Theta Bodies: p | S ≥ 0 ← − p ∈ Q k ( G ) Definition The k -th theta body of G = { g 1 , . . . , g m } is defined as TH k ( G ) := { x ∈ R n | p ( x ) ≥ 0 , ∀ p ∈ Q k ( G ) ∩ R [ X ] 1 } . ◮ We have TH 1 ( G ) ⊇ TH 2 ( G ) ⊇ · · · ⊇ TH k +1 ( G ) ⊇ · · · ⊇ cl(co( S )) .
Theta Bodies: p | S ≥ 0 ← − p ∈ Q k ( G ) Definition The k -th theta body of G = { g 1 , . . . , g m } is defined as TH k ( G ) := { x ∈ R n | p ( x ) ≥ 0 , ∀ p ∈ Q k ( G ) ∩ R [ X ] 1 } . ◮ We have TH 1 ( G ) ⊇ TH 2 ( G ) ⊇ · · · ⊇ TH k +1 ( G ) ⊇ · · · ⊇ cl(co( S )) . ◮ When Q ( G ) is Archimedean, by Putinar’s Positivstellensatz, p | S > 0 = ⇒ p ∈ Q k ( G ) for some k ∈ N . Hence, we have ∞ � cl(co( S )) = TH k ( G ) . k =1
Theta Bodies: p | S ≥ 0 ← − p ∈ Q k ( G ) Definition The k -th theta body of G = { g 1 , . . . , g m } is defined as TH k ( G ) := { x ∈ R n | p ( x ) ≥ 0 , ∀ p ∈ Q k ( G ) ∩ R [ X ] 1 } . ◮ We have TH 1 ( G ) ⊇ TH 2 ( G ) ⊇ · · · ⊇ TH k +1 ( G ) ⊇ · · · ⊇ cl(co( S )) . ◮ When Q ( G ) is Archimedean, by Putinar’s Positivstellensatz, p | S > 0 = ⇒ p ∈ Q k ( G ) for some k ∈ N . Hence, we have ∞ � cl(co( S )) = TH k ( G ) . k =1 ◮ If PP-BDR holds for S with fixed order k then cl (co( S )) = TH k ( G ) .
Theta Bodies: p | S ≥ 0 ← − p ∈ Q k ( G ) Definition The k -th theta body of G = { g 1 , . . . , g m } is defined as TH k ( G ) := { x ∈ R n | p ( x ) ≥ 0 , ∀ p ∈ Q k ( G ) ∩ R [ X ] 1 } . ◮ We have TH 1 ( G ) ⊇ TH 2 ( G ) ⊇ · · · ⊇ TH k +1 ( G ) ⊇ · · · ⊇ cl(co( S )) . ◮ When Q ( G ) is Archimedean, by Putinar’s Positivstellensatz, p | S > 0 = ⇒ p ∈ Q k ( G ) for some k ∈ N . Hence, we have ∞ � cl(co( S )) = TH k ( G ) . k =1 ◮ If PP-BDR holds for S with fixed order k then cl (co( S )) = TH k ( G ) . See [Gouveia, Parrilo, Thomas] for V R ( I ) being compact .
Example [Gouveia, Thomas] Given two curves cut out by g 1 = X 4 1 − X 2 2 − X 2 3 , g 2 = X 4 1 + X 2 1 + X 2 2 − 1 . Its first theta body is an ellipsoid { x ∈ R 3 | x 2 1 + 2 x 2 2 + x 2 3 ≤ 1 }
Example [Gouveia, Thomas] Given two curves cut out by g 1 = X 4 1 − X 2 2 − X 2 3 , g 2 = X 4 1 + X 2 1 + X 2 2 − 1 . Its first theta body is an ellipsoid { x ∈ R 3 | x 2 1 + 2 x 2 2 + x 2 3 ≤ 1 } The second and third theta bodies:
Lasserre’s Semidefinite Relaxations of cl (co( S )) Given y = { y α } , let L y : R [ X ] → R be the linear functional �� � � q α X α �→ q α y α . L y α α
Lasserre’s Semidefinite Relaxations of cl (co( S )) Given y = { y α } , let L y : R [ X ] → R be the linear functional �� � � q α X α �→ q α y α . L y α α Moment matrix M k ( y ) with rows and columns indexed in the basis X α M k ( y )( α, β ) := L y ( X α X β ) = y α + β , α, β ∈ N n k , | α | , | β | ≤ k.
