On Exact Polya, Hilbert-Artin and Putinars Representations Victor - PowerPoint PPT Presentation

On Exact Polya, Hilbert-Artin and Putinar’s Representations Victor Magron , LAAS CNRS Joint work with Mohab Safey El Din (Sorbonne Univ. -INRIA-LIP6 CNRS) JNCF 04 th February 2019 p p ε 1 4 ( 1 + x 2 + x 4 ) x

Deciding Non-negativity X = ( X 1 , . . . , X n ) co-NP hard problem: check f � 0 on K f ∈ Q [ X ] Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 1 / 27

Deciding Non-negativity X = ( X 1 , . . . , X n ) co-NP hard problem: check f � 0 on K f ∈ Q [ X ] 1 Unconstrained � K = R n 2 Constrained � K = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } g j ∈ Q [ X ] deg f , deg g j � d [Collins 75] CAD doubly exp. in n poly. in d [Grigoriev-Vorobjov 88, Basu-Pollack-Roy 98] Critical points singly exponential time ( m + 1 ) τ d O ( n ) Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 1 / 27

Deciding Non-negativity Sums of squares (SOS) σ = h 12 + · · · + h p 2 Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 2 / 27

Deciding Non-negativity Sums of squares (SOS) σ = h 12 + · · · + h p 2 H ILBERT 17 TH P ROBLEM : f SOS of rational functions? [Artin 27] YES ! [Lasserre/Parrilo 01] Numerical solvers compute σ Semidefinite programming (SDP) � approximate certificates → ≃ = The Question of Exact Certification How to go from approximate to exact certification? Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 2 / 27

What is Semidefinite Programming? Linear Programming (LP): ⊤ z min c z s.t. A z � d . Linear cost c Polyhedron Linear inequalities “ ∑ i A ij z j � d i ” Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 3 / 27

What is Semidefinite Programming? Semidefinite Programming (SDP): ⊤ z min c z ∑ s.t. F i z i � F 0 . i Linear cost c Symmetric matrices F 0 , F i Spectrahedron Linear matrix inequalities “ F � 0 ” ( F has nonnegative eigenvalues) Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 3 / 27

What is Semidefinite Programming? Semidefinite Programming (SDP): ⊤ z min c z ∑ s.t. F i z i � F 0 , A z = d . i Linear cost c Symmetric matrices F 0 , F i Spectrahedron Linear matrix inequalities “ F � 0 ” ( F has nonnegative eigenvalues) Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 3 / 27

Lasserre’s Hierarchy Prove polynomial inequalities with SDP: f ( a , b ) : = a 2 − 2 ab + b 2 � 0 . � � � � � � z 1 z 2 a Find z s.t. f ( a , b ) = a b . z 2 z 3 b � �� � � 0 Find z s.t. a 2 − 2 ab + b 2 = z 1 a 2 + 2 z 2 ab + z 3 b 2 ( A z = d ) � z 1 � � 1 � � 0 � � 0 � � 0 � z 2 0 1 0 0 = z 1 + z 2 + z 3 � z 2 z 3 0 0 1 0 0 1 0 0 � �� � � �� � � �� � � �� � F 1 F 2 F 3 F 0 Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 4 / 27

Lasserre’s Hierarchy Choose a cost c e.g. ( 1, 0, 1 ) and solve: ⊤ z min c z ∑ A z = d . s.t. F i z i � F 0 , i � 1 � z 1 � � − 1 z 2 Solution = � 0 (eigenvalues 0 and 2) z 2 z 3 − 1 1 � � 1 � � a � − 1 a 2 − 2 ab + b 2 = � = ( a − b ) 2 . a b − 1 1 b � �� � � 0 Solving SDP = ⇒ Finding S UMS OF S QUARES certificates Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 5 / 27

Lasserre’s Hierarchy NP hard General Problem : f ∗ : = min x ∈ K f ( x ) Semialgebraic set K : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 6 / 27

Lasserre’s Hierarchy NP hard General Problem : f ∗ : = min x ∈ K f ( x ) Semialgebraic set K : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } � : = [ 0, 1 ] 2 = { x ∈ R 2 : x 1 ( 1 − x 1 ) � 0, x 2 ( 1 − x 2 ) � 0 } Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 6 / 27

Lasserre’s Hierarchy NP hard General Problem : f ∗ : = min x ∈ K f ( x ) Semialgebraic set K : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } � : = [ 0, 1 ] 2 = { x ∈ R 2 : x 1 ( 1 − x 1 ) � 0, x 2 ( 1 − x 2 ) � 0 } σ 0 σ 1 σ 2 � �� � g 1 g 2 ���� ���� f � � 2 x 1 x 2 = − 1 1 x 1 + x 2 − 1 1 � �� � 1 � �� � ���� 8 + + x 1 ( 1 − x 1 ) + x 2 ( 1 − x 2 ) 2 2 2 2 Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 6 / 27

Lasserre’s Hierarchy NP hard General Problem : f ∗ : = min x ∈ K f ( x ) Semialgebraic set K : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } � : = [ 0, 1 ] 2 = { x ∈ R 2 : x 1 ( 1 − x 1 ) � 0, x 2 ( 1 − x 2 ) � 0 } σ 0 σ 1 σ 2 � �� � g 1 g 2 ���� ���� f � � 2 x 1 x 2 = − 1 1 x 1 + x 2 − 1 1 � �� � 1 � �� � ���� 8 + + x 1 ( 1 − x 1 ) + x 2 ( 1 − x 2 ) 2 2 2 2 Σ = Sums of squares (SOS) σ i Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 6 / 27

