SDP for Polynomial Optimization 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 Sums of squares (SOS) σ i Bounded degree: � � σ 0 + ∑ m Q k ( K ) : = j = 1 σ j g j , with deg σ j g j � 2 k Victor Magron On Exact Polynomial Optimization 8 / 46
SDP for Polynomial Optimization Hierarchy of SDP relaxations : � � λ k : = sup λ : f − λ ∈ Q k ( K ) λ Convergence guarantees λ k ↑ f ∗ [Lasserre 01] Can be computed with SDP solvers ( CSDP , SDPA ) “No Free Lunch” Rule : ( n + 2 k n ) SDP variables Victor Magron On Exact Polynomial Optimization 9 / 46
One Answer when K = R n Hybrid S YMBOLIC /N UMERIC methods [Peyrl-Parrilo 08] [Kaltofen et. al 08] f ( X ) ≃ v DT ( X ) ˜ Q v D ( X ) 0 � ˜ Q ∈ R D × D v D ( X ) = ( 1, X 1 , . . . , X n , X 2 1 , . . . , X D n ) Victor Magron On Exact Polynomial Optimization 10 / 46
One Answer when K = R n Hybrid S YMBOLIC /N UMERIC methods [Peyrl-Parrilo 08] [Kaltofen et. al 08] f ( X ) ≃ v DT ( X ) ˜ Q v D ( X ) 0 � ˜ Q ∈ R D × D v D ( X ) = ( 1, X 1 , . . . , X n , X 2 1 , . . . , X D n ) → ≃ = ˜ Q Rounding Q Projection ∏ ( Q ) f ( X ) = v DT ( X ) ∏ ( Q ) v D ( X ) ∏ ( Q ) � 0 when ε → 0 Victor Magron On Exact Polynomial Optimization 10 / 46
One Answer when K = R n Hybrid S YMBOLIC /N UMERIC methods [Peyrl-Parrilo 08] [Kaltofen et. al 08] f ( X ) ≃ v DT ( X ) ˜ Q v D ( X ) 0 � ˜ Q ∈ R D × D v D ( X ) = ( 1, X 1 , . . . , X n , X 2 1 , . . . , X D n ) → ≃ = ˜ 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 Polynomial Optimization 10 / 46
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 Polynomial Optimization 11 / 46
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 Polynomial Optimization 11 / 46
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 Polynomial Optimization 12 / 46
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 Polynomial Optimization 12 / 46
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] n > 1 deg f = d SOS with Exact LMIs f ( X ) = v dT ( X ) G v dT ( X ) G � 0 Critical point methods [Greuet et. al 11] CAD [Iwane 13] � τ d O ( n ) Solving over the rationals [Guo et. al 13] Determinantial varieties [Henrion et. al 16] Victor Magron On Exact Polynomial Optimization 12 / 46
Contribution: n = 1 f ∈ Q [ X ] ∩ ˚ Σ [ X ] (interior of the SOS cone) with bit size τ Existence Question Does there exist f i ∈ Q [ X ] , c i ∈ Q > 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 On Exact Polynomial Optimization 13 / 46
Contribution: n = 1 Two methods answering the questions: f = c 1 h 12 + · · · + c d h d 2 [Schweighofer 99] 3 d � Algorithm univsos1 with output size τ 1 = O (( d 2 ) 2 τ ) f = c 1 h 12 + · · · + c d + 3 h d + 32 [Chevillard et. al 11] � Algorithm univsos2 with output size τ 2 = O ( d 4 τ ) Maple package https://github.com/magronv/univsos Victor Magron On Exact Polynomial Optimization 13 / 46