Lasserre’s Semidefinite Relaxations of cl (co( S )) Given y = { y α } , let L y : R [ X ] → R be the linear functional �� � � q α X α �→ q α y α . L y α α Moment matrix M k ( y ) with rows and columns indexed in the basis X α M k ( y )( α, β ) := L y ( X α X β ) = y α + β , α, β ∈ N n k , | α | , | β | ≤ k. For instance, in R 2 1 1 X 1 X 2 y 00 y 10 y 01 � � X 2 1 = �→ M 1 ( y ) = X 1 X 1 X 2 X 1 X 1 X 2 y 10 y 20 y 11 1 X 2 X 2 X 2 X 1 X 2 y 01 y 11 y 02 2
Lasserre’s Semidefinite Relaxations of cl (co( S )) Given y = { y α } , let L y : R [ X ] → R be the linear functional �� � � q α X α �→ q α y α . L y α α Moment matrix M k ( y ) with rows and columns indexed in the basis X α M k ( y )( α, β ) := L y ( X α X β ) = y α + β , α, β ∈ N n k , | α | , | β | ≤ k. For instance, in R 2 1 1 X 1 X 2 y 00 y 10 y 01 � � X 2 1 = �→ M 1 ( y ) = X 1 X 1 X 2 X 1 X 1 X 2 y 10 y 20 y 11 1 X 2 X 2 X 2 X 1 X 2 y 01 y 11 y 02 2 ◮ We have M k ( y ) � 0 ⇐ ⇒ L ( h 2 ) ≥ 0 , ∀ h ∈ R [ X ] k
Lasserre’s Semidefinite Relaxations of cl (co( S )) Given a polynomial p ( X ) = � γ p γ X γ , let d p = ⌈ deg( p ) / 2 ⌉ .
Lasserre’s Semidefinite Relaxations of cl (co( S )) Given a polynomial p ( X ) = � γ p γ X γ , let d p = ⌈ deg( p ) / 2 ⌉ . Localizing Moment Matrix M k ( py ) with rows and columns indexed in the basis X α � M k ( py )( α, β ) = L y ( pX α X β ) = p γ y α + β + γ , α, β ∈ N n k , | α | , | β | ≤ k γ ∈ N n
Lasserre’s Semidefinite Relaxations of cl (co( S )) Given a polynomial p ( X ) = � γ p γ X γ , let d p = ⌈ deg( p ) / 2 ⌉ . Localizing Moment Matrix M k ( py ) with rows and columns indexed in the basis X α � M k ( py )( α, β ) = L y ( pX α X β ) = p γ y α + β + γ , α, β ∈ N n k , | α | , | β | ≤ k γ ∈ N n For instance, in R 2 , with p ( X 1 , X 2 ) = 1 − X 2 1 − X 2 2 1 − y 20 − y 02 y 10 − y 30 − y 12 y 01 − y 21 − y 03 M 1 ( py ) = y 10 − y 30 − y 12 y 20 − y 40 − y 22 y 11 − y 31 − y 13 y 01 − y 21 − y 03 y 11 − y 31 − y 13 y 02 − y 22 − y 04
Lasserre’s Semidefinite Relaxations of cl (co( S )) Given a polynomial p ( X ) = � γ p γ X γ , let d p = ⌈ deg( p ) / 2 ⌉ . Localizing Moment Matrix M k ( py ) with rows and columns indexed in the basis X α � M k ( py )( α, β ) = L y ( pX α X β ) = p γ y α + β + γ , α, β ∈ N n k , | α | , | β | ≤ k γ ∈ N n For instance, in R 2 , with p ( X 1 , X 2 ) = 1 − X 2 1 − X 2 2 1 − y 20 − y 02 y 10 − y 30 − y 12 y 01 − y 21 − y 03 M 1 ( py ) = y 10 − y 30 − y 12 y 20 − y 40 − y 22 y 11 − y 31 − y 13 y 01 − y 21 − y 03 y 11 − y 31 − y 13 y 02 − y 22 − y 04 We have ⇒ L y ( h 2 p ) ≥ 0 , ∀ h ∈ R [ X ] , deg( h ) ≤ k − d p . ◮ M k − d p ( py ) � 0 ⇐
Lasserre’s Semidefinite Representation of cl (co( S )) � n + k � Let G = { g 1 , . . . , g m } , s ( k ) := and k j := ⌈ deg g j / 2 ⌉ . n
Lasserre’s Semidefinite Representation of cl (co( S )) � n + k � Let G = { g 1 , . . . , g m } , s ( k ) := and k j := ⌈ deg g j / 2 ⌉ . n Definition The k -th Lasserre’s relaxation is defined as: ∃ y ∈ R s (2 k ) , s.t. L y (1) = 1 , x ∈ R n Ω k ( G ) := L y ( X i ) = x i , i = 1 , . . . , n, M k ( y ) � 0 , . M k − k j ( g j y ) � 0 , j = 1 , . . . , m,
Lasserre’s Semidefinite Representation of cl (co( S )) � n + k � Let G = { g 1 , . . . , g m } , s ( k ) := and k j := ⌈ deg g j / 2 ⌉ . n Definition The k -th Lasserre’s relaxation is defined as: ∃ y ∈ R s (2 k ) , s.t. L y (1) = 1 , x ∈ R n Ω k ( G ) := L y ( X i ) = x i , i = 1 , . . . , n, M k ( y ) � 0 , . M k − k j ( g j y ) � 0 , j = 1 , . . . , m, ◮ When Q ( G ) is Archimedean, by Putinar’s Positivstellensatz ( dual side ), ∞ � cl(co( S )) = cl (Ω k ( G )) . k =1
Lasserre’s Semidefinite Representation of cl (co( S )) � n + k � Let G = { g 1 , . . . , g m } , s ( k ) := and k j := ⌈ deg g j / 2 ⌉ . n Definition The k -th Lasserre’s relaxation is defined as: ∃ y ∈ R s (2 k ) , s.t. L y (1) = 1 , x ∈ R n Ω k ( G ) := L y ( X i ) = x i , i = 1 , . . . , n, M k ( y ) � 0 , . M k − k j ( g j y ) � 0 , j = 1 , . . . , m, ◮ When Q ( G ) is Archimedean, by Putinar’s Positivstellensatz ( dual side ), ∞ � cl(co( S )) = cl (Ω k ( G )) . k =1 ◮ If PP-BDR ( p > 0 on S then p ∈ Q k ( G ) ) holds for S with order k , then co( S ) = Ω k ( G ) .
