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

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Frank Vallentin University of Cologne, Germany

partially supported by

Solving semidefinite programs

Maria Dostert (Cologne), Cristob´ al Guzm´ an (Santiago de Chile), David de Laat (CWI), Fernando Oliveira (Sa˜

• Paulo)

for packing problems

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• 1. Introduction

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equal spheres

different spheres M&Ms

tetrahedra

Densest packings

Applications: information theory, materials science

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Rich history Hilbert’s 18th problem Arrange most densely equal solids of a given form, e.g. spheres, regular tetrahedra wide open, maximum density between 0.85 and 1 − 10−26

(Chen, Engel, Glotzer (2010), Gravel, Elser, Kallus (2011))

solved by Hales (1998, 2014) Extremely difficult n = 3: n = 8, 24: Viazovska et al. (2016) best upper bounds by de Laat, Oliveira, Vallentin (2014) n = 4, 5, . . . :

Conjecture (Torquato, Jiao, 2009): “Kepler’s conjecture for the 21st century”

Densest packings of centrally symmetric Platonic, Archimedean solids, and of lp-unit balls are given by the corresponding lattice packings

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• 2. Methods

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I ✓ G independent: 8x, y 2 I, x 6= y, x 6⇠ y which are as “large” as possible find indep. sets in Cayley(G, Σ)

Independent sets in Cayley graphs

Cayley(Z/5Z, {1, 4})

1 2

3

4 G finite: α(G) = max{|I| : I independent} Cayley(G, Σ) undirected graph on G x ∼ y ⇐ ⇒ x − y ∈ Σ Abelian group Σ ⊆ G, Σ = −Σ G = Rn, Σ = K K ⊆ Rn centrally symmetric convex body G infinite: α(Cayley(Rn, Σ)) = maximal density of packing of translates of 1 2K

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Complete SDP proof system apply Lasserre’s hierarchy for polynomial optimization polynomial optimization formulation

Cayley(Z/5Z, {1, 4})

1 2

3

4

α(G) = max P

i∈V xi

xi ≥ 0 x2

i − xi = 0 for i ∈ V

xixj = 0 if ij ∈ E

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First step of Lasserre’s hierarchy

Cayley(Z/5Z, {1, 4})

1

2 3 4 α(G) = max P

i∈V xi

xi ≥ 0 x2

i − xi = 0 for i ∈ V

xixj = 0 if ij ∈ E ≤ max y0 + y1 + y2 + y3 + y4 y∅ = 1, y0, y1, y2, y3, y4 ≥ 0         y∅ y0 y1 y2 y3 y4 y0 y0 y02 y03 y1 y1 y13 y14 y2 y02 y2 y24 y3 y03 y13 y3 y4 y14 y24 y4         ⌫ 0 linearize xixj to yij

?

✓ y∅ yi yi yi ◆ ⌫ 0 = ) 1 · yi y2

i =

) y2

i yi  0

?

first step = ϑ0(G)

?

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(Mt(y))J,J0 = ( yJ[J0 if J ∪ J0 ∈ I2t,

• therwise.

moment matrix

          y∅ y1 y2 y3 y12 y13 y23 y1 y1 y12 y13 y12 y13 y123 y2 y12 y2 y23 y12 y123 y23 y3 y13 y23 y3 y123 y13 y23 y12 y12 y12 y123 y12 y123 y123 y13 y13 y123 y13 y123 y13 y123 y23 y123 y23 y23 y123 y123 y23          

∅ 1 2 3 12 13 23

∅ 1 2 3 12 13 23

I2t = independent sets with ≤ 2t elements last(G) = max n X

x∈V

y{x} : y 2 RI2t

≥0, y∅ = 1, Mt(y) ⌫ 0

• ,

t-th step of Lasserre’s hierarchy

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Properties of Lasserre’s hierarchy

the t-th step last(G) is a semidefinite program

?

can be generalized to infinite graphs

?

SDP proof system is complete:

ϑ0(G) = las1(G) ≥ las2(G) ≥ . . . ≥ lasα(G)(G) = α(G) ?

every intermediate step gives rigorous upper bound

(Laurent, 2003)

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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

need topological assumptions

last(G) = sup n λ(I=1) : λ ∈ M(I2t)0, λ({∅}) = 1, A⇤

t λ ∈ M(It × It)⌫0

• .

last(G) = max n X

x∈V

y{x} : y 2 RI2t

≥0, y∅ = 1, Mt(y) ⌫ 0

• ,

Borel measure

Complete SDP proof system for infinite graphs

SDP proof system complete if G compact top. packing graph

(de Laat, Vallentin, 2015)

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• 3. Explicit Computations

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= min M 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, M 2 R, B 2 RV ×V is symmetric. α(G)  ϑ(G) = max hJ, Ai A 2 RV ×V is positive semidefinite, hI, Ai = 1 A(x, y) = 0 for all {x, y} 2 E.