Contribution: n � 1 f ∈ Q [ X ] ∩ ˚ f Σ [ X ] (interior of the SOS cone) Σ deg f = d bit size τ Complexity Cost Algorithm intsos � OUTPUT B IT S IZE = τ d O ( n ) σ f = 1 Polya ’s representation ( X 1 + ··· + X n ) 2 D (positive definite forms) Algorithm Polyasos � OUTPUT B IT S IZE = 2 τ d O ( n ) f = σ 0 + σ 1 g 1 + · · · + σ m g m 2 Putinar ’s representation ( f > 0 + K compact) deg σ i � 2 D Algorithm Putinarsos � OUTPUT B IT S IZE = O ( 2 τ d n C K ) Victor Magron On Exact Polynomial Optimization 14 / 46
Certify Polynomial Non-negativity The Question(s) Exact SOS Representations: n = 1 Exact SOS Representations: n � 1 Exact Polya’s Representations Exact Putinar’s Representations Conclusion and Perspectives
univsos1 : Outline [Schweighofer 99] f f ∈ Q [ X ] and f > 0 Minimizer a may not be in Q . . . 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 On Exact Polynomial Optimization 15 / 46
univsos1 : Outline [Schweighofer 99] f f ∈ Q [ X ] and f > 0 f t Minimizer a may not be in Q . . . Find f t ∈ Q [ 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 ∈ Q √ 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 On Exact Polynomial Optimization 15 / 46
univsos1 : Outline [Schweighofer 99] f f ∈ Q [ X ] and f > 0 Minimizer a may not be in Q . . . 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 On Exact Polynomial Optimization 15 / 46
univsos1 : Algorithm [Schweighofer 99] Input : f � 0 ∈ Q [ X ] of degree d � 2 Output : SOS decomposition with coefficients in Q f f t ← parab ( f ) h , f t while ( g , h ) ← sqrfree ( f − f t ) deg f > 2 f ← g Victor Magron On Exact Polynomial Optimization 16 / 46
univsos1 : Local Inequality Lemma [Schweighofer 99] f t : = f ( t ) + f ′ ( t )( X − t ) + f ′ ( t ) 2 4 f ( t )( X − t ) 2 ∈ Q [ X ] . f > 0, ∃ neighborhood U of local min a s.t. f t ( x ) � f ( x ) ∀ t , x ∈ U Victor Magron On Exact Polynomial Optimization 17 / 46
univsos1 : Local Inequality Lemma [Schweighofer 99] f t : = f ( t ) + f ′ ( t )( X − t ) + f ′ ( t ) 2 4 f ( t )( X − t ) 2 ∈ Q [ X ] . f > 0, ∃ neighborhood U of local min a s.t. f t ( x ) � f ( x ) ∀ t , x ∈ U Proof. d = 2 Rolle’s Theorem d � 4 Taylor decomposition of f at t Victor Magron On Exact Polynomial Optimization 17 / 46
univsos1 : Global Inequality Lemma [Schweighofer 99] f t : = f ( t ) + f ′ ( t )( X − t ) + f ′ ( t ) 2 4 f ( t )( X − t ) 2 ∈ Q [ X ] . f > 0, ∃ neighborhood U of smallest global min a s.t. f t ( x ) � f ( x ) ∀ t ∈ U , ∀ x ∈ R Victor Magron On Exact Polynomial Optimization 18 / 46