Lasserre’s Semidefinite Representation of cl (co( S )) � n + k � Let G = { g 1 , . . . , g m } , s ( k ) := and k j := ⌈ deg g j / 2 ⌉ . n Definition The k -th Lasserre’s relaxation is defined as: ∃ y ∈ R s (2 k ) , s.t. L y (1) = 1 , x ∈ R n Ω k ( G ) := L y ( X i ) = x i , i = 1 , . . . , n, M k ( y ) � 0 , . M k − k j ( g j y ) � 0 , j = 1 , . . . , m, ◮ When Q ( G ) is Archimedean, by Putinar’s Positivstellensatz ( dual side ), ∞ � cl(co( S )) = cl (Ω k ( G )) . k =1 ◮ If PP-BDR ( p > 0 on S then p ∈ Q k ( G ) ) holds for S with order k , then co( S ) = Ω k ( G ) . ◮ co( S ) ⊆ Ω k ( G ) ⊆ TH k ( G ) . If Q k ( G ) is closed , TH k ( G ) = cl(Ω k ( G )) .
When S is not Compact Consider the basic semialgebraic set S := { ( x 1 , x 2 ) ∈ R 2 | x 1 ≥ 0 , x 2 1 − x 3 2 ≥ 0 } .
When S is not Compact Consider the basic semialgebraic set S := { ( x 1 , x 2 ) ∈ R 2 | x 1 ≥ 0 , x 2 1 − x 3 2 ≥ 0 } . For any linear function in Q k ( G ) , c 1 X 1 + c 2 X 2 + c 0 = σ 0 ( X 1 , X 2 ) + σ 1 ( X 1 , X 2 ) X 1 + σ 2 ( X 1 , X 2 )( X 2 1 − X 3 2 ) ⇒ c 1 0 + c 2 X 2 + c 0 = σ 0 (0 , X 2 ) + σ 1 (0 , X 2 )0 + σ 2 (0 , X 2 )(0 2 − X 3 = 2 ) TH k ( G ) = cl ( Ω k ( G )) = { ( x 1 , x 2 ) ∈ R 2 | x 1 ≥ 0 } . = ⇒ c 2 = 0 = ⇒ cl (co( S )) cl (Ω k ( G )) = TH k ( G )
When S is not Compact Consider the basic semialgebraic set S := { ( x 1 , x 2 ) ∈ R 2 | x 1 ≥ 0 , x 2 1 − x 3 2 ≥ 0 } . For any linear function in Q k ( G ) , c 1 X 1 + c 2 X 2 + c 0 = σ 0 ( X 1 , X 2 ) + σ 1 ( X 1 , X 2 ) X 1 + σ 2 ( X 1 , X 2 )( X 2 1 − X 3 2 ) ⇒ c 1 0 + c 2 X 2 + c 0 = σ 0 (0 , X 2 ) + σ 1 (0 , X 2 )0 + σ 2 (0 , X 2 )(0 2 − X 3 = 2 ) TH k ( G ) = cl ( Ω k ( G )) = { ( x 1 , x 2 ) ∈ R 2 | x 1 ≥ 0 } . = ⇒ c 2 = 0 = ⇒
When S is not Compact Consider the basic semialgebraic set S := { ( x 1 , x 2 ) ∈ R 2 | x 1 ≥ 0 , x 2 1 − x 3 2 ≥ 0 } . For any linear function in Q k ( G ) , c 1 X 1 + c 2 X 2 + c 0 = σ 0 ( X 1 , X 2 ) + σ 1 ( X 1 , X 2 ) X 1 + σ 2 ( X 1 , X 2 )( X 2 1 − X 3 2 ) ⇒ c 1 0 + c 2 X 2 + c 0 = σ 0 (0 , X 2 ) + σ 1 (0 , X 2 )0 + σ 2 (0 , X 2 )(0 2 − X 3 = 2 ) TH k ( G ) = cl ( Ω k ( G )) = { ( x 1 , x 2 ) ∈ R 2 | x 1 ≥ 0 } . = ⇒ c 2 = 0 = ⇒ cl (co( S )) cl (Ω k ( G )) = TH k ( G )
Semidefinite Representation of a Noncompact Set S ◮ Homogenization f ( � ˜ X ) = X deg( f ) f ( X/X 0 ) ∈ R [ X 0 , X 1 , . . . , X n ] = R [ � X ] . 0
Semidefinite Representation of a Noncompact Set S ◮ Homogenization f ( � ˜ X ) = X deg( f ) f ( X/X 0 ) ∈ R [ X 0 , X 1 , . . . , X n ] = R [ � X ] . 0 ◮ Lifting S to a cone � S 1 : S := { x ∈ R n | g 1 ( x ) ≥ 0 , . . . , g m ( x ) ≥ 0 } , x ∈ R n +1 | ˜ � S 1 := { ˜ g 1 (˜ x ) ≥ 0 , . . . , ˜ g m (˜ x ) ≥ 0 , x 0 > 0 } .
