Nichtnegativstellenstze for Univariate Polynomials Victor Magron , - PowerPoint PPT Presentation

Nichtnegativstellensätze for Univariate Polynomials Victor Magron , CNRS Verimag Joint work with Mohab Safey El Din (INRIA/UPMC/LIP6) Markus Schweighofer (Konstanz University) JNCF f p 17 January 2017 p ε f t 4 ( 1 + x 2 + x 4 ) 1 x x a t

The Question(s) Let f ∈ R [ X ] and f � 0 on R Theorem [Hilbert 1888] There exist f 1 , f 2 ∈ R [ X ] s.t. f = f 12 + f 22 . Victor Magron Nichtnegativstellensätze for Univariate Polynomials 1 / 28

The Question(s) Let f ∈ R [ X ] and f � 0 on R Theorem [Hilbert 1888] There exist f 1 , f 2 ∈ R [ X ] s.t. f = f 12 + f 22 . Proof. f = h 2 ( q + ir )( q − ir ) Victor Magron Nichtnegativstellensätze for Univariate Polynomials 1 / 28

The Question(s) Let f ∈ R [ X ] and f � 0 on R Theorem [Hilbert 1888] There exist f 1 , f 2 ∈ R [ X ] s.t. f = f 12 + f 22 . Proof. f = h 2 ( q + ir )( q − ir ) Examples � √ � 2 � 2 � X + 1 3 1 + X + X 2 = + 2 2 √ � 2 � 2 X + 1 + X 2 + 1 5 1 + X + X 2 + X 3 + X 4 = + 4 √ √ √ � 2 � � � � 10 + 2 5 + 10 − 2 10 − 2 5 5 X + 4 4 Victor Magron Nichtnegativstellensätze for Univariate Polynomials 1 / 28

The Question(s) Ordered real field K Let f ∈ K [ X ] with bitsize τ and f � 0 on R Existence Question Does there exist f i ∈ K [ X ] , c i ∈ K > 0 s.t. f = ∑ i c i f i 2 ? Victor Magron Nichtnegativstellensätze for Univariate Polynomials 2 / 28

The Question(s) Ordered real field K Let f ∈ K [ X ] with bitsize τ and f � 0 on R Existence Question Does there exist f i ∈ K [ X ] , c i ∈ K > 0 s.t. f = ∑ i c i f i 2 ? Examples � √ � 2 � 2 � 2 � � X + 1 3 X + 1 + 3 1 + X + X 2 = 4 ( 1 ) 2 + = 1 2 2 2 √ � 2 � X 2 + 1 2 X + 1 + 5 1 + X + X 2 + X 3 + X 4 = + 4 √ √ √ � 2 � � � � 10 + 2 5 + 10 − 2 5 10 − 2 5 X + = ??? 4 4 Victor Magron Nichtnegativstellensätze for Univariate Polynomials 2 / 28

Motivation Nichtnegativstellensätze (Nonnegativity certificates): Stability proofs of critical control systems (Lyapunov) Certified function evaluation [Chevillard et. al 11] Formal verification of real inequalities [Hales et. al 15]: C OQ H OL - LIGHT Victor Magron Nichtnegativstellensätze for Univariate Polynomials 3 / 28

Related work Existence Question Does there exist f i ∈ K [ X ] , c i ∈ K > 0 s.t. f = ∑ i c i f i 2 ? f = c 1 f 12 + c 2 f 22 + c 3 f 32 + c 4 f 42 + c 5 f 52 [Pourchet 72] f = c 1 f 12 + · · · + c n f n 2 [Schweighofer 99] f = c 1 f 12 + · · · + c n + 3 f n + 32 [Chevillard et. al 11] n n 2 ) T G ( 1 x . . . x 2 ) G � 0 f = ( 1 x . . . x SOS with Exact LMIs Critical point methods [Greuet et. al 11] CAD [Iwane 13] Solving over the rationals [Guo et. al 13] � output size = τ O ( 1 ) 2 O ( n 3 ) Determinantial varieties [Henrion et. al 16] Victor Magron Nichtnegativstellensätze for Univariate Polynomials 4 / 28

Contribution Ordered real field K Let f ∈ K [ X ] with bitsize τ and f � 0 on R Existence Question Does there exist f i ∈ K [ X ] , c i ∈ K > 0 s.t. f = ∑ i c i f i 2 ? Complexity Question What is the output bitsize of ∑ i c i f i 2 ? Victor Magron Nichtnegativstellensätze for Univariate Polynomials 5 / 28