univsos1 : Global Inequality Lemma [Schweighofer 99] f t : = f ( t ) + f ′ ( t )( X − t ) + f ′ ( t ) 2 4 f ( t )( X − t ) 2 ∈ Q [ 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 ′′ d = 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 On Exact Polynomial Optimization 18 / 46
univsos1 : Global Inequality Lemma [Schweighofer 99] f t : = f ( t ) + f ′ ( t )( X − t ) + f ′ ( t ) 2 4 f ( t )( X − t ) 2 ∈ Q [ 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) d � 4 � � | a dt | , . . . , | a ( d − 1 ) t | 1, | a 0 t | Cauchy bound: C t : = max � C | a dt | Smallest global min a : � 5 cases ( − ∞ , C ] [ − C , a − ǫ ] [ a − ǫ , a ) [ a , C ) [ C , ∞ ) Victor Magron On Exact Polynomial Optimization 18 / 46
univsos1 : Nichtnegativstellensätz Theorem [Schweighofer 99] Let f ∈ Q [ X ] , deg f = d . f � 0 on R ⇔ ∃ c i ∈ Q � 0 , f i ∈ Q [ X ] s.t. f = c 1 f 12 + · · · + c d f d 2 Victor Magron On Exact Polynomial Optimization 19 / 46
univsos1 : Nichtnegativstellensätz Theorem [Schweighofer 99] Let f ∈ Q [ X ] , deg f = d . f � 0 on R ⇔ ∃ c i ∈ Q � 0 , f i ∈ Q [ X ] s.t. f = c 1 f 12 + · · · + c d f d 2 Proof by induction. d = 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 On Exact Polynomial Optimization 19 / 46
univsos1 : Nichtnegativstellensätz Theorem [Schweighofer 99] Let f ∈ Q [ X ] , deg f = d . f � 0 on R ⇔ ∃ c i ∈ Q � 0 , f i ∈ Q [ X ] s.t. f = c 1 f 12 + · · · + c d f d 2 Proof by induction. d � 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 On Exact Polynomial Optimization 19 / 46
univsos1 : Bitsize of t Lemma Let 0 < f ∈ Z [ X ] with bitsize τ , deg f = d . 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 ( d 2 τ ) Victor Magron On Exact Polynomial Optimization 20 / 46
univsos1 : Bitsize of t Lemma Let 0 < f ∈ Z [ X ] with bitsize τ , deg f = d . 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 ( d 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 ( d 2 τ ) Quantifier elimination/CAD [BPR 06]: Victor Magron On Exact Polynomial Optimization 20 / 46
univsos1 : Bitsize of Square-free Part Lemma Let 0 < f ∈ Z [ X ] with bitsize τ , deg f = d . 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 ∃ ˆ f , ˆ f t , g ∈ Z [ X ] s.t. ˆ f − ˆ f t = ( X − t ) 2 g τ ( f t ) = τ ( g ) = O ( d 3 τ ) Victor Magron On Exact Polynomial Optimization 21 / 46
univsos1 : Bitsize of Square-free Part Lemma Let 0 < f ∈ Z [ X ] with bitsize τ , deg f = d . 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 ∃ ˆ f , ˆ f t , g ∈ Z [ X ] s.t. ˆ f − ˆ f t = ( X − t ) 2 g τ ( f t ) = τ ( g ) = O ( d 3 τ ) Proof. t = t 1 f : = t 2 d ˆ f t : = t 2 d ˆ 2 f ( t ) f ( X ) 2 f ( t ) f t ( X ) t 2 Square-free part: τ ( g ) � d − 2 + τ ( ˆ f − ˆ f t ) + log 2 ( d + 1 ) Victor Magron On Exact Polynomial Optimization 21 / 46