Semidefinite Representation of a Noncompact Set S ◮ Homogenization f ( � ˜ X ) = X deg( f ) f ( X/X 0 ) ∈ R [ X 0 , X 1 , . . . , X n ] = R [ � X ] . 0 ◮ Lifting S to a cone � S 1 : S := { x ∈ R n | g 1 ( x ) ≥ 0 , . . . , g m ( x ) ≥ 0 } , x ∈ R n +1 | ˜ � S 1 := { ˜ g 1 (˜ x ) ≥ 0 , . . . , ˜ g m (˜ x ) ≥ 0 , x 0 > 0 } . � � ˜ � ◮ f ( x ) ≥ 0 on S ⇐ ⇒ f ( ˜ x ) ≥ 0 on cl . S 1
Semidefinite Representation of a Noncompact Set S ◮ Homogenization f ( � ˜ X ) = X deg( f ) f ( X/X 0 ) ∈ R [ X 0 , X 1 , . . . , X n ] = R [ � X ] . 0 ◮ Lifting S to a cone � S 1 : S := { x ∈ R n | g 1 ( x ) ≥ 0 , . . . , g m ( x ) ≥ 0 } , x ∈ R n +1 | ˜ � S 1 := { ˜ g 1 (˜ x ) ≥ 0 , . . . , ˜ g m (˜ x ) ≥ 0 , x 0 > 0 } . � � ˜ � ◮ f ( x ) ≥ 0 on S ⇐ ⇒ f ( ˜ x ) ≥ 0 on cl . S 1 ◮ Compactification: x ∈ R n +1 | ˜ � x � 2 S := { ˜ g 1 (˜ x ) ≥ 0 , . . . , ˜ g m (˜ x ) ≥ 0 , x 0 ≥ 0 , � ˜ 2 = 1 } .
Semidefinite Representation of a Noncompact Set S ◮ Homogenization f ( � ˜ X ) = X deg( f ) f ( X/X 0 ) ∈ R [ X 0 , X 1 , . . . , X n ] = R [ � X ] . 0 ◮ Lifting S to a cone � S 1 : S := { x ∈ R n | g 1 ( x ) ≥ 0 , . . . , g m ( x ) ≥ 0 } , x ∈ R n +1 | ˜ � S 1 := { ˜ g 1 (˜ x ) ≥ 0 , . . . , ˜ g m (˜ x ) ≥ 0 , x 0 > 0 } . � � ˜ � ◮ f ( x ) ≥ 0 on S ⇐ ⇒ f ( ˜ x ) ≥ 0 on cl . S 1 ◮ Compactification: x ∈ R n +1 | ˜ � x � 2 S := { ˜ g 1 (˜ x ) ≥ 0 , . . . , ˜ g m (˜ x ) ≥ 0 , x 0 ≥ 0 , � ˜ 2 = 1 } . x ) ≥ 0 on � ◮ f ( x ) ≥ 0 on S � ˜ f ( ˜ S .
Semidefinite Representation of a Noncompact Set S ◮ Homogenization f ( � ˜ X ) = X deg( f ) f ( X/X 0 ) ∈ R [ X 0 , X 1 , . . . , X n ] = R [ � X ] . 0 ◮ Lifting S to a cone � S 1 : S := { x ∈ R n | g 1 ( x ) ≥ 0 , . . . , g m ( x ) ≥ 0 } , x ∈ R n +1 | ˜ � S 1 := { ˜ g 1 (˜ x ) ≥ 0 , . . . , ˜ g m (˜ x ) ≥ 0 , x 0 > 0 } . � � ˜ � ◮ f ( x ) ≥ 0 on S ⇐ ⇒ f ( ˜ x ) ≥ 0 on cl . S 1 ◮ Compactification: x ∈ R n +1 | ˜ � x � 2 S := { ˜ g 1 (˜ x ) ≥ 0 , . . . , ˜ g m (˜ x ) ≥ 0 , x 0 ≥ 0 , � ˜ 2 = 1 } . x ) ≥ 0 on � ◮ f ( x ) ≥ 0 on S � ˜ f ( ˜ S . x 2 ≥ 0 on { ( x 1 , x 2 ) ∈ R 2 | x 2 − x 2 1 ≥ 0 } but x 2 can be < 0 on � S .
Semidefinite Representation of a Noncompact Set S Definition � � � = � S is closed at ∞ if cl S 1 S 2 where x ∈ R n +1 | ˜ � { ˜ x ) ≥ 0 , . . . , ˜ x ) ≥ 0 , x 0 > 0 } , S 1 := g 1 (˜ g m (˜ x ∈ R n +1 | ˜ � S 2 := { ˜ g 1 (˜ x ) ≥ 0 , . . . , ˜ g m (˜ x ) ≥ 0 , x 0 ≥ 0 } .
