Certifying Non-negativity with Lasserre’s Hierarchy and Semidefinite Programming Victor Magron , LAAS–CNRS 5 March 2019 Faculty of Mechanical Engineering University of Ljubljana Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 0 / 40
Introduction V ERIFICATION OF NONLINEAR SYSTEMS . . . SAFETY of critical parts for computing � physical devices Control Software/Hardware Cars x x j x i Space Smart Systems Grids . . . CAST AS C ERTIFIED OPTIMIZATION � S OLVE OFFLINE Input: linear semidefinite polynomial Output: value + numerical/symbolic/formal certificate Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 1 / 40
SDP for Polynomial Optimization NP-hard NON CONVEX Problem p ⋆ = inf p ( x ) Theory (Primal) (Dual) � inf p d µ sup λ µ proba ⇒ ⇐ with p − λ � 0 with INFINITE LP Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 2 / 40
SDP for Polynomial Optimization NP-hard NON CONVEX Problem p ⋆ = inf p ( x ) Practice (Primal Relaxation ) (Dual Strengthening ) � x α d µ p − λ = sum of squares moments finite number ⇒ SDP ⇐ fixed degree L ASSERRE ’ S H IERARCHY of CONVEX P ROBLEMS ↑ p ∗ [Lasserre/Parrilo 01] degree d ⇒ ( n + d = n ) SDP VARIABLES n vars Numeric = ⇒ Approx Certificate Solvers Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 2 / 40
Success Stories: Lasserre’s Hierarchy M ODELING P OWER : Cast as ∞ -dimensional LP over measures S TATIC Polynomial Optimization Optimal Powerflow n ≃ 10 3 [Josz et al 16] Roundoff Error n ≃ 10 2 [Magron et al 17] D YNAMICAL Polynomial Optimization Regions of attraction [Henrion et al 14] Reachable sets [Magron et al 17] △ ! APPROXIMATE O PTIMIZATION B OUNDS ! Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 3 / 40
Success Stories: Certified Optimization Kepler’s Conjecture(1611) √ The max density of sphere packings is π / 18 F lys p ec k : F ormalizing the p roof of K epler by T.Hales (1994) Verification of thousands of “tight” nonlinear inequalities Seminal Paper: Hales, Adams, Bauer, Dang, Harrison, Hoang, Kaliszyk, M., Mclaughlin, Nguyen, Nguyen, Nipkow, Obua, Pleso, Rute, Solovyev, Ta, Tran, Trieu, Urban, Vu & Zumkeller, Forum of Mathematics, Pi , 5 2017 ∼ 120 citations M Y CONTRIBUTION : (Non)-Polynomial optimization to verify F lys p ec k inequalities Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 4 / 40
Certification Challenges A PPROXIMATE SOLUTIONS sum of squares of a 2 − 2 ab + b 2 ? ( 1.00001 a − 0.99998 b ) 2 ! a 2 − 2 ab + b 2 ≃ ( 1.00001 a − 0.99998 b ) 2 a 2 − 2 ab + b 2 � = 1.0000200001 a 2 − 1.9999799996 ab + 0.9999600004 b 2 → = ? ≃ Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 5 / 40