univsos1 : Output Bitsize Theorem Let 0 < f ∈ Q [ X ] with bitsize τ , deg f = d . 3 d The output bitsize τ 1 of univsos1 on f is O (( d 2 ) 2 τ ) . Victor Magron On Exact Polynomial Optimization 22 / 46
univsos1 : Output Bitsize Theorem Let 0 < f ∈ Q [ X ] with bitsize τ , deg f = d . 3 d The output bitsize τ 1 of univsos1 on f is O (( d 2 ) 2 τ ) . Proof. Worst-case: k = d /2 induction steps � � τ + k 3 τ + ( k − 1 ) 3 k 3 τ + · · · + ( k ! ) 3 τ = ⇒ τ 1 = O Victor Magron On Exact Polynomial Optimization 22 / 46
univsos1 : Bit Complexity Theorem Let 0 < f ∈ Q [ X ] with bitsize τ , deg f = d . ∼ 3 d O (( d 2 τ ) . The bit complexity of univsos1 on f is 2 ) Victor Magron On Exact Polynomial Optimization 23 / 46
univsos1 : Bit Complexity Theorem Let 0 < f ∈ Q [ X ] with bitsize τ , deg f = d . ∼ 3 d O (( d 2 τ ) . The bit complexity of univsos1 on f is 2 ) All involved polynomials have a global min in Z ∼ O ( d 4 + d 3 τ ) . = ⇒ the bit complexity is Victor Magron On Exact Polynomial Optimization 23 / 46
univsos1 : Bit Complexity Theorem Let 0 < f ∈ Q [ X ] with bitsize τ , deg f = d . ∼ 3 d O (( d 2 τ ) . The bit complexity of univsos1 on f is 2 ) All involved polynomials have a global min in Z ∼ O ( d 4 + d 3 τ ) . = ⇒ the bit complexity is Proof. Root bitsize: τ ( t ) = O ( τ ) Square-free part: τ ( g ) = O ( d + τ ( f − f t )) = O ( d + τ ) Output bisize: τ 1 = O ( d 3 + d τ ) Victor Magron On Exact Polynomial Optimization 23 / 46
univsos2 : Outline [Chevillard et. al 11] Algorithm from [Chevillard et. al 11] p p ∈ Z [ X ] , deg p = d = 2 k , p > 0 x p = 1 + X + X 2 + X 3 + X 4 Victor Magron On Exact Polynomial Optimization 24 / 46
univsos2 : Outline [Chevillard et. al 11] Algorithm from [Chevillard et. al 11] p p ε p ∈ Z [ 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 Polynomial Optimization 24 / 46
univsos2 : Outline [Chevillard et. al 11] Algorithm from [Chevillard et. al 11] p p ∈ Z [ 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 Polynomial Optimization 24 / 46
univsos2 : Outline [Chevillard et. al 11] 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 Polynomial Optimization 25 / 46
univsos2 : Absorbtion � ( X + 1 ) 2 − 1 − X 2 � X = 1 2 � ( X − 1 ) 2 − 1 − X 2 � − X = 1 2 Victor Magron On Exact Polynomial Optimization 26 / 46
univsos2 : Absorbtion � ( X + 1 ) 2 − 1 − X 2 � X = 1 2 � ( X − 1 ) 2 − 1 − X 2 � − X = 1 2 u 2 i + 1 X 2 i + 1 = | u 2 i + 1 | � ( X i + 1 + sgn ( u 2 i + 1 ) X i ) 2 − X 2 i − X 2 i + 2 � 2 Victor Magron On Exact Polynomial Optimization 26 / 46