Semidefinite Representation of a Noncompact Set S Definition � � � = � S is closed at ∞ if cl S 1 S 2 where x ∈ R n +1 | ˜ � { ˜ x ) ≥ 0 , . . . , ˜ x ) ≥ 0 , x 0 > 0 } , S 1 := g 1 (˜ g m (˜ x ∈ R n +1 | ˜ � S 2 := { ˜ g 1 (˜ x ) ≥ 0 , . . . , ˜ g m (˜ x ) ≥ 0 , x 0 ≥ 0 } . S = { ( x 1 , x 2 ) ∈ R 2 | x 2 − x 2 1 ≥ 0 } , S 2 = { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 0 x 2 − x 2 � 1 ≥ 0 , x 0 ≥ 0 } , � � = { (0 , 0 , x 2 ) ∈ R 3 | x 2 < 0 } = � � S 2 \ cl S 1 ⇒ S is not closed at ∞ .
Modified Lasserre’s Hierarchy and Theta Body x ∈ R n +1 | ˜ � x � 2 S := { ˜ x ) ≥ 0 , . . . , ˜ x ) ≥ 0 , x 0 ≥ 0 , � ˜ 2 = 1 } , g 1 (˜ g m (˜ � g m , X 0 , � � 2 − 1 , 1 − � � X � 2 X � 2 G := { ˜ g 1 , . . . , ˜ 2 } .
Modified Lasserre’s Hierarchy and Theta Body x ∈ R n +1 | ˜ � x � 2 S := { ˜ x ) ≥ 0 , . . . , ˜ x ) ≥ 0 , x 0 ≥ 0 , � ˜ 2 = 1 } , g 1 (˜ g m (˜ � g m , X 0 , � � 2 − 1 , 1 − � � X � 2 X � 2 G := { ˜ g 1 , . . . , ˜ 2 } . Modified Lasserre’s Hierarchy ∃ y ∈ R ˜ s (2 k ) , s.t. L y ( X 0 ) = 1 , L y ( X i ) = x i , i = 1 , . . . , n, Ω k ( � � x ∈ R n G ) := . M k ( y ) � 0 , M k − k j (˜ g j y ) � 0 , j = 1 , . . . , m M k − 1 ( X 0 y ) � 0 , M k − 1 (( � � X � 2 2 − 1) y ) = 0
Modified Lasserre’s Hierarchy and Theta Body x ∈ R n +1 | ˜ � x � 2 S := { ˜ x ) ≥ 0 , . . . , ˜ x ) ≥ 0 , x 0 ≥ 0 , � ˜ 2 = 1 } , g 1 (˜ g m (˜ � g m , X 0 , � � 2 − 1 , 1 − � � X � 2 X � 2 G := { ˜ g 1 , . . . , ˜ 2 } . Modified Lasserre’s Hierarchy ∃ y ∈ R ˜ s (2 k ) , s.t. L y ( X 0 ) = 1 , L y ( X i ) = x i , i = 1 , . . . , n, Ω k ( � � x ∈ R n G ) := . M k ( y ) � 0 , M k − k j (˜ g j y ) � 0 , j = 1 , . . . , m M k − 1 ( X 0 y ) � 0 , M k − 1 (( � � X � 2 2 − 1) y ) = 0 Modified Theta Body G ) := { x ∈ R n | ˜ TH k ( � � ∀ ˜ l ∈ Q k ( � G ) ∩ P [ � l (1 , x ) ≥ 0 , X ] 1 } . where P [ � X ] 1 is a set of homogeneous polynomials of degree one in R [ � X ] .
Pointed Convex Cone A convex cone K is pointed if it is closed and contains no lines .
Pointed Convex Cone A convex cone K is pointed if it is closed and contains no lines .
Pointed Convex Cone A convex cone K is pointed if it is closed and contains no lines . ⇒ { ∃ c ∈ R n s.t. � c , x � > 0 for all x ∈ K \{ 0 } } , ◮ K is pointed ⇐ Remark: � c , x � = c T x = c 1 x 1 + · · · + c n x n .
Modified Lasserre’s Hierarchy and Theta Body Assumptions [Guo, Wang, Zhi] � � � = � ◮ S is closed at ∞ , i.e., cl S 1 S 2 .
Modified Lasserre’s Hierarchy and Theta Body Assumptions [Guo, Wang, Zhi] � � � = � ◮ S is closed at ∞ , i.e., cl S 1 S 2 . ◮ The convex cone co(cl( � S 1 )) is pointed .
Modified Lasserre’s Hierarchy and Theta Body Assumptions [Guo, Wang, Zhi] � � � = � ◮ S is closed at ∞ , i.e., cl S 1 S 2 . ◮ The convex cone co(cl( � S 1 )) is pointed . Under the assumptions [Guo, Wang, Zhi] � � Ω k ( � � ⊆ � TH k ( � ◮ cl(co( S )) ⊆ cl G ) G ) for every k ∈ N and � � ∞ ∞ � � Ω k ( � � TH k ( � � cl(co( S )) = cl G ) = G ) . k =1 k =1
Modified Lasserre’s Hierarchy and Theta Body Assumptions [Guo, Wang, Zhi] � � � = � ◮ S is closed at ∞ , i.e., cl S 1 S 2 . ◮ The convex cone co(cl( � S 1 )) is pointed . Under the assumptions [Guo, Wang, Zhi] � � Ω k ( � � ⊆ � TH k ( � ◮ cl(co( S )) ⊆ cl G ) G ) for every k ∈ N and � � ∞ ∞ � � Ω k ( � � TH k ( � � cl(co( S )) = cl G ) = G ) . k =1 k =1 ◮ If the PP-BDR property holds for � S with order k , then � � Ω k ( � � = � TH k ( � cl(co( S )) = cl G ) G ) .