? exhibiting feasible solution for dual gives rigorous bound

Lasserre’s first step / Lov´ asz ϑ

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4 1

2

3

ϑ(G) = min M 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, M 2 R, B 2 RV ×V is symmetric. C5 = Cayley(Z/5Z, {1, 4}) ϑ(C5) = min M B − J is positive semidefinite, B =       M b01 b04 b01 M b12 b12 M b23 b23 M b34 b04 b34 M      

Exercise: Computing ϑ(C5)

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Exploiting cyclic symmetry

1

2

3 4

C5 = Cayley(Z/5Z, {1, 4}) cyclic permutation matrix C =       1 1 1 1 1       consider group average: 1 5

4

X

k=0

(Ck)TBCk = B B B B @ M b b b M b b M b b M b b b M 1 C C C C A

b = 1 5(b01 + b12 + b23 + b34 + b04)

CTBC =       M b12 b01 b12 M b23 b23 M b34 b34 M b04 b01 b04 M       If B is feasible, then also CTBC (with same objective value): B =       M b01 b04 b01 M b12 b12 M b23 b23 M b34 b04 b34 M      

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1 5

4

X

k=0

(Ck)T(B − J)Ck = circ(M − 1, b − 1, −1, −1, b − 1)

Circulant matrices

=      y0 y1 . . . yn−1 yn−1 y0 . . . yn−2 ... ... ... ... y1 y2 . . . y0      Y = circ(y0, y1, . . . , yn−1) = (Yrs)r,s = (ys−r)r,s ∈ Cn×n Practically every matrix theoretic question for circulants can be solved in closed form. χa = (ω−a·0

n

, ω−a·1

n

, . . . , ω−a·(n−1)

n

)T ∈ Cn e.g. eigenvalues and eigenvectors ωn = e2πi/n a = 0, . . . , n − 1 are eigenvectors

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Eigenvalues are discrete Fourier coefficients

=      y0 y1 . . . yn−1 yn−1 y0 . . . yn−2 ... ... ... ... y1 y2 . . . y0      Y = circ(y0, y1, . . . , yn−1) = (Yrs)r,s = (ys−r)r,s ∈ Cn×n χa = (ω−a·0

n

, ω−a·1

n

, . . . , ω−a·(n−1)

n

)T ∈ Cn ωn = e2πi/n a = 0, . . . , n − 1

(Y χa)r =

n−1

X

s=0

ys−rω−a·s

n

= ω−a·r

n n−1

X

s=0

ys−rω−a·(s−r)

n

= ω−a·r

n n−1

X

s=0

ysω−a·s

n

= ω−a·r

n

b y(a)

where b y(a) = Pn−1

s=0 y(s)e−2πias/n

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Back to ϑ(C5)

1 2

3 4

ϑ(C5) = min M circ(y0, y1, y2, y3, y4) is positive semidefinite, with y = (M − 1, b − 1, −1, −1, b − 1). b y(0) = M + 2b − 5 ≥ 0 ⇐ ⇒ M + 2b ≥ 5 = M + b(cos(−2π/5) + i sin(−2π/5) + cos(−8π/5) + i(sin −8π/5)) = M + 2b cos(−2π/5) ≥ 0 b y(2) = M + 2b cos(−4π/5) ≥ 0 b y(3) = b y(2), b y(4) = b y(1) LP with 2 variables and 3 constraints

• ptimum at b

y(0) = b y(2) M = √ 5 b y(1) = M + ω−1·1

n

b + ω−1·4

n

b −

4

X

k=0

ω−1·k

n

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?

ϑ(C5) = √ 5

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Proof generalizes in many ways

for infinite Cayley graphs on Abelian groups: G = Rn, Σ = K α(Cayley(Rn, K)) = maximal density of packing of translates of 1/2K for other cyclic graphs: ϑ0(Cayley(Z/nZ, Σ)) = min y0 b y(0) n, b y(a) 0, a = 1, . . . , n 1, yi  0 if i 62 Σ. b y(a) =

n−1

X

s=0

yse−2πis·a/n where f ∈ L1(Rn) and continuous b f(a) = Z

Rn f(x)e−2πix·adx

Cohn-Elkies (2003) = min f(0) b f(0) vol 1

2K,

b f(a) 0, a 2 Rn \ {0}, f(x)  0 if x 62 K. ≤ ϑ0(Cayley(Rn, K))

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approximate by semi-infinite linear program and set b f(u) = p(u)eπkuk2

• ptimize over polynomials p ∈ R[u1, . . . , un]≤2d

Infinite dimensional linear program

where f ∈ L1(Rn) and continuous = min f(0) b f(0) vol 1

2K,

b f(a) 0, a 2 Rn \ {0}, f(x)  0 if x 62 K. α(Cayley(Rn, K)) ≤ ϑ0(Cayley(Rn, K)) α(Cayley(G, K))  inf R