univsos2 : Absorbtion � ( X + 1 ) 2 − 1 − X 2 � X = 1 2 � ( X − 1 ) 2 − 1 − X 2 � − X = 1 2 u 2 i + 1 X 2 i + 1 = | u 2 i + 1 | � ( X i + 1 + sgn ( u 2 i + 1 ) X i ) 2 − X 2 i − X 2 i + 2 � 2 u · · · · · · 2 i − 2 2 i − 1 2 i 2 i + 1 2 i + 2 ε ∑ k i = 0 X 2 i ε ε ε Victor Magron On Exact Polynomial Optimization 26 / 46
univsos2 : Absorbtion � ( X + 1 ) 2 − 1 − X 2 � X = 1 2 � ( X − 1 ) 2 − 1 − X 2 � − X = 1 2 u 2 i + 1 X 2 i + 1 = | u 2 i + 1 | � ( X i + 1 + sgn ( u 2 i + 1 ) X i ) 2 − X 2 i − X 2 i + 2 � 2 u · · · · · · 2 i − 2 2 i − 1 2 i 2 i + 1 2 i + 2 ε ∑ k i = 0 X 2 i ε ε ε k ε � | u 2 i + 1 | + | u 2 i − 1 | X 2 i + u ∑ − u 2 i = ⇒ ε SOS 2 i = 0 Victor Magron On Exact Polynomial Optimization 26 / 46
univsos2 : Nichtnegativstellensätz Theorem [Chevillard et. al 11] Let 0 � f ∈ Z [ X ] , deg f = d . f � 0 on R ⇔ ∃ c i ∈ Q � 0 , f i ∈ Q [ X ] s.t. f = c 1 f 12 + · · · + c d + 3 f d + 3 2 Victor Magron On Exact Polynomial Optimization 27 / 46
univsos2 : Nichtnegativstellensätz Theorem [Chevillard et. al 11] Let 0 � f ∈ Z [ X ] , deg f = d . f � 0 on R ⇔ ∃ c i ∈ Q � 0 , f i ∈ Q [ X ] s.t. f = c 1 f 12 + · · · + c d + 3 f d + 3 2 Proof. f = p h 2 = ⇒ 0 < p ∈ Z [ X ] , deg p = 2 k , p ε : = p − ε ∑ k i = 0 X 2 i > 0 Root isolation: p = ls 12 + ls 22 + ε ∑ k i = 0 X 2 i + u at precision δ X 2 j + 1 = ( X j + 1 + X j 2 ) 2 − ( X 2 j + 2 + X 2 j 4 ) = − ( X j + 1 − X j 2 ) 2 + ( X 2 j + 2 + X 2 j 4 ) Smallest δ s.t. ε � | u 2 i + 1 | − u 2 i + | u 2 i − 1 | 4 i = 0 X 2 i + u ⇒ weighted SOS decomposition of ε ∑ k = Victor Magron On Exact Polynomial Optimization 27 / 46
univsos2 : Bitsize of Perturbed Polynomials Lemma Let 0 < p ∈ Z [ X ] with bitsize τ , deg p = d = 2 k . Then ∃ ε s.t. p ε > 0 and τ ( ε ) = d log 2 d + d τ Victor Magron On Exact Polynomial Optimization 28 / 46
univsos2 : Bitsize of Perturbed Polynomials Lemma Let 0 < p ∈ Z [ X ] with bitsize τ , deg p = d = 2 k . Then ∃ ε s.t. p ε > 0 and τ ( ε ) = d log 2 d + d τ Proof. ⇒ ∃ R s.t. p ε ( x ) > 0 for | x | > R = 2 d 2 τ (Cauchy) ε : = 1/2 = inf | x | � R p 1 Smallest N s.t. ε = 2 N < 1 + R 2 ··· + R 2 k ⇒ 1 + R 2 + · · · + R 2 k < kR 2 k R > 1 = inf x ∈ R p ( x ) > ( d 2 τ ) − d + 2 2 − d log 2 d − d τ [Melczer et. al 16] Victor Magron On Exact Polynomial Optimization 28 / 46
univsos2 : Bitsize of Remainder Lemma Let 0 < p ∈ Z [ X ] with bitsize τ , deg p = d = 2 k . Then k ∃ ε , s 1 , s 2 , u s.t. p = ls 12 + ls 22 + ε X 2 i + u SOS ∑ i = 0 with approx. root precision δ of p ε s.t. τ ( δ ) = d 2 + d τ Victor Magron On Exact Polynomial Optimization 29 / 46