Modified Lasserre’s Hierarchy and Theta Body Assumptions [Guo, Wang, Zhi] � � � = � ◮ S is closed at ∞ , i.e., cl S 1 S 2 . ◮ The convex cone co(cl( � S 1 )) is pointed . Under the assumptions [Guo, Wang, Zhi] � � Ω k ( � � ⊆ � TH k ( � ◮ cl(co( S )) ⊆ cl G ) G ) for every k ∈ N and � � ∞ ∞ � � Ω k ( � � TH k ( � � cl(co( S )) = cl G ) = G ) . k =1 k =1 ◮ If the PP-BDR property holds for � S with order k , then � � Ω k ( � � = � TH k ( � cl(co( S )) = cl G ) G ) . � � ◮ If Q k ( � G ) is closed, then � TH k ( � Ω k ( � � G ) = cl G ) .
Example (continued) Consider the set S := { ( x 1 , x 2 ) ∈ R 2 | x 1 ≥ 0 , x 2 1 − x 3 2 ≥ 0 } , we have S 1 = { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 1 ≥ 0 , x 0 x 2 � 1 − x 3 2 ≥ 0 , x 0 > 0 } , S 2 := { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 1 ≥ 0 , x 0 x 2 � 1 − x 3 2 ≥ 0 , x 0 ≥ 0 } , S := { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 1 ≥ 0 , x 0 x 2 � 1 − x 3 2 ≥ 0 , x 0 ≥ 0 , x 2 0 + x 2 1 + x 2 2 = 1 } .
Example (continued) Consider the set S := { ( x 1 , x 2 ) ∈ R 2 | x 1 ≥ 0 , x 2 1 − x 3 2 ≥ 0 } , we have S 1 = { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 1 ≥ 0 , x 0 x 2 � 1 − x 3 2 ≥ 0 , x 0 > 0 } , S 2 := { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 1 ≥ 0 , x 0 x 2 � 1 − x 3 2 ≥ 0 , x 0 ≥ 0 } , S := { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 1 ≥ 0 , x 0 x 2 � 1 − x 3 2 ≥ 0 , x 0 ≥ 0 , x 2 0 + x 2 1 + x 2 2 = 1 } . � � � = � cl S 1 S 2 ⇒ S is closed at ∞
Example (continued) Consider the set S := { ( x 1 , x 2 ) ∈ R 2 | x 1 ≥ 0 , x 2 1 − x 3 2 ≥ 0 } , we have S 1 = { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 1 ≥ 0 , x 0 x 2 � 1 − x 3 2 ≥ 0 , x 0 > 0 } , S 2 := { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 1 ≥ 0 , x 0 x 2 � 1 − x 3 2 ≥ 0 , x 0 ≥ 0 } , S := { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 1 ≥ 0 , x 0 x 2 � 1 − x 3 2 ≥ 0 , x 0 ≥ 0 , x 2 0 + x 2 1 + x 2 2 = 1 } . � � � = � cl S 1 S 2 ⇒ S is closed at ∞ 2 X 0 + 2 X 1 − 3 X 2 > 0 on co( � S 2 ) \{ 0 } ⇒ co( � S 2 ) is pointed
Example (continued) Consider the set S := { ( x 1 , x 2 ) ∈ R 2 | x 1 ≥ 0 , x 2 1 − x 3 2 ≥ 0 } , we have S 1 = { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 1 ≥ 0 , x 0 x 2 � 1 − x 3 2 ≥ 0 , x 0 > 0 } , S 2 := { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 1 ≥ 0 , x 0 x 2 � 1 − x 3 2 ≥ 0 , x 0 ≥ 0 } , S := { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 1 ≥ 0 , x 0 x 2 � 1 − x 3 2 ≥ 0 , x 0 ≥ 0 , x 2 0 + x 2 1 + x 2 2 = 1 } . � � � = � cl S 1 S 2 ⇒ S is closed at ∞ 2 X 0 + 2 X 1 − 3 X 2 > 0 on co( � S 2 ) \{ 0 } ⇒ co( � S 2 ) is pointed � � Ω 3 ( � S 2 S G )
Essentiality of Closedness at Infinity The convergence might fail if co(cl( � S 1 ) is pointed but S is not closed at infinity .
Essentiality of Closedness at Infinity The convergence might fail if co(cl( � S 1 ) is pointed but S is not closed at infinity . Consider the set S := { ( x 1 , x 2 ) ∈ R 2 | x 2 − x 2 1 ≥ 0 } . S 1 = { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 0 x 2 − x 2 � 1 ≥ 0 , x 0 > 0 } , S 2 = { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 0 x 2 − x 2 � 1 ≥ 0 , x 0 ≥ 0 } .
Essentiality of Closedness at Infinity The convergence might fail if co(cl( � S 1 ) is pointed but S is not closed at infinity . Consider the set S := { ( x 1 , x 2 ) ∈ R 2 | x 2 − x 2 1 ≥ 0 } . S 1 = { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 0 x 2 − x 2 � 1 ≥ 0 , x 0 > 0 } , S 2 = { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 0 x 2 − x 2 � 1 ≥ 0 , x 0 ≥ 0 } . � � = { (0 , 0 , x 2 ) ∈ R 3 | x 2 < 0 } � = ∅ = ◮ � � S 2 \ cl S 1 ⇒ S is not closed at ∞ .
