Energy minimization via conic programming hierarchies David de Laat - PowerPoint PPT Presentation

Energy minimization via conic programming hierarchies David de Laat (TU Delft) IFORS July 14, 2014, Barcelona

Energy minimization ◮ What is the minimal potential energy E when we distribute N particles in a container V with pair potential w ?

Energy minimization ◮ What is the minimal potential energy E when we distribute N particles in a container V with pair potential w ? ◮ Example: For the Thomson problem we take 1 V = S 2 w ( { x, y } ) = and � x − y �

Energy minimization ◮ What is the minimal potential energy E when we distribute N particles in a container V with pair potential w ? ◮ Example: For the Thomson problem we take 1 V = S 2 w ( { x, y } ) = and � x − y � ◮ Optimization problem: � E = inf w ( P ) S ∈ ( V N ) P ∈ ( S 2 )

Approach ◮ Configurations provide upper bounds on the optimal energy E

Approach ◮ Configurations provide upper bounds on the optimal energy E ◮ To prove a configuration is good (or optimal) we need good lower bounds for E

Approach ◮ Configurations provide upper bounds on the optimal energy E ◮ To prove a configuration is good (or optimal) we need good lower bounds for E ◮ For this we use infinite dimensional moment hierarchies and semidefinite programming

Approach 0 E

Approach Difficult minimization problem 0 E

Approach Difficult minimization problem E t 0 E

Approach Difficult minimization problem E t 0 E Relaxation to a conic program: Infinite dimensional minimization problem

Approach Difficult minimization problem E ∗ E t 0 E t Relaxation to a conic program: Infinite dimensional minimization problem

Approach Conic dual: Infinite dimensional maximization problem Difficult minimization problem E ∗ E t 0 E t Relaxation to a conic program: Infinite dimensional minimization problem

Approach Conic dual: Infinite dimensional maximization problem Difficult minimization problem E ∗ E ∗ E t 0 E t t,d Relaxation to a conic program: Infinite dimensional minimization problem

Approach Conic dual: Infinite dimensional maximization problem Difficult minimization problem E ∗ E ∗ E t 0 E t t,d Relaxation to a conic program: Infinite dimensional minimization problem Semi-infinite semidefinite program

Finite container ◮ If V = { 1 , . . . , n } is a finite set, then E is a polynomial optimization problem: � � � � w ( { i, j } ) x i x j : x ∈ { 0 , 1 } n , E = min x i = N { i,j }∈ ( V 2 ) i ∈ V

Finite container ◮ If V = { 1 , . . . , n } is a finite set, then E is a polynomial optimization problem: � � � � w ( { i, j } ) x i x j : x ∈ { 0 , 1 } n , E = min x i = N { i,j }∈ ( V 2 ) i ∈ V ◮ The Lasserre hierarchy gives a chain E 1 ≤ E 2 ≤ · · · ≤ E n of lower bounds to the optimal energy E :

Finite container ◮ If V = { 1 , . . . , n } is a finite set, then E is a polynomial optimization problem: � � � � w ( { i, j } ) x i x j : x ∈ { 0 , 1 } n , E = min x i = N { i,j }∈ ( V 2 ) i ∈ V ◮ The Lasserre hierarchy gives a chain E 1 ≤ E 2 ≤ · · · ≤ E n of lower bounds to the optimal energy E : w ( S ) y ( S ) : y ∈ R ( V � � ≤ 2 t ) , y ( ∅ ) = 1 , � � y ( A ∪ B ) ≤ t ) � 0 , E t = min A,B ∈ ( V S ∈ ( V 2 ) �� � V � y ( T ∪ { x } ) = Ny ( T ) for T ∈ ≤ 2 t − 1 x ∈ V

Infinite container ◮ Assume V is a compact Hausdorff space and w continuous

Infinite container ◮ Assume V is a compact Hausdorff space and w continuous ◮ � V � \ {∅} gets its topology as a quotient of V t ≤ t

Infinite container ◮ Assume V is a compact Hausdorff space and w continuous ◮ � V � \ {∅} gets its topology as a quotient of V t ≤ t ◮ Generalization (here s = min { 2 t, N } ): � V � V � V � ) ≥ 0 , A ∗ � � � E t = min λ ( w ) : λ ∈ M ( t λ ∈ M ( × ) � 0 , ≤ s ≤ t ≤ t � � V � N � � λ ( ) = for i = 0 , . . . , s i i

Infinite container ◮ Assume V is a compact Hausdorff space and w continuous ◮ � V � \ {∅} gets its topology as a quotient of V t ≤ t ◮ Generalization (here s = min { 2 t, N } ): � V � V � V � ) ≥ 0 , A ∗ � � � E t = min λ ( w ) : λ ∈ M ( t λ ∈ M ( × ) � 0 , ≤ s ≤ t ≤ t � � V � N � � λ ( ) = for i = 0 , . . . , s i i ◮ λ generalizes the moment vector y

