Semialgebraic Relaxations using Moment-SOS Hierarchies Victor Magron - PowerPoint PPT Presentation

Semialgebraic Relaxations using Moment-SOS Hierarchies Victor Magron , Postdoc LAAS-CNRS 17 September 2014 S IERRA Seminar Laboratoire d’Informatique de l’Ecole Normale Superieure y par + b 3 √ par + b �→ sin ( b ) b 1 b b 1 b 3 = 500 1 b 2 par + par − b 2 b 3 par − b 2 par − b 1 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 1 / 47

Personal Background 2008 − 2010: Master at Tokyo University H IERARCHICAL D OMAIN D ECOMPOSITION M ETHODS (S. Yoshimura) 2010 − 2013: PhD at Inria Saclay LIX/CMAP F ORMAL P ROOFS FOR N ONLINEAR O PTIMIZATION (S. Gaubert and B. Werner) 2014 − now: Postdoc at LAAS-CNRS M OMENT -SOS APPLICATIONS (J.B. Lasserre and D. Henrion) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 2 / 47

Errors and Proofs Mathematicians 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, . . . ) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 3 / 47

Errors and Proofs Possible workaround: proof assistants C OQ (Coquand, Huet 1984) H OL - LIGHT (Harrison, Gordon 1980) Built in top of OC AML V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 3 / 47

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. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 4 / 47

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 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 5 / 47

...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 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 6 / 47

...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!! V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 6 / 47

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 ) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 7 / 47

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 ) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 7 / 47

A “Simple” Example In the computational part: Transcendental functions T : composition of semialgebraic functions with arctan, exp, sin, + , − , × , . . . V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 7 / 47

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 ) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 7 / 47

Existing Formal Frameworks Formal proofs for Global Optimization: Bernstein polynomial methods [Zumkeller 08] restricted to polynomials Taylor + Interval arithmetic [Melquiond 12, Solovyev 13] robust but subject to the C URSE OF D IMENSIONALITY V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 8 / 47

Existing Formal Frameworks Lemma 9922699028 from Flyspeck: � ∂ 4 ∆ x � √ 4 x 1 ∆ x ∀ x ∈ K , arctan + 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 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 8 / 47

Introduction Moment-SOS relaxations Another look at Nonnegativity New Applications of Moment-SOS Hierarchies Conclusion

Polynomial Optimization Problems Semialgebraic set K : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } p ∗ : = min x ∈ K p ( x ) : NP hard Sums of squares Σ [ x ] e.g. x 2 1 − 2 x 1 x 2 + x 2 2 = ( x 1 − x 2 ) 2 � � σ 0 ( x ) + ∑ m Q ( K ) : = j = 1 σ j ( x ) g j ( x ) , with σ j ∈ Σ [ x ] V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 9 / 47

Polynomial Optimization Problems Archimedean module The set K is compact and the polynomial N − � x � 2 2 belongs to Q ( K ) for some N > 0. Assume that K is a box: product of closed intervals Normalize the feasibility set to get K ′ : = [ − 1, 1 ] n K ′ : = { x ∈ R n : g 1 : = 1 − x 2 1 � 0, · · · , g n : = 1 − x 2 n � 0 } n − � x � 2 2 belongs to Q ( K ′ ) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 10 / 47

Convexification and the K Moment Problem Borel σ -algebra B (generated by the open sets of R n ) M + ( K ) : set of probability measures supported on K . If µ ∈ M + ( K ) then 1 µ : B → [ 0, 1 ] , µ ( ∅ ) = 0, µ ( R n ) < ∞ 2 µ ( � i B i ) = ∑ i µ ( B i ) , for any countable ( B i ) ⊂ B 3 � K µ ( d x ) = 1 supp ( µ ) is the smallest set K such that µ ( R n \ K ) = 0 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 11 / 47

Convexification and the K Moment Problem � p ∗ = inf x ∈ K p ( x ) = K p d µ inf µ ∈M + ( K ) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 11 / 47

Convexification and the K Moment Problem Let ( x α ) α ∈ N n be the monomial basis Definition A sequence y has a representing measure on K if there exists a finite measure µ supported on K such that � ∀ α ∈ N n . K x α µ ( d x ) , y α = V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 11 / 47

Convexification and the K Moment Problem L y ( q ) : q ∈ R [ x ] �→ ∑ α q α y α Theorem [Putinar 93] Let K be compact and Q ( K ) be Archimedean. Then y has a representing measure on K iff L y ( σ ) � 0 , L y ( g j σ ) � 0 , ∀ σ ∈ Σ [ x ] . V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 11 / 47

Lasserre’s Hierarchy of SDP relaxations Moment matrix M ( y ) u , v : = L y ( u · v ) , u , v monomials Localizing matrix M ( g j y ) associated with g j M ( g j y ) u , v : = L y ( u · v · g j ) , u , v monomials V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 12 / 47

Lasserre’s Hierarchy of SDP relaxations M k ( y ) contains ( n + 2 k n ) variables, has size ( n + k n ) Truncated matrix of order k = 2 with variables x 1 , x 2 : x 2 x 2 | | 1 x 1 x 2 x 1 x 2 1 2   | | 1 1 y 1,0 y 0,1 y 2,0 y 1,1 y 0,2 −  − − − − − − −      x 1 y 1,0 | y 2,0 y 1,1 | y 3,0 y 2,1 y 1,2     x 2 y 0,1 | y 1,1 y 0,2 | y 2,1 y 1,2 y 0,3   M 2 ( y ) =   − − − − − − − − −     x 2   y 2,0 | y 3,0 y 2,1 | y 4,0 y 3,1 y 2,2   1   x 1 x 2 y 1,1 | y 2,1 y 1,2 | y 3,1 y 2,2 y 1,3   x 2 | | y 0,2 y 1,2 y 0,3 y 2,2 y 1,3 y 0,4 2 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 12 / 47

Lasserre’s Hierarchy of SDP relaxations Consider g 1 ( x ) : = 2 − x 2 1 − x 2 2 . Then v 1 = ⌈ deg g 1 /2 ⌉ = 1. 1 x 1 x 2   2 − y 2,0 − y 0,2 2 y 1,0 − y 3,0 − y 1,2 2 y 0,1 − y 2,1 − y 0,3 1 M 1 ( g 1 y ) = 2 y 1,0 − y 3,0 − y 1,2 2 y 2,0 − y 4,0 − y 2,2 2 y 1,1 − y 3,1 − y 1,3 x 1   x 2 2 y 0,1 − y 2,1 − y 0,3 2 y 1,1 − y 3,1 − y 1,3 2 y 0,2 − y 2,2 − y 0,4 M 1 ( g 1 y )( 3, 3 ) = L ( g 1 ( x ) · x 2 · x 2 ) = L ( 2 x 2 2 − x 2 1 x 2 2 − x 4 2 ) = 2 y 0,2 − y 2,2 − y 0,4 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 12 / 47

Lasserre’s Hierarchy of SDP relaxations Truncation with moments of order at most 2 k v j : = ⌈ deg g j /2 ⌉ Hierarchy of semidefinite relaxations:  � K p α x α µ ( d x ) = ∑ α p α y α inf y L y ( p ) = ∑ α     M k ( y ) 0 , � M k − v j ( g j y ) 0 , 1 � j � m , �     = y 1 1 . V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 12 / 47