Essentiality of Closedness at Infinity The convergence might fail if co(cl( � S 1 ) is pointed but S is not closed at infinity . Consider the set S := { ( x 1 , x 2 ) ∈ R 2 | x 2 − x 2 1 ≥ 0 } . S 1 = { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 0 x 2 − x 2 � 1 ≥ 0 , x 0 > 0 } , S 2 = { ( x 0 , x 1 , x 2 ) ∈ R 3 | x 0 x 2 − x 2 � 1 ≥ 0 , x 0 ≥ 0 } . � � = { (0 , 0 , x 2 ) ∈ R 3 | x 2 < 0 } � = ∅ = ◮ � � S 2 \ cl S 1 ⇒ S is not closed at ∞ . � � = R 2 � = cl (co( S )) . ◮ � TH k ( � Ω k ( � � G ) = cl G )
Closedness at Infinity We notice that the property of closedness at ∞ depends not only on S but also on its generators .
Closedness at Infinity We notice that the property of closedness at ∞ depends not only on S but also on its generators . ◮ Let S ′ := { ( x 1 , x 2 ) ∈ R 2 | x 2 − x 2 1 ≥ 0 , 1 + x 2 ≥ 0 } .
Closedness at Infinity We notice that the property of closedness at ∞ depends not only on S but also on its generators . ◮ Let S ′ := { ( x 1 , x 2 ) ∈ R 2 | x 2 − x 2 1 ≥ 0 , 1 + x 2 ≥ 0 } . ◮ S = S ′ since 1 + X 2 > 0 on S.
Closedness at Infinity We notice that the property of closedness at ∞ depends not only on S but also on its generators . ◮ Let S ′ := { ( x 1 , x 2 ) ∈ R 2 | x 2 − x 2 1 ≥ 0 , 1 + x 2 ≥ 0 } . ◮ S = S ′ since 1 + X 2 > 0 on S. ◮ S ′ is closed at ∞ .
Essentiality of Pointedness The convergence might fail if S is closed at infinity but co(cl( � S 1 )) is not pointed .
Essentiality of Pointedness The convergence might fail if S is closed at infinity but co(cl( � S 1 )) is not pointed . Example Consider the set S = { ( x 1 , x 2 ) ∈ R 2 | x 3 2 − x 2 1 ≥ 0 } , we have � S 1 = { x 3 2 − x 2 1 x 0 ≥ 0 , x 0 > 0 } , S 2 = { x 3 � 2 − x 2 1 x 0 ≥ 0 , x 0 ≥ 0 } .
Essentiality of Pointedness The convergence might fail if S is closed at infinity but co(cl( � S 1 )) is not pointed . Example Consider the set S = { ( x 1 , x 2 ) ∈ R 2 | x 3 2 − x 2 1 ≥ 0 } , we have � S 1 = { x 3 2 − x 2 1 x 0 ≥ 0 , x 0 > 0 } , S 2 = { x 3 � 2 − x 2 1 x 0 ≥ 0 , x 0 ≥ 0 } . ◮ The convex cone co(cl( � S 1 )) is not pointed since � � √ ǫ ) = (0 , ± 1 , 0) and ( 0 , ± 1 , 0 ) ∈ cl � lim ǫ → 0 ( ǫ, ± 1 , ⇒ 3 S 1 = c 0 X 0 + c 1 X 1 + c 2 X 2 will be ± c 1 at (0 , ± 1 , 0) .
Essentiality of Pointedness The convergence might fail if S is closed at infinity but co(cl( � S 1 )) is not pointed . Example Consider the set S = { ( x 1 , x 2 ) ∈ R 2 | x 3 2 − x 2 1 ≥ 0 } , we have � S 1 = { x 3 2 − x 2 1 x 0 ≥ 0 , x 0 > 0 } , S 2 = { x 3 � 2 − x 2 1 x 0 ≥ 0 , x 0 ≥ 0 } . ◮ The convex cone co(cl( � S 1 )) is not pointed since � � √ ǫ ) = (0 , ± 1 , 0) and ( 0 , ± 1 , 0 ) ∈ cl � lim ǫ → 0 ( ǫ, ± 1 , ⇒ 3 S 1 = c 0 X 0 + c 1 X 1 + c 2 X 2 will be ± c 1 at (0 , ± 1 , 0) . � � = R 2 � = cl (co( S )) . ◮ We have � TH k ( � Ω k ( � � G ) = cl G )
Summary We have shown ◮ how to compute semidefinite approximations of a noncompact semialgebraic set;
Summary We have shown ◮ how to compute semidefinite approximations of a noncompact semialgebraic set; ◮ under assumptions that S is closed at ∞ and co(cl( � S 1 )) is pointed , � � TH k ( � � Ω k ( � � G ) and cl G ) will converge to cl(co( S )) .
Summary We have shown ◮ how to compute semidefinite approximations of a noncompact semialgebraic set; ◮ under assumptions that S is closed at ∞ and co(cl( � S 1 )) is pointed , � � TH k ( � � Ω k ( � � G ) and cl G ) will converge to cl(co( S )) . ◮ the assumptions of pointedness and closedness at infinity are essential.
