A semidefinite programming hierarchy for geometric packing problems - PowerPoint PPT Presentation

A semidefinite programming hierarchy for geometric packing problems David de Laat Joint work with Frank Vallentin 4th SDP days – March 2013

Geometric packing problems ◮ Spherical codes (spherical cap packings): What is the largest number of points that one can place on S n − 1 such that the pairwise inner products are at most t ? ◮ Model geometric packing problems as maximum independent set problems ◮ G = ( S n − 1 , x ∼ y if x · y > t ) Definition A packing graph is a graph where - the vertex set is a Hausdorff topological space - each finite clique is contained in an open clique ◮ We will consider compact packing graphs

Upper bounds for the max independent set problem Finite Graphs Infinite Graphs Delsarte, 1973 Delsarte, 1977 Kabatiansky, Levenshtein, 1978 Lov´ asz, 1979 Bachoc, Nebe, de Oliveira, Vallentin, 2009 Schrijver, 1979 McEliece, Rodemich, Rumsey, 1978 Lasserre, 2001 Laurent, 2003 This talk: generalize the Lasserre hierarchy to infinite graphs and prove finite convergence

The Lasserre hierarchy for finite graphs ∅ 1 2 · · · n { 1 , 2 } · · · { 3 , 5 } · · · 1 ∅ y 1 1 y 2 2 ... . . . y n � � n n α ≤ max i =1 y i : � 0 , { 1 , 2 } . . . y { 3 , 4 , 5 } { 3 , 4 } . . . � y S = 0 if S has an edge

Finite subset spaces ◮ Sub ( V, t ) is the collection of nonempty subsets of V with at most t elements ◮ Quotient map: q : V t → Sub ( V, t ) , ( v 1 , . . . , v t ) �→ { v 1 , . . . , v t } ◮ Sub ( V, t ) is a compact Hausdorff space ◮ I t ⊆ Sub ( V, t ) is the collection of nonempty independent sets with at most t elements ◮ V t = Sub ( V, t ) ∪ {∅} is part of the semigroup (2 V , ∪ )

Finite subset spaces I t is the collection of nonempty independent sets with ≤ t elements Lemma I t is compact x x y y { x, y } ∈ I 2 { x, y } �∈ I 2

Finite subset spaces ◮ If the topology on V comes from a metric, then the topology on Sub ( V, t ) is given by the Hausdorff distance ◮ Example: the sets { x, y } and { u, v, w } are close in Sub ( V, t ) u y w x v Lemma I t → Z ≥ 0 , S �→ | S | is continuous ◮ The sets { x, y } and { u, v, w } are in different connected components in I t

Positive kernels ◮ A function f ∈ C ( V t × V t ) sym is a positive kernel if ( f ( x i , x j )) m i,j =1 is positive semidefinite for all m and x 1 , . . . , x m ∈ V t ◮ Cone of positive (definite) kernels: C ( V t × V t ) � 0

Measures of positive type ◮ M ( V t × V t ) � 0 is the cone dual to C ( V t × V t ) � 0 ◮ The elements in M ( V t × V t ) � 0 are called positive definite measures ◮ Define the operator A t : C ( V t × V t ) sym → C ( I 2 t ) by � f ( J, J ′ ) A t f ( S ) = J,J ′ ∈ V t : J ∪ J ′ = S ◮ The measures of positive type on I 2 t : � � λ ∈ M ( I 2 t ): A ∗ t λ ∈ M ( V t × V t ) � 0 ◮ A measure λ on a locally compact group Γ is of positive type if it defines a positive linear functional on the group algebra: λ ( f ∗ ∗ f ) ≥ 0 for all f ∈ C (Γ) ◮ For f, g ∈ C ( V t ) , let f ∗ = f and f ∗ g = A t ( f ⊗ g )

The hierarchie ◮ Generalization of the Lasserre hierarchy to infinite graphs: � ϑ t = inf f ( ∅ , ∅ ): f ∈ C ( V t × V t ) � 0 , � ✶ I 1 + A t f ∈ C ( I 2 t ) ≤ 0 ◮ Conic duality gives the dual chain � ϑ ∗ t = sup λ ( I 1 ): λ ∈ M ( I 2 t ) ≥ 0 , � δ ∅ ⊗ δ ∅ + A ∗ t λ ∈ M ( V t × V t ) � 0 Theorem 1. ϑ t = ϑ ∗ t for all t 2. α ≤ . . . ≤ ϑ ∗ 3 ≤ ϑ ∗ 2 ≤ ϑ ∗ 1 3. ϑ ∗ α = α

Strong duality ◮ To prove strong duality we use a closed cone condition ◮ We need to show that the cone K = { ( A ∗ t λ − µ, λ ( I 1 )): µ ∈ M ( V t × V t ) � 0 , λ ∈ M ( I 2 t ) ≥ 0 } is closed in M ( V t × V t ) sym × R ◮ Idea: K = K 1 − K 2 (Minkowski difference) Lemma (Klee 1955) If K 1 and K 2 are closed convex cones in a topological vector space, K 1 is locally compact, and K 1 ∩ K 2 = { 0 } , then K 1 − K 2 is closed.

Finite convergence ◮ We write the α th step of the hierarchy as Θ = max { λ ( I 1 ): λ ∈ M ( I ) , λ ( {∅} ) = 1 , A ∗ λ � 0 } where I is the collection of all independent sets ◮ Claim: Θ = α ◮ Given an independent set S , χ S = � J ⊆ S δ J is feasible for Θ ◮ λ feasible ⇒ λ = � χ S dσ ( S ) ◮ We show that σ is a probability measure ◮ Θ = max { � | S | dσ ( S ): σ ∈ P ( I ) }

Thank you! D. de Laat, F. Vallentin, A semidefinite programming hierarchy for geometric packing problems , in preparation.