Contribution Two methods answering the questions: f = c 1 f 12 + · · · + c n f n 2 [Schweighofer 99] 3 n � Algorithm univsos1 with output size τ 1 = O (( n 2 ) 2 τ ) ∼ 3 n O (( n 2 ) 2 τ ) bit complexity f = c 1 f 12 + · · · + c n + 3 f n + 32 [Chevillard et. al 11] � Algorithm univsos2 with output size τ 2 = O ( n 4 τ ) ∼ O ( n 4 τ ) bit complexity Maple package https://github.com/magronv/univsos � Integration in RAGlib Victor Magron Nichtnegativstellensätze for Univariate Polynomials 5 / 28

The Question(s) univsos1 : Quadratic Approximations univsos2 : Perturbed Polynomials Benchmarks Conclusion and Perspectives

univsos1 : Outline [Schweighofer 99] f f ∈ K [ X ] and f > 0 Minimizer a may not be in K . . . x a f = 1 + X + X 2 + X 3 + X 4 √ 6 ) 1/3 6 ) 1/3 − 4 ( 135 + 60 5 − 1 a = √ 12 4 4 ( 135 + 60 Victor Magron Nichtnegativstellensätze for Univariate Polynomials 6 / 28

univsos1 : Outline [Schweighofer 99] f f ∈ K [ X ] and f > 0 f t Minimizer a may not be in K . . . Find f t ∈ K [ X ] s.t. : deg f t � 2 f t � 0 x f � f t t a f = 1 + X + X 2 + X 3 + X 4 f − f t has a root t ∈ K √ 6 ) 1/3 6 ) 1/3 − 4 ( 135 + 60 5 − 1 a = √ 12 4 4 ( 135 + 60 f t = X 2 t = − 1 Victor Magron Nichtnegativstellensätze for Univariate Polynomials 6 / 28

univsos1 : Outline [Schweighofer 99] f f ∈ K [ X ] and f > 0 Minimizer a may not be in K . . . f t Square-free decomposition: f − f t = gh 2 deg g � deg f − 2 x t a g > 0 f = 1 + X + X 2 + X 3 + X 4 f t = X 2 Do it again on g f − f t = ( X 2 + 2 X + 1 )( X + 1 ) 2 Victor Magron Nichtnegativstellensätze for Univariate Polynomials 6 / 28

univsos1 : Algorithm [Schweighofer 99] Input : K , f � 0 ∈ K [ X ] of degree n � 2 Output : SOS decomposition with coefficients in K f f t ← parab ( f ) h , f t while ( g , h ) ← sqrfree ( f − f t ) deg f > 2 f ← g Victor Magron Nichtnegativstellensätze for Univariate Polynomials 7 / 28

univsos1 : Local Inequality Lemma [Schweighofer 99] f t : = f ( t ) + f ′ ( t )( X − t ) + f ′ ( t ) 2 4 f ( t )( X − t ) 2 ∈ K [ X ] . f > 0, ∃ neighborhood U of local min a s.t. f t ( x ) � f ( x ) ∀ t , x ∈ U Victor Magron Nichtnegativstellensätze for Univariate Polynomials 8 / 28

univsos1 : Local Inequality Lemma [Schweighofer 99] f t : = f ( t ) + f ′ ( t )( X − t ) + f ′ ( t ) 2 4 f ( t )( X − t ) 2 ∈ K [ X ] . f > 0, ∃ neighborhood U of local min a s.t. f t ( x ) � f ( x ) ∀ t , x ∈ U Proof. n = 2 Rolle’s Theorem n � 4 Taylor decomposition of f at t Victor Magron Nichtnegativstellensätze for Univariate Polynomials 8 / 28

univsos1 : Global Inequality Lemma [Schweighofer 99] f t : = f ( t ) + f ′ ( t )( X − t ) + f ′ ( t ) 2 4 f ( t )( X − t ) 2 ∈ K [ X ] . f > 0, ∃ neighborhood U of smallest global min a s.t. f t ( x ) � f ( x ) ∀ t ∈ U , ∀ x ∈ R Victor Magron Nichtnegativstellensätze for Univariate Polynomials 9 / 28