Lasserre’s Hierarchy NP hard General Problem : f ∗ : = min x ∈ K f ( x ) Semialgebraic set K : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } � : = [ 0, 1 ] 2 = { x ∈ R 2 : x 1 ( 1 − x 1 ) � 0, x 2 ( 1 − x 2 ) � 0 } σ 0 σ 1 σ 2 � �� � g 1 g 2 ���� ���� f � � 2 x 1 x 2 = − 1 1 x 1 + x 2 − 1 1 � �� � 1 � �� � ���� 8 + + x 1 ( 1 − x 1 ) + x 2 ( 1 − x 2 ) 2 2 2 2 Σ = Sums of squares (SOS) σ i Bounded degree: � � σ 0 + ∑ m Q d ( K ) : = j = 1 σ j g j , with deg σ j g j � 2 d Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 6 / 27

Lasserre’s Hierarchy Hierarchy of SDP relaxations : � � λ d : = sup λ : f − λ ∈ Q d ( K ) Convergence guarantees λ d ↑ f ∗ [Lasserre 01] Can be computed with SDP solvers ( CSDP , SDPA ) “No Free Lunch” Rule : ( n + 2 d n ) SDP variables Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 7 / 27

Certifying Non-negativity A PPROXIMATE SOLUTIONS sum of squares of a 2 − 2 ab + b 2 ? ( 1.00001 a − 0.99998 b ) 2 ! a 2 − 2 ab + b 2 ≃ ( 1.00001 a − 0.99998 b ) 2 a 2 − 2 ab + b 2 � = 1.0000200001 a 2 − 1.9999799996 ab + 0.9999600004 b 2 → = ? ≃ Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 8 / 27

Certifying Non-negativity σ f = 1 Polya ’s representation ( X 2 1 + ··· + X 2 n ) D positive definite form f [Reznick 95] f = σ 2 Hilbert-Artin ’s representation h 2 f � 0 [Artin 27] 3 Putinar ’s representation f = σ 0 + σ 1 g 1 + · · · + σ m g m f > 0 on compact K deg σ i � 2 D [Putinar 93] Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 9 / 27

One Answer when K = R n Hybrid S YMBOLIC /N UMERIC methods [Peyrl-Parrilo 08] [Kaltofen-Yang-Zhi 08] � can handle degenerate situations when f ∈ ∂ Σ f ( X ) ≃ v DT ( X ) ˜ ˜ Q v D ( X ) Q � 0 v D ( X ) : vector of monomials of deg � D Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 10 / 27

One Answer when K = R n Hybrid S YMBOLIC /N UMERIC methods [Peyrl-Parrilo 08] [Kaltofen-Yang-Zhi 08] � can handle degenerate situations when f ∈ ∂ Σ f ( X ) ≃ v DT ( X ) ˜ ˜ Q v D ( X ) Q � 0 v D ( X ) : vector of monomials of deg � D → ≃ = ˜ Q Rounding Q Projection ∏ ( Q ) f ( X ) = v DT ( X ) ∏ ( Q ) v D ( X ) ∏ ( Q ) � 0 when ε → 0 Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 10 / 27

One Answer when K = R n Hybrid S YMBOLIC /N UMERIC methods [Peyrl-Parrilo 08] [Kaltofen-Yang-Zhi 08] � can handle degenerate situations when f ∈ ∂ Σ f ( X ) ≃ v DT ( X ) ˜ ˜ Q v D ( X ) Q � 0 v D ( X ) : vector of monomials of deg � D → ≃ = ˜ Q Rounding Q Projection ∏ ( Q ) f ( X ) = v DT ( X ) ∏ ( Q ) v D ( X ) ∏ ( Q ) � 0 when ε → 0 C OMPLEXITY ? Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 10 / 27

One Answer when K = { x ∈ R n : g j ( x ) � 0 } Hybrid S YMBOLIC /N UMERIC methods Magron-Allamigeon-Gaubert-Werner 14 f ≃ ˜ σ 0 + ˜ σ 1 g 1 + · · · + ˜ σ m g m u = f − ˜ σ 0 + ˜ σ 1 g 1 + · · · + ˜ σ m g m Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 11 / 27

One Answer when K = { x ∈ R n : g j ( x ) � 0 } Hybrid S YMBOLIC /N UMERIC methods Magron-Allamigeon-Gaubert-Werner 14 Compact K ⊆ [ 0, 1 ] n f ≃ ˜ σ 0 + ˜ σ 1 g 1 + · · · + ˜ σ m g m u = f − ˜ σ 0 + ˜ σ 1 g 1 + · · · + ˜ σ m g m → ≃ = ∀ x ∈ [ 0, 1 ] n , u ( x ) � − ε min K f � ε when ε → 0 C OMPLEXITY ? Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 11 / 27

Related Work: Exact Methods Existence Question Does there exist h i ∈ Q [ X ] , c i ∈ Q > 0 s.t. f = ∑ i c i h i 2 ? Victor Magron On Exact Polya, Hilbert-Artin and Putinar’s Representations 12 / 27