Flyspeck Inequalities and Semidefinite Programming Victor Magron , - PowerPoint PPT Presentation

Flyspeck Inequalities and Semidefinite Programming Victor Magron , RA Imperial College Memory Optimization and Co-Design Meeting 29 June 2015 Victor Magron Flyspeck Inequalities and Semidefinite Programming 1 / 18

Errors and Proofs Mathematicians and Computer Scientists want to eliminate all the uncertainties on their results. Why? Victor Magron Flyspeck Inequalities and Semidefinite Programming 2 / 18

Errors and Proofs Mathematicians and Computer Scientists want to eliminate all the uncertainties on their results. Why? M. Lecat, Erreurs des Mathématiciens des origines à nos jours, 1935. ❀ 130 pages of errors! (Euler, Fermat, Sylvester, . . . ) Victor Magron Flyspeck Inequalities and Semidefinite Programming 2 / 18

Errors and Proofs Mathematicians and Computer Scientists want to eliminate all the uncertainties on their results. Why? M. Lecat, Erreurs des Mathématiciens des origines à nos jours, 1935. ❀ 130 pages of errors! (Euler, Fermat, Sylvester, . . . ) Ariane 5 launch failure, Pentium FDIV bug Victor Magron Flyspeck Inequalities and Semidefinite Programming 2 / 18

Errors and Proofs Possible workaround: proof assistants C OQ (Coquand, Huet 1984) H OL - LIGHT (Harrison, Gordon 1980) Built in top of OC AML Victor Magron Flyspeck Inequalities and Semidefinite Programming 3 / 18

Complex Proofs Complex mathematical proofs / mandatory computation K. Appel and W. Haken , Every Planar Map is Four-Colorable, 1989. T. Hales, A Proof of the Kepler Conjecture, 1994. Victor Magron Flyspeck Inequalities and Semidefinite Programming 4 / 18

From Oranges Stack... Kepler Conjecture (1611): π The maximal density of sphere packings in 3D-space is √ 18 Face-centered cubic Packing Hexagonal Compact Packing Victor Magron Flyspeck Inequalities and Semidefinite Programming 5 / 18

...to Flyspeck Nonlinear Inequalities The proof of T. Hales (1998) contains mathematical and computational parts Computation: check thousands of nonlinear inequalities Robert MacPherson, editor of The Annals of Mathematics: “[...] the mathematical community will have to get used to this state of affairs.” F lys p ec k [Hales 06]: F ormal P roof of K epler Conjecture Victor Magron Flyspeck Inequalities and Semidefinite Programming 6 / 18

...to Flyspeck Nonlinear Inequalities The proof of T. Hales (1998) contains mathematical and computational parts Computation: check thousands of nonlinear inequalities Robert MacPherson, editor of The Annals of Mathematics: “[...] the mathematical community will have to get used to this state of affairs.” F lys p ec k [Hales 06]: F ormal P roof of K epler Conjecture Project Completion on 10 August by the Flyspeck team!! Victor Magron Flyspeck Inequalities and Semidefinite Programming 6 / 18

...to Flyspeck Nonlinear Inequalities Nonlinear inequalities: quantified reasoning with “ ∀ ” ∀ x ∈ K , f ( x ) � 0 NP-hard optimization problem Victor Magron Flyspeck Inequalities and Semidefinite Programming 7 / 18

A “Simple” Example In the computational part: Multivariate Polynomials: ∆ x : = x 1 x 4 ( − x 1 + x 2 + x 3 − x 4 + x 5 + x 6 ) + x 2 x 5 ( x 1 − x 2 + x 3 + x 4 − x 5 + x 6 ) + x 3 x 6 ( x 1 + x 2 − x 3 + x 4 + x 5 − x 6 ) − x 2 ( x 3 x 4 + x 1 x 6 ) − x 5 ( x 1 x 3 + x 4 x 6 ) Victor Magron Flyspeck Inequalities and Semidefinite Programming 8 / 18

A “Simple” Example In the computational part: Semialgebraic functions: composition of polynomials with | · | , √ , + , − , × , /, sup, inf, . . . p ( x ) : = ∂ 4 ∆ x q ( x ) : = 4 x 1 ∆ x � r ( x ) : = p ( x ) / q ( x ) 2 + 1.6294 − 0.2213 ( √ x 2 + √ x 3 + √ x 5 + √ x 6 − l ( x ) : = − π 8.0 ) + 0.913 ( √ x 4 − 2.52 ) + 0.728 ( √ x 1 − 2.0 ) Victor Magron Flyspeck Inequalities and Semidefinite Programming 8 / 18

A “Simple” Example In the computational part: Transcendental functions T : composition of semialgebraic functions with arctan, exp, sin, + , − , × , . . . Victor Magron Flyspeck Inequalities and Semidefinite Programming 8 / 18