univsos2 : Bitsize of Remainder Lemma Let 0 < p ∈ Z [ X ] with bitsize τ , deg p = d = 2 k . Then k ∃ ε , s 1 , s 2 , u s.t. p = ls 12 + ls 22 + ε X 2 i + u SOS ∑ i = 0 with approx. root precision δ of p ε s.t. τ ( δ ) = d 2 + d τ Proof. i = 0 a i X i = ∏ d p ε = ∑ d e = 2 − δ i = 1 ( X − z i ) | ˆ z i | � z i ( 1 + e ) Vieta’s formula: ∑ 1 � i 1 < ··· < i j � d z i 1 . . . z i j = ( − 1 ) j a d − j l Smallest δ s.t. ε � | u 2 i + 1 | − u 2 i + | u 2 i − 1 | 4 Victor Magron On Exact Polynomial Optimization 29 / 46
univsos2 : Output Bitsize Theorem Let 0 � f ∈ Z [ X ] with bitsize τ , deg f = d . The max coeff bitsize τ 2 of univsos2 on f is O ( d 3 + d 2 τ ) . Victor Magron On Exact Polynomial Optimization 30 / 46
univsos2 : Output Bitsize Theorem Let 0 � f ∈ Z [ X ] with bitsize τ , deg f = d . The max coeff bitsize τ 2 of univsos2 on f is O ( d 3 + d 2 τ ) . Proof. i = 0 a i X i = ∏ d p ε = ∑ d e = 2 − δ i = 1 ( X − z i ) | ˆ z i | � z i ( 1 + e ) Square-free part: τ ( p ) = O ( d + τ ) z j | = | z j | ( 1 + 2 − δ ) � 2 τ ( p ε ) + 1 ( 1 + 2 − δ ) | [Melczer et.al 16] 1 | ˆ Victor Magron On Exact Polynomial Optimization 30 / 46
univsos2 : Bit Complexity Theorem Let 0 � f ∈ Z [ X ] with bitsize τ , deg f = d . ∼ O ( d 4 + d 3 τ ) . The bit complexity of univsos2 on f is Victor Magron On Exact Polynomial Optimization 31 / 46
univsos2 : Bit Complexity Theorem Let 0 � f ∈ Z [ X ] with bitsize τ , deg f = d . ∼ O ( d 4 + d 3 τ ) . The bit complexity of univsos2 on f is Proof. Root isolation with radius O ( δ + τ ( p ε )) [Melczer et.al 16]: ∼ O ( d 3 + d 2 τ ( p ε ) + d ( δ + τ ( p ε ))) Victor Magron On Exact Polynomial Optimization 31 / 46
Benchmarks Maple version 16, Intel Core i7-5600U CPU (2.60 GHz) Averaging over five runs 1 univsos1 : sqrfree , real root isolation in Maple 2 univsos2 : PARI/GP implementation [Chevillard et. al 11] � sqrfree , sturm , polroots (interface Maple-PARI/GP) 3 univsos3 : SDPA-GMP solver (arbitrary precision) � sqrfree , sturm , sdp Victor Magron On Exact Polynomial Optimization 32 / 46
Benchmarks: [Chevillard et. al 11] Approximation f ∈ Q [ X ] of mathematical function f math Validation of sup norm � f math − f � ∞ on a rational interval univsos1 univsos2 Id d τ (bits) τ 1 (bits) t 1 (ms) τ 2 (bits) t 2 (ms) # 1 13 22 682 3 403 023 2 352 51 992 824 # 5 34 117 307 7 309 717 82 583 265 330 5 204 # 7 43 67 399 18 976 562 330 288 152 277 11 190 # 9 20 30 414 641 561 928 68 664 1 605 Victor Magron On Exact Polynomial Optimization 33 / 46
Benchmarks: [Chevillard et. al 11] Approximation f ∈ Q [ X ] of mathematical function f math Validation of sup norm � f math − f � ∞ on a rational interval univsos1 univsos2 Id d τ (bits) τ 1 (bits) t 1 (ms) τ 2 (bits) t 2 (ms) # 1 13 22 682 3 403 023 2 352 51 992 824 # 5 34 117 307 7 309 717 82 583 265 330 5 204 # 7 43 67 399 18 976 562 330 288 152 277 11 190 # 9 20 30 414 641 561 928 68 664 1 605 = ⇒ τ 1 > τ 2 t 1 > t 2 Victor Magron On Exact Polynomial Optimization 33 / 46
