New Applications of Semidefinite Programming Victor Magron , RA Imperial College 3 Février 2015 Journées GDRIM y par + b 3 √ par + b �→ sin ( b ) b 1 b b 3 = 500 1 b 1 b 2 par + par − b 2 b 3 par − b 2 par − b 1 Victor Magron New Applications of Semidefinite Programming 1 / 22
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 New Applications of Semidefinite Programming 2 / 22
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 New Applications of Semidefinite Programming 2 / 22
What is Semidefinite Programming? Semidefinite Programming (SDP): ⊤ z min c z ∑ A z = d . 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 New Applications of Semidefinite Programming 2 / 22
Applications of SDP Combinatorial optimization Control theory Matrix completion Unique Games Conjecture (Khot ’02) : “A single concrete algorithm provides optimal guarantees among all efficient algorithms for a large class of computational problems.” (Barak and Steurer survey at ICM’14) Solving polynomial optimization (Lasserre ’01) Victor Magron New Applications of Semidefinite Programming 3 / 22
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 New Applications of Semidefinite Programming 4 / 22
SDP for Polynomial Optimization Choose a cost c e.g. ( 1, 0, 1 ) and solve: ⊤ z min c z ∑ s.t. F i z i � F 0 , A z = d . i � 1 � z 1 � � z 2 − 1 Solution = � 0 (eigenvalues 0 and 1) − 1 z 2 z 3 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 New Applications of Semidefinite Programming 4 / 22
SDP for Polynomial Optimization General case : Semialgebraic set S : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } p ∗ : = min x ∈ S p ( x ) : NP hard Sums of squares (SOS) Σ [ x ] (e.g. ( x 1 − x 2 ) 2 ) � � σ 0 ( x ) + ∑ m Q ( S ) : = j = 1 σ j ( x ) g j ( x ) , with σ j ∈ Σ [ x ] Fix the degree 2 k of sums of squares Q k ( S ) : = Q ( S ) ∩ R 2 k [ x ] Victor Magron New Applications of Semidefinite Programming 4 / 22
SDP for Polynomial Optimization Hierarchy of SDP relaxations : � � λ k : = sup λ : p − λ ∈ Q k ( S ) λ Convergence guarantees λ k ↑ p ∗ [Lasserre 01] Can be computed with SDP solvers ( CSDP , SDPA ) � Extension to semialgebraic functions r ( x ) = p ( x ) / q ( x ) [Lasserre-Putinar 10] Victor Magron New Applications of Semidefinite Programming 4 / 22
Introduction SDP for Nonlinear (Formal) Optimization SDP for Real Algebraic Geometry SDP for Program Verification Conclusion
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 New Applications of Semidefinite Programming 5 / 22
...to Flyspeck Nonlinear Inequalities The proof of T. Hales (1998) contains mathematical and computational parts Computation: check thousands of nonlinear inequalities F lys p ec k [Hales 06]: F ormal P roof of K epler Conjecture Victor Magron New Applications of Semidefinite Programming 6 / 22
...to Flyspeck Nonlinear Inequalities The proof of T. Hales (1998) contains mathematical and computational parts Computation: check thousands of nonlinear inequalities 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 New Applications of Semidefinite Programming 6 / 22
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 New Applications of Semidefinite Programming 7 / 22
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 − 8.0 ) + l ( x ) : = − π 0.913 ( √ x 4 − 2.52 ) + 0.728 ( √ x 1 − 2.0 ) Victor Magron New Applications of Semidefinite Programming 7 / 22
A “Simple” Example In the computational part: Transcendental functions T : composition of semialgebraic functions with arctan, exp, sin, + , − , × , . . . Victor Magron New Applications of Semidefinite Programming 7 / 22