Rn p(u)eπkuk2 du

p 2 R[u]2d p(0) vol 1

2K

p(u) 0 for all u 2 Rn \ {0} R

Rn p(u)eπkuk2e2πiu·x du  0 for all x 62 K

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Solving the optimization problem

α(Cayley(G, K))  inf R

Rn p(u)eπkuk2 du

p 2 R[u]2d p(0) vol 1

2K

p(u) 0 for all u 2 Rn \ {0} R

Rn p(u)eπkuk2e2πiu·x du  0 for all x 62 K

symmetrize again: p(u) =

1 |S(K)|

P

A∈S(K) p(A−1u)

symmetry group of K: S(K) = {A ∈ O(n) : AK = K}

for K with octahedral symmetry

— checking that p is globally nonnegative: NP-hard — semidefinite relaxation: p is a sum of squares (SOS), p = p2

1 + · · · + p2 m

— p with deg p = 2d is SOS ⇐ ⇒ ∃Q ∈ S(

n+d d )

⌫0

: p(u) = [u]T

dQ[u]d

— if n = 3, d = 15, then Q ∈ S816

⌫0 ; too big for current SDP solvers

— idea: can assume that p is invariant under symmetry group of K

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  −1 1 1   ,   1 −1 −1   ,   1 1 1  

reflection group B3, 48 elements

Finite reflection groups

p(±xσ(1), ±xσ(2), ±xσ(3)) = p(x1, x2, x3)

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invariant ring: C[x]G = {p ∈ C[x] : p(g−1x) = p(x) for all g ∈ G}

θ1 = x2 + y2 + z2, θ2 = x4 + y4 + z4, θ3 = x6 + y6 + z6

C[x]G = C[θ1, . . . , θn]

generated by basic invariants:

Chevalley-Shephard-Todd-Serre theory

finite reflection group: G ⊆ GL(Cn) G = S(K) dimension of C[x]≤30 is 5456 can assume p(x1, x2, x3) ∈ C[x]G but p has to be a SOS dimension of C[x]G

≤30 is only 174

dimension reduction

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C[x]G = C[x]/I, where I = (θ1, . . . , θn)

C[x] = C[x]G ⊗ C[x]G

has dimension |G| and is isomorphic to regular representation of G coinvariant algebra:

ϕπ

ij, with π ∈ b

G, 1 ≤ i, j ≤ dπ

gϕπ

ij = (π(g)j)T

   ϕπ

i1

. . . ϕπ

idπ

   , i = 1, . . . , dπ,

C[x]G has basis ϕπ

ij with

How to ensure that p is SOS?

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8 < :p 2 R[x] : p = X

π∈ b G

hP π, Qπi, P π is Hermitian SOS matrix polynomial in θi 9 = ; .

[Qπ]kl =

dπ

X

i=1

ϕπ

kiϕπ li.

• Theorem. (Gatermann, Parrilo (2004), DGOV (2015))

The cone of SOS polynomials which are G-invariant equals where hA, Bi = Tr(B∗A) P π = (Lπ)∗Lπ with matrix Lπ having entries in C[x]G advantages: — substantial size reduction: one semdefinite matrix for every π ∈ b G — only computation of matrix Qπ needed (independent of degree) n = 3, d = 15: 10 matrices (31, 23, 11, 7, 27, 39, 34, 50, 50, 70) vs. 876

Invariant SOS polynomials

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A1g 1 A1u θ3

1 − 3θ1θ2 + 2θ3

A2g −θ6

1 + 9θ4 1θ2 − 8θ3 1θ3 − 21θ2 1θ2 2 + 36θ1θ2θ3 + 3θ3 2 − 18θ2 3

A2u −θ9

1 + 12θ7 1θ2 − 10θ6 1θ3 − 48θ5 1θ2 2 + 78θ4 1θ2θ3 + 66θ3 1θ3 2 − 34θ3 1θ2 3 − 150θ2 1θ2 2θ3