Benchmarks: Power Sums f = 1 + X + X 2 + · · · + X d j = 1 (( X − cos θ j ) 2 + sin 2 θ j ) , with θ j : = 2 j π f = ∏ k d + 1 univsos1 univsos2 d τ 1 (bits) t 1 (ms) τ 2 (bits) t 2 (ms) 10 823 8 567 264 20 9 003 16 1 598 485 40 91 903 45 6 034 2 622 60 301 841 92 12 326 6 320 100 1 717 828 516 31 823 19 466 200 146 140 792 130 200 120 831 171 217 − − 500 2 263 423 520 5 430 000 Victor Magron On Exact Polynomial Optimization 34 / 46
Benchmarks: Power Sums f = 1 + X + X 2 + · · · + X d j = 1 (( X − cos θ j ) 2 + sin 2 θ j ) , with θ j : = 2 j π f = ∏ k d + 1 univsos1 univsos2 d τ 1 (bits) t 1 (ms) τ 2 (bits) t 2 (ms) 10 823 8 567 264 20 9 003 16 1 598 485 40 91 903 45 6 034 2 622 60 301 841 92 12 326 6 320 100 1 717 828 516 31 823 19 466 200 146 140 792 130 200 120 831 171 217 − − 500 2 263 423 520 5 430 000 = ⇒ τ 1 > τ 2 t 1 < t 2 Victor Magron On Exact Polynomial Optimization 34 / 46
Benchmarks: Modified Wilkinson Polynomials k ( X − j ) 2 ∏ f = 1 + j = 1 k ( X − j ) 2 ∏ a = t = 1 f t = 1 f − f t = j = 1 Relatively closed roots 1, . . . , k Victor Magron On Exact Polynomial Optimization 35 / 46
Benchmarks: Modified Wilkinson Polynomials k ( X − j ) 2 ∏ f = 1 + j = 1 k ( X − j ) 2 ∏ a = t = 1 f t = 1 f − f t = j = 1 Relatively closed roots 1, . . . , k univsos1 univsos2 d τ (bits) τ 1 (bits) t 1 (ms) τ 2 (bits) t 2 (ms) 10 140 47 17 2 373 751 20 737 198 31 12 652 3 569 40 3 692 939 35 65 404 47 022 100 29 443 7 384 441 − − 500 1 022 771 255 767 73 522 Victor Magron On Exact Polynomial Optimization 35 / 46
Benchmarks: Modified Wilkinson Polynomials k ( X − j ) 2 ∏ f = 1 + j = 1 k ( X − j ) 2 ∏ a = t = 1 f t = 1 f − f t = j = 1 Relatively closed roots 1, . . . , k univsos1 univsos2 d τ (bits) τ 1 (bits) t 1 (ms) τ 2 (bits) t 2 (ms) 10 140 47 17 2 373 751 20 737 198 31 12 652 3 569 40 3 692 939 35 65 404 47 022 100 29 443 7 384 441 − − 500 1 022 771 255 767 73 522 = ⇒ τ 1 < τ 2 t 1 < t 2 Victor Magron On Exact Polynomial Optimization 35 / 46
Certify Polynomial Non-negativity The Question(s) Exact SOS Representations: n = 1 Exact SOS Representations: n � 1 Exact Polya’s Representations Exact Putinar’s Representations Conclusion and Perspectives
intsos n = 1 & Root Approximation: univsos2 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 Polynomial Optimization 36 / 46
intsos n = 1 & 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 ε < | u 2 i + 1 | + | u 2 i − 1 | p ε ≤ 0 − u 2 i 2 Victor Magron On Exact Polynomial Optimization 37 / 46
intsos with n � 1 : Perturbation f Σ P ERTURBATION idea Approximate SOS Decomposition f ( X ) - ε ∑ α ∈P /2 X 2 α = ˜ σ + u Victor Magron On Exact Polynomial Optimization 38 / 46
