On Exact Polya & Putinars Representations Victor Magron , CNRS - PowerPoint PPT Presentation

On Exact Polya & Putinar’s Representations Victor Magron , CNRS Joint work with Mohab Safey El Din (Sorbonne Univ. -INRIA-LIP6 CNRS) ISSAC 17 th July 2018 p p ε 1 4 ( 1 + x 2 + x 4 ) x

Deciding Non-negativity co-NP hard problem: check f � 0 on K X = ( X 1 , . . . , X n ) f ∈ Q [ X ] Victor Magron On Exact Polya & Putinar’s Representations 1 / 21

Deciding Non-negativity co-NP hard problem: check f � 0 on K X = ( X 1 , . . . , X n ) 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 & Putinar’s Representations 1 / 21

Certifying Non-negativity Sums of squares (SOS) σ = h 12 + · · · + h p 2 Victor Magron On Exact Polya & Putinar’s Representations 2 / 21

Certifying 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 f = 4 X 4 1 + 4 X 3 1 X 2 − 7 X 2 1 X 2 2 − 2 X 1 X 3 2 + 10 X 4 2 2 ) 2 + ( 4 2 ) 2 + ( 2 f ≃ σ = ( 2 X 2 1 + X 1 X 2 − 8 3 X 2 3 X 1 X 2 + 3 2 X 2 7 X 2 2 ) 2 Victor Magron On Exact Polya & Putinar’s Representations 2 / 21

Certifying 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 f = 4 X 4 1 + 4 X 3 1 X 2 − 7 X 2 1 X 2 2 − 2 X 1 X 3 2 + 10 X 4 2 2 ) 2 + ( 4 2 ) 2 + ( 2 f ≃ σ = ( 2 X 2 1 + X 1 X 2 − 8 3 X 2 3 X 1 X 2 + 3 2 X 2 7 X 2 2 ) 2 f = σ + 8 2 − 2 2 + 983 9 X 2 1 X 2 3 X 1 X 3 1764 X 4 2 Victor Magron On Exact Polya & Putinar’s Representations 2 / 21

Certifying 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 f = 4 X 4 1 + 4 X 3 1 X 2 − 7 X 2 1 X 2 2 − 2 X 1 X 3 2 + 10 X 4 2 2 ) 2 + ( 4 2 ) 2 + ( 2 f ≃ σ = ( 2 X 2 1 + X 1 X 2 − 8 3 X 2 3 X 1 X 2 + 3 2 X 2 7 X 2 2 ) 2 f = σ + 8 2 − 2 2 + 983 9 X 2 1 X 2 3 X 1 X 3 1764 X 4 2 → ≃ = The Question of Exact Certification How to go from approximate to exact certification? Victor Magron On Exact Polya & Putinar’s Representations 2 / 21

Certifying Non-negativity σ 1 Polya ’s representation f = ( X 1 + ··· + X n ) 2 D positive definite form f [Reznick 95] 2 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 & Putinar’s Representations 3 / 21

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 & Putinar’s Representations 4 / 21

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 & Putinar’s Representations 4 / 21

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 & Putinar’s Representations 4 / 21

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 & Putinar’s Representations 5 / 21

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 & Putinar’s Representations 5 / 21

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 & Putinar’s Representations 6 / 21

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 ? n = 1 deg f = d f = c 1 h 12 + c 2 h 22 + c 3 h 32 + c 4 h 42 + c 5 h 52 [Pourchet 72] f = c 1 h 12 + · · · + c d h d 2 [Schweighofer 99] f = c 1 h 12 + · · · + c d + 3 h d + 32 [Chevillard et. al 11] Victor Magron On Exact Polya & Putinar’s Representations 6 / 21

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 ? n = 1 deg f = d f = c 1 h 12 + c 2 h 22 + c 3 h 32 + c 4 h 42 + c 5 h 52 [Pourchet 72] f = c 1 h 12 + · · · + c d h d 2 [Schweighofer 99] f = c 1 h 12 + · · · + c d + 3 h d + 32 [Chevillard et. al 11] deg f = d n > 1 SOS with Exact LMIs f = v dT ( X ) G v dT ( X ) G � 0 Solving over the rationals [Guo-Safey El Din-Zhi 13] Solving over the reals [Henrion-Naldi-Safey El Din 16] Victor Magron On Exact Polya & Putinar’s Representations 6 / 21