A “Simple” Example In the computational part: Feasible set K : = [ 4, 6.3504 ] 3 × [ 6.3504, 8 ] × [ 4, 6.3504 ] 2 Lemma 9922699028 from Flyspeck: � p ( x ) � ∀ x ∈ K , arctan + l ( x ) � 0 � q ( x ) Victor Magron Flyspeck Inequalities and Semidefinite Programming 8 / 18

Existing Formal Frameworks Formal proofs for Global Optimization: Bernstein polynomial methods [Zumkeller’s PhD 08] SMT methods [Gao et al. 12] Interval analysis and Sums of squares Victor Magron Flyspeck Inequalities and Semidefinite Programming 9 / 18

Existing Formal Frameworks Interval analysis Certified interval arithmetic in C OQ [Melquiond 12] Taylor methods in HOL Light [Solovyev thesis 13] Formal verification of floating-point operations robust but subject to the Curse of Dimensionality Victor Magron Flyspeck Inequalities and Semidefinite Programming 9 / 18

Existing Formal Frameworks Lemma 9922699028 from Flyspeck: � ∂ 4 ∆ x � ∀ x ∈ K , arctan √ 4 x 1 ∆ x + l ( x ) � 0 Dependency issue using Interval Calculus: One can bound ∂ 4 ∆ x / √ 4 x 1 ∆ x and l ( x ) separately Too coarse lower bound: − 0.87 Subdivide K to prove the inequality K 3 K K 0 K 1 K 4 = ⇒ K 2 Victor Magron Flyspeck Inequalities and Semidefinite Programming 9 / 18

Introduction Flyspeck Inequalities and Semidefinite Programming

Semidefinite Programming Linear Programming (LP): ⊤ z min c z A z � d . s.t. Linear cost c Polyhedron Linear inequalities “ ∑ i A ij z j � d i ” Victor Magron Flyspeck Inequalities and Semidefinite Programming 10 / 18

Semidefinite Programming Semidefinite Programming (SDP): ⊤ z min c z ∑ F i z i � F 0 . s.t. i Linear cost c Symmetric matrices F 0 , F i Spectrahedron Linear matrix inequalities “ F � 0” ( F has nonnegative eigenvalues) Victor Magron Flyspeck Inequalities and Semidefinite Programming 11 / 18

Semidefinite Programming Semidefinite Programming (SDP): ⊤ z min c z ∑ F i z i � F 0 , A z = d . s.t. i Linear cost c Symmetric matrices F 0 , F i Spectrahedron Linear matrix inequalities “ F � 0” ( F has nonnegative eigenvalues) Victor Magron Flyspeck Inequalities and Semidefinite Programming 12 / 18

SDP for Polynomial Optimization Prove polynomial inequalities with SDP: p ( a , b ) : = a 2 − 2 ab + b 2 � 0 . � � � � � � z 1 z 2 a Find z s.t. p ( 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 Flyspeck Inequalities and Semidefinite Programming 13 / 18

SDP for Polynomial Optimization Choose a cost c e.g. ( 1, 0, 1 ) and solve: ⊤ z min c z ∑ F i z i � F 0 , A z = d . s.t. i � 1 � z 1 � � z 2 − 1 Solution = � 0 (eigenvalues 0 and 1) 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 Flyspeck Inequalities and Semidefinite Programming 14 / 18

Polynomial Optimization   1 a b Semidefinite Programming  � 0 a 1 c  b c 1 � control, polynomial optim (Henrion, Lasserre, Parrilo) � combinatorial optim. electrical engineering (Laurent, Steurers) Victor Magron Flyspeck Inequalities and Semidefinite Programming 15 / 18

Polynomial Optimization   1 a b Semidefinite Programming  � 0 a 1 c  b c 1 � control, polynomial optim (Henrion, Lasserre, Parrilo) � combinatorial optim. electrical engineering (Laurent, Steurers) Theoretical Approach p ∗ : = inf R n p ( x ) ? sup λ ⇐ with p − λ � 0 I NFINITE LP Victor Magron Flyspeck Inequalities and Semidefinite Programming 15 / 18

Polynomial Optimization   1 a b Semidefinite Programming  � 0 a 1 c  b c 1 � control, polynomial optim (Henrion, Lasserre, Parrilo) � combinatorial optim. electrical engineering (Laurent, Steurers) Practical Approach p ∗ : = inf R n p ( x ) ? sup λ ⇐ with p − λ = sums of squares F INITE SDP of fixed degree Victor Magron Flyspeck Inequalities and Semidefinite Programming 15 / 18

Polynomial Optimization   1 a b Semidefinite Programming  � 0 a 1 c  b c 1 � control, polynomial optim (Henrion, Lasserre, Parrilo) � combinatorial optim. electrical engineering (Laurent, Steurers) Practical Approach p ∗ : = inf R n p ( x ) ? sup λ ⇐ with p − λ = sums of squares F INITE SDP of fixed degree SDP bounds Hierarchy ↑ p ∗ degree d ⇒ ( n + 2 d n ) variables SDP = n variables Victor Magron Flyspeck Inequalities and Semidefinite Programming 15 / 18