Certification Challenges S CALABILITY [Joswig et al 16] 20000+ terms, d = 39 , n = 6 2 d 2 1 d 2 2 k 12 1 k 4 5 x 6 1 x 13 9 + 46 d 2 1 d 2 2 k 12 1 k 4 5 x 5 1 x 14 9 + 2 d 2 1 d 2 2 k 11 1 k 5 5 x 5 1 x 14 9 + 46 d 2 1 d 2 2 k 11 1 k 5 5 x 4 1 x 15 9 + 11 d 1 d 2 k 13 1 k 4 5 x 6 1 x 14 9 + 297 d 1 d 2 k 13 1 k 4 5 x 5 1 x 15 9 + 11 d 1 d 2 k 12 1 k 5 5 x 5 1 x 15 9 + 297 d 1 d 2 k 12 1 k 5 5 x 4 1 x 16 9 + 242 k 14 1 k 4 5 x 5 1 x 16 9 + 242 k 13 1 k 5 5 x 4 1 x 17 9 + 2 d 3 1 d 3 2 k 11 1 k 4 5 x 6 1 x 11 9 + 46 d 3 1 d 3 2 k 11 1 k 4 5 x 5 1 x 12 9 + 2 d 3 1 d 3 2 k 10 1 k 5 5 x 5 1 x 12 9 + 46 d 3 1 d 3 2 k 10 1 k 5 5 x 4 1 x 13 9 + 6 d 3 1 d 2 2 k 11 1 k 4 5 x 5 1 x 13 9 + 138 d 3 1 d 2 2 k 11 1 k 4 5 x 4 1 x 14 9 + 4 d 3 1 d 2 2 k 10 1 k 5 5 x 4 1 x 14 9 + 92 d 3 1 d 2 2 k 10 1 k 5 5 x 3 1 x 15 9 + 8 d 2 1 d 3 2 k 11 1 k 4 5 x 6 1 x 12 9 + 184 d 2 1 d 3 2 k 11 1 k 4 5 x 5 1 x 13 9 + 6 d 2 1 d 3 2 k 10 1 k 5 5 x 5 1 x 13 9 + 138 d 2 1 d 3 2 k 10 1 k 5 5 x 4 1 x 14 9 + 2 d 2 1 d 2 2 k 12 1 k 4 5 x 7 1 x 11 9 + 73 d 2 1 d 2 2 k 12 1 k 4 5 x 6 1 x 12 9 + 617 d 2 1 d 2 2 k 12 1 k 4 5 x 5 1 x 13 9 + 2 d 2 1 d 2 2 k 12 1 k 3 5 x 6 1 x 13 9 + 46 d 2 1 d 2 2 k 12 1 k 3 5 x 5 1 x 14 9 + 2 d 2 1 d 2 2 k 11 1 k 5 5 x 6 1 x 12 9 + 73 d 2 1 d 2 2 k 11 1 k 5 5 x 5 1 x 13 9 + 617 d 2 1 d 2 2 k 11 1 k 5 5 x 4 1 x 14 9 + 4 d 2 1 d 2 2 k 11 1 k 4 5 x 5 1 x 14 9 + 92 d 2 1 d 2 2 k 11 1 k 4 5 x 4 1 x 15 9 + 2 d 2 1 d 2 2 k 10 1 k 5 5 x 4 1 x 15 9 + 46 d 2 1 d 2 2 k 10 1 k 5 5 x 3 1 x 16 9 + 45 d 2 1 d 2 k 12 1 k 4 5 x 5 1 x 14 9 + 1215 d 2 1 d 2 k 12 1 k 4 5 x 4 1 x 15 9 + 34 d 2 1 d 2 k 11 1 k 5 5 x 4 1 x 15 9 + 918 d 2 1 d 2 k 11 1 k 5 5 x 3 1 x 16 9 + d 1 d 2 2 k 12 1 k 4 5 x 7 1 x 12 9 + 91 d 1 d 2 2 k 12 1 k 4 5 x 6 1 x 13 9 + 1760 d 1 d 2 2 k 12 1 k 4 5 x 5 1 x 14 9 + d 1 d 2 2 k 11 1 k 5 5 x 6 1 x 13 9 + 80 d 1 d 2 2 k 11 1 k 5 5 x 5 1 x 14 9 + 1463 d 1 d 2 2 k 11 1 k 5 5 x 4 1 x 15 9 + 12 d 1 d 2 k 13 1 k 4 5 x 7 1 x 12 9 + 467 d 1 d 2 k 13 1 k 4 5 x 6 1 x 13 9 + 3575 d 1 d 2 k 13 1 k 4 5 x 5 1 x 14 9 + 11 d 1 d 2 k 13 1 k 3 5 x 6 1 x 14 9 + 297 d 1 d 2 k 13 1 k 3 5 x 5 1 x 15 9 + 12 d 1 d 2 k 12 1 k 5 5 x 6 1 x 13 9 + 467 d 1 d 2 k 12 1 k 5 5 x 5 1 x 14 9 + 3575 d 1 d 2 k 12 1 k 5 5 x 4 1 x 15 9 + 22 d 1 d 2 k 12 1 k 4 5 x 5 1 x 15 9 + 594 d 1 d 2 k 12 1 k 4 5 x 4 1 x 16 9 + 11 d 1 d 2 k 11 1 k 5 5 x 4 1 x 16 9 + 297 d 1 d 2 k 11 1 k 5 5 x 3 1 x 17 9 + 1254 d 1 k 13 1 k 4 5 x 4 1 x 16 9 + 1012 d 1 k 12 1 k 5 5 x 3 1 x 17 9 + + 43 d 2 k 13 1 k 4 5 x 6 1 x 14 9 + 1834 d 2 k 13 1 k 4 5 x 5 1 x 15 9 + 43 d 2 k 12 1 k 5 5 x 5 1 x 15 9 + 1592 d 2 k 12 1 k 5 5 x 4 1 x 16 9 + 286 k 14 1 k 4 5 x 6 1 x 14 9 + 2904 k 14 1 k 4 5 x 5 1 x 15 9 + 242 k 14 1 k 3 5 x 5 1 x 16 9 + 286 k 13 1 k 5 5 x 5 1 x 15 9 + . . . Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 6 / 40