The Cost of Exact Polynomial Optimization f ∈ Q [ X ] ∩ ˚ f Σ [ X ] (interior of the SOS cone) Σ deg f = d bit size τ Complexity Question(s) What is the output bit size of ∑ i c i h i 2 ? σ f = 1 Polya ’s representation ( X 1 + ··· + X n ) 2 D positive definite form f f = σ 0 + σ 1 g 1 + · · · + σ m g m 2 Putinar ’s representation f > 0 on compact K deg σ i � 2 D Exact algorithm ? B OUNDS on D , τ ( σ i ) ? Victor Magron On Exact Polya & Putinar’s Representations 7 / 21

Contributions f ∈ Q [ X ] ∩ ˚ f Σ [ X ] (interior of the SOS cone) Σ bit size τ deg f = d Complexity cost of certifying non-negativity Algorithm intsos � OUTPUT B IT S IZE = τ d O ( n ) Similar complexity cost d O ( n ) for Deciding 1 Polya ’s representation Algorithm Polyasos OUTPUT B IT S IZE = 2 τ d O ( n ) positive definite form f 2 Putinar ’s representation Algorithm Putinarsos OUTPUT B IT S IZE = O ( 2 τ d n C K ) f > 0 on compact K Victor Magron On Exact Polya & Putinar’s Representations 8 / 21

Deciding Non-negativity Exact SOS Representations Exact Polya’s Representations Exact Putinar’s Representations Benchmarks Conclusion and Perspectives

intsos with n = 1 and Root Approximation Algorithm from [Chevillard-Harrison-Joldes-Lauter 11] p p ∈ Q [ X ] , deg p = d = 2 k , p > 0 x p = 1 + X + X 2 + X 3 + X 4 Victor Magron On Exact Polya & Putinar’s Representations 9 / 21

intsos with n = 1 and Root Approximation Algorithm from [Chevillard-Harrison-Joldes-Lauter 11] p p ε p ∈ Q [ X ] , deg p = d = 2 k , p > 0 P ERTURB : find ε ∈ Q s.t. 4 ( 1 + x 2 + x 4 ) 1 k X 2 i > 0 ∑ p ε : = p − ε x i = 0 p = 1 + X + X 2 + X 3 + X 4 ε = 1 4 p > 1 4 ( 1 + X 2 + X 4 ) Victor Magron On Exact Polya & Putinar’s Representations 9 / 21

intsos with n = 1 and Root Approximation Algorithm from [Chevillard-Harrison-Joldes-Lauter 11] p p ∈ Q [ X ] , deg p = d = 2 k , p > 0 p ε P ERTURB : find ε ∈ Q s.t. k X 2 i > 0 ∑ p ε : = p − ε i = 0 4 ( 1 + x 2 + x 4 ) 1 Root isolation: x k X 2 i = s 12 + s 22 + u ∑ p − ε p = 1 + X + X 2 + X 3 + X 4 i = 0 ε = 1 A BSORB : small enough u i 4 i = 0 X 2 i + u SOS ⇒ ε ∑ k = p > 1 4 ( 1 + X 2 + X 4 ) Victor Magron On Exact Polya & Putinar’s Representations 9 / 21

intsos with n = 1 and Root Approximation Input : f � 0 ∈ Q [ X ] of degree d � 2 , ε ∈ Q > 0 , δ ∈ N > 0 Output : SOS decomposition with coefficients in Q h , s 1 , s 2 , ε , u f ( p , h ) ← sqrfree ( f ) k ∑ X 2 i ( s 1 , s 2 ) ← sum2squares ( p ε , δ ) p ε ← p − ε u ← p ε − s 12 − s 22 i = 0 ε ← ε δ ← 2 δ 2 while while p ε ≤ 0 ε < | u 2 i + 1 | + | u 2 i − 1 | − u 2 i 2 Victor Magron On Exact Polya & Putinar’s Representations 10 / 21

intsos with n = 1 and SDP Approximation Input : f � 0 ∈ Q [ X ] of degree d � 2 , ε ∈ Q > 0 , δ ∈ N > 0 Output : SOS decomposition with coefficients in Q h , ˜ σ , ε , u f ( p , h ) ← sqrfree ( f ) k X 2 i ∑ p ε ← p − ε σ ← sdp ( p ε , δ ) ˜ i = 0 u ← p ε − ˜ σ ε ← ε δ ← 2 δ 2 while while p ε ≤ 0 ε < | u 2 i + 1 | + | u 2 i − 1 | − u 2 i 2 Victor Magron On Exact Polya & Putinar’s Representations 11 / 21