Infinite container ◮ Assume V is a compact Hausdorff space and w continuous ◮ � V � \ {∅} gets its topology as a quotient of V t ≤ t ◮ Generalization (here s = min { 2 t, N } ): � V � V � V � ) ≥ 0 , A ∗ � � � E t = min λ ( w ) : λ ∈ M ( t λ ∈ M ( × ) � 0 , ≤ s ≤ t ≤ t � � V � N � � λ ( ) = for i = 0 , . . . , s i i ◮ λ generalizes the moment vector y � V � V � V � V ◮ M ( � × � ) � 0 is dual to the cone C ( � × � ) � 0 of ≤ t ≤ t ≤ t ≤ t positive definite kernels

Infinite container ◮ Assume V is a compact Hausdorff space and w continuous ◮ � V � \ {∅} gets its topology as a quotient of V t ≤ t ◮ Generalization (here s = min { 2 t, N } ): � V � V � V � ) ≥ 0 , A ∗ � � � E t = min λ ( w ) : λ ∈ M ( t λ ∈ M ( × ) � 0 , ≤ s ≤ t ≤ t � � V � N � � λ ( ) = for i = 0 , . . . , s i i ◮ λ generalizes the moment vector y � V � V � V � V ◮ M ( � × � ) � 0 is dual to the cone C ( � × � ) � 0 of ≤ t ≤ t ≤ t ≤ t positive definite kernels ◮ Relaxation: If S is an N subset of V , then � χ S = δ R R ∈ ( S ≤ 2 t ) is feasible for E t

Infinite container ◮ Assume V is a compact Hausdorff space and w continuous ◮ � V � \ {∅} gets its topology as a quotient of V t ≤ t ◮ Generalization (here s = min { 2 t, N } ): � V � V � V � ) ≥ 0 , A ∗ � � � E t = min λ ( w ) : λ ∈ M ( t λ ∈ M ( × ) � 0 , ≤ s ≤ t ≤ t � � V � N � � λ ( ) = for i = 0 , . . . , s i i ◮ λ generalizes the moment vector y � V � V � V � V ◮ M ( � × � ) � 0 is dual to the cone C ( � × � ) � 0 of ≤ t ≤ t ≤ t ≤ t positive definite kernels ◮ Relaxation: If S is an N subset of V , then � χ S = δ R R ∈ ( S ≤ 2 t ) is feasible for E t ◮ We have E N = E

Infinite container ◮ Assume V is a compact Hausdorff space and w continuous ◮ � V � \ {∅} gets its topology as a quotient of V t ≤ t ◮ Generalization (here s = min { 2 t, N } ): � V � V � V � ) ≥ 0 , A ∗ � � � E t = min λ ( w ) : λ ∈ M ( t λ ∈ M ( × ) � 0 , ≤ s ≤ t ≤ t � � V � N � � λ ( ) = for i = 0 , . . . , s i i ◮ λ generalizes the moment vector y � V � V � V � V ◮ M ( � × � ) � 0 is dual to the cone C ( � × � ) � 0 of ≤ t ≤ t ≤ t ≤ t positive definite kernels ◮ Relaxation: If S is an N subset of V , then � χ S = δ R R ∈ ( S ≤ 2 t ) is feasible for E t ◮ We have E N = E ◮ Uses techniques from [de Laat-Vallentin 2013]: hierarchy for packing problems in discrete geometry

Dual hierarchy ◮ For lower bounds we need feasible solutions of the dual

Dual hierarchy ◮ For lower bounds we need feasible solutions of the dual ◮ In the dual hierarchy optimization is over scalars a i and � V � V � � positive definite kernels K ∈ C ( × ) � 0 : ≤ t ≤ t

Dual hierarchy ◮ For lower bounds we need feasible solutions of the dual ◮ In the dual hierarchy optimization is over scalars a i and � V � V � � positive definite kernels K ∈ C ( × ) � 0 : ≤ t ≤ t s � V � V � E ∗ � � N � � � t = sup a i : a 0 , . . . , a s ∈ R , K ∈ C ( × ) � 0 , i ≤ t ≤ t i =0 � � V a i − A t K ≤ w on � for i = 0 , . . . , s i

Dual hierarchy ◮ For lower bounds we need feasible solutions of the dual ◮ In the dual hierarchy optimization is over scalars a i and � V � V � � positive definite kernels K ∈ C ( × ) � 0 : ≤ t ≤ t s � V � V � E ∗ � � N � � � t = sup a i : a 0 , . . . , a s ∈ R , K ∈ C ( × ) � 0 , i ≤ t ≤ t i =0 � � V a i − A t K ≤ w on � for i = 0 , . . . , s i � V ◮ Techniquality: we only put a linear constraint for S ∈ � if i the points in S are not too close

Dual hierarchy ◮ For lower bounds we need feasible solutions of the dual ◮ In the dual hierarchy optimization is over scalars a i and � V � V � � positive definite kernels K ∈ C ( × ) � 0 : ≤ t ≤ t s � V � V � E ∗ � � N � � � t = sup a i : a 0 , . . . , a s ∈ R , K ∈ C ( × ) � 0 , i ≤ t ≤ t i =0 � � V a i − A t K ≤ w on � for i = 0 , . . . , s i � V ◮ Techniquality: we only put a linear constraint for S ∈ � if i the points in S are not too close ◮ Strong duality holds: E t = E ∗ t