A “Simple” Example In the computational part: Feasible set S : = [ 4, 6.3504 ] 3 × [ 6.3504, 8 ] × [ 4, 6.3504 ] 2 Lemma 9922699028 from Flyspeck: � p ( x ) � ∀ x ∈ S , arctan + l ( x ) � 0 � q ( x ) Victor Magron New Applications of Semidefinite Programming 7 / 22
New Framework (in my PhD thesis) Certificates for Nonlinear Optimization using SDP and: Maxplus approximation (Optimal Control) Nonlinear templates (Static Analysis) Verification of these certificates inside C OQ : p = σ 0 + ∑ j σ j g j = ⇒ ∀ x ∈ S , p ( x ) � 0 . Victor Magron New Applications of Semidefinite Programming 8 / 22
Contribution: Publications and Software V. M., X. Allamigeon, S. Gaubert and B. Werner. Formal Proofs for Nonlinear Optimization, arxiv:1404.7282, 2015. Journal of Formalized Reasoning . Software Implementation NLCertify : https://forge.ocamlcore.org/projects/nl-certify/ 15 000 lines of OC AML code 4000 lines of C OQ code V. M. NLCertify: A Tool for Formal Nonlinear Optimization, arxiv:1405.5668, 2014. ICMS . Victor Magron New Applications of Semidefinite Programming 9 / 22
Introduction SDP for Nonlinear (Formal) Optimization SDP for Real Algebraic Geometry SDP for Program Verification Conclusion
Projections of Semialgebraic Sets Semialgebraic set S : = { x ∈ R n : g 1 ( x ) � 0, . . . , g l ( x ) � 0 } A polynomial map f : R n → R m , x �→ f ( x ) : = ( f 1 ( x ) , . . . , f m ( x )) F : = f ( S ) ⊆ B , with B ⊂ R m a box or a ball Tractable approximations of F ? Victor Magron New Applications of Semidefinite Programming 10 / 22
Projections of Semialgebraic Sets f 1 : = ( x 1 + x 2 − 7.5 ) 2 /4 + ( − x 1 + x 2 + 3 ) 2 , g 1 : = − ( x 1 − 2 ) 3 /2 − x 2 + 2.5 , g 2 : = − x 1 − x 2 + 8 ( − x 1 + x 2 + 0.65 ) 2 + 3.85 , f 2 : = ( x 1 − 1 ) 2 /4 + ( x 2 − 4 ) 2 /4 . S : = { x ∈ R 2 : g 1 ( x ) � 0, g 2 ( x ) � 0 } . F = f ( S ) S Victor Magron New Applications of Semidefinite Programming 10 / 22
Projections of Semialgebraic Sets Includes important special cases: 1 m = 1: polynomial optimization F ⊆ [ min x ∈ S f ( x ) , max x ∈ S f ( x )] 2 Approximate projections of S when f ( x ) : = ( x 1 , . . . , x m ) Victor Magron New Applications of Semidefinite Programming 10 / 22
Existential Quantifier Elimination Another point of view: F = { y ∈ B : ∃ x ∈ S s.t. f ( x ) = y } , Victor Magron New Applications of Semidefinite Programming 11 / 22
Existential Quantifier Elimination Another point of view: F = { y ∈ B : ∃ x ∈ S s.t. � y − f ( x ) � 2 2 = 0 } , Victor Magron New Applications of Semidefinite Programming 11 / 22
Existential Quantifier Elimination Another point of view: F = { y ∈ B : ∃ x ∈ S s.t. h f ( x , y ) � 0 } , with h f ( x , y ) : = −� y − f ( x ) � 2 2 . Define h ( y ) : = sup x ∈ S h f ( x , y ) Victor Magron New Applications of Semidefinite Programming 11 / 22
Existential Quantifier Elimination Hierarchy of SDP : � � � B ( q − h ) d y : q − h f ∈ Q k ( S × B )) inf . q Existential QE: approximate F as closely as desired [Lasserre 14] F k : = { y ∈ B : q k ( y ) � 0 } , for some polynomials q k ∈ R 2 k [ y ] . Victor Magron New Applications of Semidefinite Programming 11 / 22
Existential Quantifier Elimination Theorem Assuming that S has non empty interior, k → ∞ vol ( F k \ F ) = 0 . lim Victor Magron New Applications of Semidefinite Programming 11 / 22
Approximating Projections f ( x ) = ( x 1 , x 2 ) : projection on R 2 of the semialgebraic set 2 � 1, 1/4 − ( x 1 + 1/2 ) 2 − x 2 S : = { x ∈ R 3 : � x � 2 2 � 0, 1/9 − ( x 1 − 1/2 ) 4 − x 4 2 � 0 } F 2 Victor Magron New Applications of Semidefinite Programming 12 / 22
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