−9θ1θ4

2 + 126θ1θ2θ2 3 + 6θ3 2θ3 − 36θ3 3

Eg −2θ5

1 + 12θ3 1θ2 − 4θ2 1θ3 − 18θ1θ2 2 + 12θ2θ3

−2θ4

1θ2 + 6θ3 1θ3 + 6θ2 1θ2 2 − 22θ1θ2θ3 + 12θ2 3

θ7

1 − 9θ5 1θ2 + 10θ4 1θ3 + 19θ3 1θ2 2 − 36θ2 1θ2θ3 − 3θ1θ3 2 + 16θ1θ2 3 + 2θ2 2θ3

Eu −2θ2

1 + 6θ2

−2θ1θ2 + 6θ3 θ4

1 − 6θ2 1θ2 + 8θ1θ3 + θ2 2

T1g 12θ1θ3 − 12θ2

2

2θ5

1 − 12θ3 1θ2 + 16θ2 1θ3 + 6θ1θ2 2 − 12θ2θ3

2θ6

1 − 12θ4 1θ2 + 10θ3 1θ3 + 12θ2 1θ2 2 − 6θ1θ2θ3 − 6θ3 2

2θ6

1 − 10θ4 1θ2 + 10θ3 1θ3 + 10θ1θ2θ3 − 12θ2 3

θ7

1 − 3θ5 1θ2 + 2θ4 1θ3 − 9θ3 1θ2 2 + 24θ2 1θ2θ3 + 3θ1θ3 2 − 12θ1θ2 3 − 6θ2 2θ3

4θ6

1θ2 − 3θ5 1θ3 − 21θ4 1θ2 2 + 32θ3 1θ2θ3 + 12θ2 1θ3 2 − 12θ2 1θ2 3 − 9θ1θ2 2θ3 − 3θ4 2

T1u −12θ3

1 + 48θ1θ2 − 36θ3

−6θ4

1 + 24θ2 1θ2 − 12θ1θ3 − 6θ2 2

−6θ3

1θ2 + 6θ2 1θ3 + 18θ1θ2 2 − 18θ2θ3

−2θ5

1 + 6θ3 1θ2 + 2θ2 1θ3 − 6θ2θ3

θ6

1 − 9θ4 1θ2 + 8θ3 1θ3 + 15θ2 1θ2 2 − 12θ1θ2θ3 − 3θ3 2

θ7

1 − 6θ5 1θ2 + 5θ4 1θ3 + 3θ3 1θ2 2 + 6θ1θ3 2 − 9θ2 2θ3

T2g 3θ2

1 − 3θ2

6θ1θ2 − 6θ3 −θ4

1 + 6θ2 1θ2 − 2θ1θ3 − 3θ2 2

−2θ4

1 + 12θ2 1θ2 − 10θ1θ3

−θ5

1 + 4θ3 1θ2 − 2θ2 1θ3 + 3θ1θ2 2 − 4θ2θ3

−2θ4

1θ2 + θ3 1θ3 + 9θ2 1θ2 2 − 7θ1θ2θ3 − 3θ3 2 + 2θ2 3

T2u 6θ1 6θ2 6θ3 6θ3 θ4

1 − 6θ2 1θ2 + 8θ1θ3 + 3θ2 2

θ5

1 − 5θ3 1θ2 + 5θ2 1θ3 + 5θ2θ3

Qπ matrices

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p 1 2 (CE, 2003) 4 6 8 lower bound 0.9473 . . . 0.7404 . . . 0.8698 . . . 0.9318 . . . 0.9582 . . . upper bound 0.9699 . . . 0.7797 . . . 0.8740 . . . 0.9338 . . . 0.9619 . . .

Jiao, Stillinger, Torquato (2009) lower bounds by

Conjecture: There is p0 < ∞ so that the bound is tight for all p ≥ p0.

Translative packings of superballs

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— Hoylman (1970): optimal lattice packing of T has density 18/49 = 0.3673 . . . — Zong (2014): δt(T ) ≤ 0.3840 . . . — DGOV (2015): δt(T ) ≤ 0.375 . . .

body lower bound upper bound truncated cube 0.9737 . . . 0.9845 . . . truncated tetrahedron 0.6809 . . . 0.7292 . . . rhombic cuboctahedron 0.8758 . . . 0.8794 . . . truncated cuboctahedron 0.8493 . . . 0.8845 . . .

Translative tetrahedra packings

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Designing a rigorous proof

— solve SDP with SDPA-GMP — gives high precision approximation of strictly feasible solution — modify solution (blow up domain K − K slightly) — check SDP conditions with rational arithmetic (C++ code, SAGE) — check nonnegativity using interval arithmetic (MPFI library)

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Tetrahedra packing with translates

References

• M. Dostert, C. Guzm´

an, F.M. de Oliveira Filho, F. Vallentin, New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry, arXiv:1510.02331 [math.MG], 29 pages SDP hierarchy for packing problems

• D. de Laat, F. Vallentin, A semidefinite programming hierarchy for packing prob-

lems in discrete geometry, Math. Program., Ser. B 151 (2015), 529–553. Cohn-Elkies bound

• H. Cohn and N.D. Elkies, New upper bounds on sphere packings I, Ann. of Math.

157 (2003), 689–714.

• H. Cohn, A. Kumar, S.D. Miller, D. Radchenko, and M.S. Viazovska, The sphere

packing problem in dimension 24, Ann. of Math., to appear. M.S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math., to appear.