intsos with n � 1 : Absorbtion f ( X ) - ε ∑ α ∈P /2 X 2 α = ˜ σ + u Choice of P ? y 6 ε 2 ( x + y 3 ) 2 − x 2 + y 6 xy 3 = 1 5 2 4 u 1,3 3 2 1 ε x 0 1 2 3 4 5 Victor Magron On Exact Polynomial Optimization 39 / 46
intsos with n � 1 : Absorbtion f ( X ) - ε ∑ α ∈P /2 X 2 α = ˜ σ + u Choice of P ? y 6 2 ( xy + y 2 ) 2 − x 2 y 2 + y 4 xy 3 = 1 5 2 ε 4 u 1,3 3 ε 2 1 x 0 1 2 3 4 5 Victor Magron On Exact Polynomial Optimization 39 / 46
intsos with n � 1 : Absorbtion f ( X ) - ε ∑ α ∈P /2 X 2 α = ˜ σ + u Choice of P ? y 6 2 ( xy 2 + y ) 2 − x 2 y 4 + y 2 xy 3 = 1 5 2 ε 4 3 u 1,3 ε 2 1 x 0 1 2 3 4 5 Victor Magron On Exact Polynomial Optimization 39 / 46
intsos with n � 1 : Absorbtion f ( X ) - ε ∑ α ∈P /2 X 2 α = ˜ σ + u Choice of P ? f = 4 x 4 y 6 + x 2 − xy 2 + y 2 spt ( f ) = { ( 4, 6 ) , ( 2, 0 ) , ( 1, 2 ) , ( 0, 2 ) } Newton Polytope P = conv ( spt ( f )) Squares in SOS decomposition ⊆ P 2 ∩ N n [Reznick 78] Victor Magron On Exact Polynomial Optimization 39 / 46
Algorithm intsos 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 ← conv ( spt ( f )) f ε ← f − ε ∑ X 2 α σ ← sdp ( f ε , δ ) ˜ α ∈P /2 u ← f ε − ˜ σ ε ← ε δ ← 2 δ 2 while while f ε ≤ 0 u + ε ∑ X 2 α / ∈ Σ α ∈P /2 Victor Magron On Exact Polynomial Optimization 40 / 46
Algorithm intsos Theorem (Exact Certification Cost in ˚ Σ ) f ∈ Q [ X ] ∩ ˚ Σ [ X ] with deg f = d = 2 k and bit size τ ⇒ intsos terminates with SOS output of bit size τ d O ( n ) = Victor Magron On Exact Polynomial Optimization 40 / 46
Algorithm intsos Theorem (Exact Certification Cost in ˚ Σ ) f ∈ Q [ X ] ∩ ˚ Σ [ X ] with deg f = d = 2 k and bit size τ ⇒ intsos terminates with SOS output of bit size τ d O ( n ) = Proof. { ε ∈ R : ∀ x ∈ R n , f ( x ) − ε ∑ α ∈P /2 x 2 α � 0 } � = ∅ ⇒ τ ( ε ) = τ d O ( n ) Quantifier Elimination [Basu et. al 06] = # Coefficients in SOS output = size( P /2) = ( n + k n ) � d n Ellipsoid algorithm for SDP [Grötschel et. al 93] Victor Magron On Exact Polynomial Optimization 40 / 46
Certify Polynomial Non-negativity The Question(s) Exact SOS Representations: n = 1 Exact SOS Representations: n � 1 Exact Polya’s Representations Exact Putinar’s Representations Conclusion and Perspectives
Algorithm Polyasos f positive definite form has Polya ’s representation: σ f = with σ ∈ Σ [ X ] ( X 1 + · · · + X n ) 2 D Victor Magron On Exact Polynomial Optimization 41 / 46
Algorithm Polyasos f positive definite form has Polya ’s representation: σ f = with σ ∈ Σ [ X ] ( X 1 + · · · + X n ) 2 D Theorem f ( X 1 + · · · + X n ) 2 D ∈ Σ [ X ] = ⇒ f ( X 1 + · · · + X n ) 2 D + 2 ∈ ˚ Σ [ X ] Victor Magron On Exact Polynomial Optimization 41 / 46
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