )  ϑ ( G ) = max h J, A i A 2 R V × V is positive semidefinite , h I, A i = 1 A ( x, y ) = 0 for all { x, y } 2 E . = min M is positive semidefinite , B � J B ( x, x ) = M for all x 2 V , B ( x, y ) = 0 for all { x, y } 62 E where x 6 = y, M 2 R , B 2 R V × V is symmetric. ? exhibiting feasible solution for dual gives rigorous bound

Exercise: Computing ϑ ( C 5 ) 3 ϑ ( G ) = min M 4 2 B � J is positive semidefinite , B ( x, x ) = M for all x 2 V , B ( x, y ) = 0 for all { x, y } 62 E where x 6 = y, 0 1 M 2 R , B 2 R V × V is symmetric. C 5 = Cayley( Z / 5 Z , { 1 , 4 } ) ϑ ( C 5 ) = min M is positive semidefinite , B − J   0 0 M b 01 b 04 0 0 b 01 M b 12     0 0 B = b 12 M b 23     0 0 b 23 M b 34   0 0 b 04 b 34 M

Exploiting cyclic symmetry 3   0 0 0 0 1 1 0 0 0 0   4 2   C = 0 1 0 0 0     0 0 1 0 0   0 1 0 0 0 1 0 C 5 = Cayley( Z / 5 Z , { 1 , 4 } ) cyclic permutation matrix If B is feasible, then also C T BC (with same objective value):   M b 12 b 01 0 0   M b 01 b 04 0 0 b 12 M b 23 0 0 b 01 M b 12 0 0     C T BC =   b 23 M b 34   0 0 B = b 12 M b 23 0 0       b 34 M b 04   0 0 0 0 b 23 M b 34     b 01 b 04 M 0 0 b 04 b 34 M 0 0 consider group average: 0 1 0 0 M b b 0 0 b M b 4 B C 1 ( C k ) T BC k = X B C 0 0 b M b B C 5 B C 0 0 b M b k =0 @ A 0 0 b b M b = 1 5( b 01 + b 12 + b 23 + b 34 + b 04 )

Circulant matrices 4 1 ( C k ) T ( B − J ) C k = circ( M − 1 , b − 1 , − 1 , − 1 , b − 1) X 5 k =0 Y = circ( y 0 , y 1 , . . . , y n − 1 ) = ( Y rs ) r,s = ( y s − r ) r,s ∈ C n × n   y 0 y 1 . . . y n − 1 y n − 1 y 0 . . . y n − 2     =   ... ... ... ...   y 1 y 2 . . . y 0 Practically every matrix theoretic question for circulants can be solved in closed form. e.g. eigenvalues and eigenvectors ) T ∈ C n χ a = ( ω − a · 0 , ω − a · 1 , . . . , ω − a · ( n − 1) ω n = e 2 π i/n n n n a = 0 , . . . , n − 1 are eigenvectors

Eigenvalues are discrete Fourier coefficients Y = circ( y 0 , y 1 , . . . , y n − 1 ) = ( Y rs ) r,s = ( y s − r ) r,s ∈ C n × n   y 0 y 1 . . . y n − 1 ω n = e 2 π i/n y n − 1 y 0 . . . y n − 2     = ) T ∈ C n χ a = ( ω − a · 0 , ω − a · 1 , . . . , ω − a · ( n − 1)   ... ... ... ...   n n n y 1 y 2 . . . y 0 a = 0 , . . . , n − 1 n − 1 X ( Y χ a ) r = y s − r ω − a · s n s =0 n − 1 X y s − r ω − a · ( s − r ) = ω − a · r n n s =0 n − 1 X = ω − a · r y s ω − a · s n n s =0 y ( a ) = P n − 1 = ω − a · r y ( a ) b s =0 y ( s ) e − 2 π ias/n where b n

3 Back to ϑ ( C 5 ) 4 2 ϑ ( C 5 ) = min M 0 1 circ( y 0 , y 1 , y 2 , y 3 , y 4 ) is positive semidefinite , with y = ( M − 1 , b − 1 , − 1 , − 1 , b − 1) . y (0) = M + 2 b − 5 ≥ 0 ⇐ ⇒ M + 2 b ≥ 5 b X 4 ω − 1 · k y (1) = M + ω − 1 · 1 b + ω − 1 · 4 b b − n n n k =0 = M + b (cos( − 2 π / 5) + i sin( − 2 π / 5) + cos( − 8 π / 5) + i (sin − 8 π / 5)) = M + 2 b cos( − 2 π / 5) ≥ 0 y (2) = M + 2 b cos( − 4 π / 5) ≥ 0 b y (3) = b y (2) , y (4) = b y (1) b b LP with 2 variables and 3 constraints optimum at b y (0) = b y (2) √ M = 5

√ ϑ ( C 5 ) = 5 ?

Proof generalizes in many ways for other cyclic graphs: ϑ 0 (Cayley( Z /n Z , Σ )) = min y 0 y (0) � n, b b y ( a ) � 0 , a = 1 , . . . , n � 1 , n − 1 X y i  0 if i 62 Σ . y s e − 2 π is · a/n y ( a ) = b s =0 for infinite Cayley graphs on Abelian groups: G = R n , Σ = K � α (Cayley( R n , K � )) = maximal density of packing of translates of 1 / 2 K ≤ ϑ 0 (Cayley( R n , K � )) = min f (0) f ( a ) � 0 , a 2 R n \ { 0 } , b b f (0) � vol 1 2 K , f ( x )  0 if x 62 K � . where f ∈ L 1 ( R n ) and continuous Z b R n f ( x ) e − 2 π ix · a dx f ( a ) = Cohn-Elkies (2003)

Infinite dimensional linear program α (Cayley( R n , K � )) ≤ ϑ 0 (Cayley( R n , K � )) = min f (0) f ( a ) � 0 , a 2 R n \ { 0 } , b b f (0) � vol 1 2 K , f ( x )  0 if x 62 K � . where f ∈ L 1 ( R n ) and continuous approximate by semi-infinite linear program optimize over polynomials p ∈ R [ u 1 , . . . , u n ] ≤ 2 d and set b f ( u ) = p ( u ) e � π k u k 2 R n p ( u ) e � π k u k 2 du α ( Cayley ( G, K � ))  inf R p 2 R [ u ]  2 d p (0) � vol 1 2 K p ( u ) � 0 for all u 2 R n \ { 0 } R n p ( u ) e � π k u k 2 e 2 π iu · x du  0 for all x 62 K � R