Outlines ◮ Semidefinite representations of the closure of the convex hull of S : � { x ∈ R n | p ( x ) ≥ 0 } . cl(co( S )) := p ∈ R [ X ] 1 ,p | S ≥ 0 ◮ Optimizing a parametric linear function over a real algebraic variety: 0 = sup x ∈ S c T x for unspecified parameters ; ◮ c ∗ ◮ S = V ∩ R n , V = { v ∈ C n | h 1 ( v ) = · · · = h p ( v ) = 0 } .
Optimizing a Parametric Linear Function We consider the optimization problem: c T x = c 1 x 1 + · · · + c n x n . c ∗ 0 := sup x ∈V∩ R n where V = { v ∈ C n | h 1 ( v ) = · · · = h p ( v ) = 0 } and c = ( c 1 , . . . , c n ) are unspecified parameters .
Optimizing a Parametric Linear Function We consider the optimization problem: c T x = c 1 x 1 + · · · + c n x n . c ∗ 0 := sup x ∈V∩ R n where V = { v ∈ C n | h 1 ( v ) = · · · = h p ( v ) = 0 } and c = ( c 1 , . . . , c n ) are unspecified parameters . ◮ Tarski-Seidenberg ’s theorem on quantifier elimination ensures that the optimal value function c ∗ 0 is a semialgebraic function.
Optimizing a Parametric Linear Function We consider the optimization problem: c T x = c 1 x 1 + · · · + c n x n . c ∗ 0 := sup x ∈V∩ R n where V = { v ∈ C n | h 1 ( v ) = · · · = h p ( v ) = 0 } and c = ( c 1 , . . . , c n ) are unspecified parameters . ◮ Tarski-Seidenberg ’s theorem on quantifier elimination ensures that the optimal value function c ∗ 0 is a semialgebraic function. The problem is how to compute a polynomial Φ ∈ R [ c 0 , c ] s.t. c ∗ 0 can be obtained by solving Φ( c 0 , γ ) = 0 for a generic γ ∈ R n ?
Previous Work ◮ Cylindrical algebraic decomposition (CAD): for any V , but limited to small n [Brown,Collins,Hong,McCallum...].
Previous Work ◮ Cylindrical algebraic decomposition (CAD): for any V , but limited to small n [Brown,Collins,Hong,McCallum...]. ◮ Using KKT equations: for V being irreducible , smooth and compact in R n [Rostalski, Sturmfels].
Previous Work ◮ Cylindrical algebraic decomposition (CAD): for any V , but limited to small n [Brown,Collins,Hong,McCallum...]. ◮ Using KKT equations: for V being irreducible , smooth and compact in R n [Rostalski, Sturmfels]. ◮ Using modified polar varieties: for the specialized optimization problem, V ∩ R n could be not compact [Greuet, Safey El Din].
Previous Work ◮ Cylindrical algebraic decomposition (CAD): for any V , but limited to small n [Brown,Collins,Hong,McCallum...]. ◮ Using KKT equations: for V being irreducible , smooth and compact in R n [Rostalski, Sturmfels]. ◮ Using modified polar varieties: for the specialized optimization problem, V ∩ R n could be not compact [Greuet, Safey El Din]. Our goal is to compute Φ for V ∩ R n being nonsmooth or noncompact .
Compact Cases The dual variety V ∗ is the Zariski closure of the set { u ∈ P n | u lies in the row space of Jac ( V ) at x ∈ V reg } .
Compact Cases The dual variety V ∗ is the Zariski closure of the set { u ∈ P n | u lies in the row space of Jac ( V ) at x ∈ V reg } . Computing V ∗ [Rostalski, Sturmfels] Suppose V = { v ∈ C n | h 1 ( v ) = · · · = h p ( v ) = 0 } is smooth and J is the ideal generated by using KKT conditions : p � ∂h j c T X − c 0 , h 1 , . . . , h p , c i − µ j , i = 1 , . . . , n. ∂X i j =1
Compact Cases The dual variety V ∗ is the Zariski closure of the set { u ∈ P n | u lies in the row space of Jac ( V ) at x ∈ V reg } . Computing V ∗ [Rostalski, Sturmfels] Suppose V = { v ∈ C n | h 1 ( v ) = · · · = h p ( v ) = 0 } is smooth and J is the ideal generated by using KKT conditions : p � ∂h j c T X − c 0 , h 1 , . . . , h p , c i − µ j , i = 1 , . . . , n. ∂X i j =1 We have V ∗ = J ∩ R [ c 0 , c 1 , . . . , c n ] .
Compact Cases The dual variety V ∗ is the Zariski closure of the set { u ∈ P n | u lies in the row space of Jac ( V ) at x ∈ V reg } . Computing V ∗ [Rostalski, Sturmfels] Suppose V = { v ∈ C n | h 1 ( v ) = · · · = h p ( v ) = 0 } is smooth and J is the ideal generated by using KKT conditions : p � ∂h j c T X − c 0 , h 1 , . . . , h p , c i − µ j , i = 1 , . . . , n. ∂X i j =1 We have V ∗ = J ∩ R [ c 0 , c 1 , . . . , c n ] . ◮ If V is irreducible , smooth and compact in R n , then V ∗ is defined by an irreducible polynomial Φ( − c 0 , c 1 , . . . , c n ) = 0 .
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