univsos1 : Global Inequality Lemma [Schweighofer 99] f t : = f ( t ) + f ′ ( t )( X − t ) + f ′ ( t ) 2 4 f ( t )( X − t ) 2 ∈ K [ X ] . f > 0, ∃ neighborhood U of smallest global min a s.t. f t ( x ) � f ( x ) ∀ t ∈ U , ∀ x ∈ R Proof. t = f ′ ( t ) 2 f ′′ n = 2 2 f ( t ) Taylor Decomposition of f at t Negative discriminant of f : f ′ ( t ) 2 − 4 f ( t ) f ′′ ( t ) < 0 2 Victor Magron Nichtnegativstellensätze for Univariate Polynomials 9 / 28

univsos1 : Global Inequality Lemma [Schweighofer 99] f t : = f ( t ) + f ′ ( t )( X − t ) + f ′ ( t ) 2 4 f ( t )( X − t ) 2 ∈ K [ X ] . f > 0, ∃ neighborhood U of smallest global min a s.t. f t ( x ) � f ( x ) ∀ t ∈ U , ∀ x ∈ R Proof. f − f t = ∑ n i = 0 a it X i U = [ a − ǫ , a + ǫ ] (Local Ineq) n � 4 � � | a nt | , . . . , | a ( n − 1 ) t | 1, | a 0 t | Cauchy bound: C t : = max � C | a nt | Smallest global min a : � 5 cases ( − ∞ , C ] [ − C , a − ǫ ] [ a − ǫ , a ) [ a , C ) [ C , ∞ ) Victor Magron Nichtnegativstellensätze for Univariate Polynomials 9 / 28

univsos1 : Nichtnegativstellensätz Theorem [Schweighofer 99] Let K be an ordered real field, f ∈ K [ X ] , deg f = n . f � 0 on R ⇔ ∃ c i ∈ K � 0 , f i ∈ K [ X ] s.t. f = c 1 f 12 + · · · + c n f n 2 Victor Magron Nichtnegativstellensätze for Univariate Polynomials 10 / 28

univsos1 : Nichtnegativstellensätz Theorem [Schweighofer 99] Let K be an ordered real field, f ∈ K [ X ] , deg f = n . f � 0 on R ⇔ ∃ c i ∈ K � 0 , f i ∈ K [ X ] s.t. f = c 1 f 12 + · · · + c n f n 2 Proof by induction. n = 2 f = a 2 X 2 + a 1 X + a 0 = a 2 ( X + a 1 2 a 2 ) 2 + ( a 0 − a 12 4 a 2 ) Discriminant a 12 − 4 a 2 a 0 � 0 Victor Magron Nichtnegativstellensätze for Univariate Polynomials 10 / 28

univsos1 : Nichtnegativstellensätz Theorem [Schweighofer 99] Let K be an ordered real field, f ∈ K [ X ] , deg f = n . f � 0 on R ⇔ ∃ c i ∈ K � 0 , f i ∈ K [ X ] s.t. f = c 1 f 12 + · · · + c n f n 2 Proof by induction. n � 4 ⇒ f = g h 2 f not square-free = ⇒ f > 0, ∃ f t � 0 s.t. f − f t = g ( X − t ) 2 f square-free = Victor Magron Nichtnegativstellensätze for Univariate Polynomials 10 / 28

univsos1 : Bitsize of t Lemma Let 0 < f ∈ Z [ X ] with bitsize τ , deg f = n . Let t ∈ Q , f t : = f ( t ) + f ′ ( t )( X − t ) + f ′ ( t ) 2 4 f ( t ) ( X − t ) 2 s.t. f − f t > 0. Then τ ( t ) = O ( n 2 τ ) Victor Magron Nichtnegativstellensätze for Univariate Polynomials 11 / 28

univsos1 : Bitsize of t Lemma Let 0 < f ∈ Z [ X ] with bitsize τ , deg f = n . Let t ∈ Q , f t : = f ( t ) + f ′ ( t )( X − t ) + f ′ ( t ) 2 4 f ( t ) ( X − t ) 2 s.t. f − f t > 0. Then τ ( t ) = O ( n 2 τ ) Proof. Bitsize B of polynomials describing: { t ∈ Q | ∀ x ∈ R , f ( t ) 2 + f ′ ( t ) f ( t )( x − t ) + f ′ ( t ) 2 ( x − t ) 2 � 4 f ( t ) f ( x ) } B = O ( n 2 τ ) Quantifier elimination/CAD [BPR 06]: Victor Magron Nichtnegativstellensätze for Univariate Polynomials 11 / 28