Certification Challenges “In theory, theory and practice are the same. In practice, they are different.” - A. Einstein C ONVERGENCE RATE 1 √ ↑ P RACTICE ? T HEORY c log STAIRS c [Nie-Schweighofer 07] Scientific challenge : bridge THEORY & PRACTICE gap Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 7 / 40
Modeling Challenges Cyber-Physical C ONTROL SYSTEMS Vehicules Collisions Fluid mechanics P ARTIAL D IFFERENTIAL E QUATIONS M IXING D ISCRETE /C ONTINUOUS E QUATIONS Discrete x t + 1 = f ( x t ) = ⇒ µ T = µ 0 + f # µ Liouville Transport Continuous x = f ( x ) ˙ = ⇒ µ T = µ 0 + div f µ Equation Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 8 / 40
Modeling Challenges Cyber-Physical F INITE - PRECISION SOFTWARE / HARDWARE a + ( b + c ) � = ( a + b ) + c Tuned Precision Approx Math Optimize Programs FPGAs Functions P ERFORMANCE A CCURACY VS M IXED P RECISION � Scalability � Loops Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 9 / 40
What is Semidefinite Optimization? 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 Certifying Non-negativity with Lasserre’s Hierarchy and SDP 10 / 40
What is Semidefinite Optimization? 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 Certifying Non-negativity with Lasserre’s Hierarchy and SDP 10 / 40
What is Semidefinite Optimization? Semidefinite Programming (SDP): ⊤ z min c z ∑ s.t. F i z i � F 0 , A z = d . i Linear cost c Symmetric matrices F 0 , F i Spectrahedron Linear matrix inequalities “ F � 0 ” ( F has nonnegative eigenvalues) Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 10 / 40
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 Certifying Non-negativity with Lasserre’s Hierarchy and SDP 10 / 40
Lasserre’s Hierarchy Prove polynomial inequalities with SDP: f ( a , b ) : = a 2 − 2 ab + b 2 � 0 . � � � � � � z 1 z 2 a Find z s.t. f ( 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 Certifying Non-negativity with Lasserre’s Hierarchy and SDP 11 / 40
Lasserre’s Hierarchy 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 � � − 1 z 2 Solution = � 0 (eigenvalues 0 and 2) − 1 z 2 z 3 1 � � 1 � � a � − 1 a 2 − 2 ab + b 2 = � = ( a − b ) 2 . a b − 1 b 1 � �� � � 0 Solving SDP = ⇒ Finding S UMS OF S QUARES certificates Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 12 / 40
Lasserre’s Hierarchy NP hard General Problem : f ∗ : = min x ∈ K f ( x ) Semialgebraic set K : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } Victor Magron Certifying Non-negativity with Lasserre’s Hierarchy and SDP 13 / 40
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