NLCertify : A Tool for Formal Nonlinear Optimization Victor Magron , Postdoc LAAS-CNRS 18 September 2014 Aric Seminar Lyon y par + b 3 √ par + b �→ sin ( b ) b 1 b b 1 b 2 b 3 = 500 1 par + par − b 2 b 3 par − b 2 par − b 1 V. Magron NLCertify : A Tool for Formal Nonlinear Optimization 1 / 44
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 NLCertify : A Tool for Formal Nonlinear Optimization 2 / 44
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 NLCertify : A Tool for Formal Nonlinear Optimization 2 / 44
Computer Science and Mathematics Tool: Formal Bounds for Global Optimization Collaboration with: Benjamin Werner (LIX Polytechnique) Stéphane Gaubert (Maxplus Team CMAP/INRIA Polytechnique) Xavier Allamigeon (Maxplus Team) V. Magron NLCertify : A Tool for Formal Nonlinear Optimization 3 / 44
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 NLCertify : A Tool for Formal Nonlinear Optimization 4 / 44
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 NLCertify : A Tool for Formal Nonlinear Optimization 5 / 44
...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 NLCertify : A Tool for Formal Nonlinear Optimization 6 / 44
...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 NLCertify : A Tool for Formal Nonlinear Optimization 6 / 44
...to Flyspeck Nonlinear Inequalities Nonlinear inequalities: quantified reasoning with “ ∀ ” ∀ x ∈ K , f ( x ) � 0 NP-hard optimization problem V. Magron NLCertify : A Tool for Formal Nonlinear Optimization 7 / 44
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 NLCertify : A Tool for Formal Nonlinear Optimization 8 / 44
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 NLCertify : A Tool for Formal Nonlinear Optimization 8 / 44
A “Simple” Example In the computational part: Transcendental functions T : composition of semialgebraic functions with arctan, exp, sin, + , − , × , . . . V. Magron NLCertify : A Tool for Formal Nonlinear Optimization 8 / 44
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 NLCertify : A Tool for Formal Nonlinear Optimization 8 / 44
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 V. Magron NLCertify : A Tool for Formal Nonlinear Optimization 9 / 44
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 V. Magron NLCertify : A Tool for Formal Nonlinear Optimization 9 / 44
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 NLCertify : A Tool for Formal Nonlinear Optimization 9 / 44
Existing Formal Frameworks Sums of squares techniques Formalized in H OL - LIGHT [Harrison 07] C OQ [Besson 07] Precise methods but scalibility and robustness issues (numerical) powerful: global optimality certificates without branching but not so robust: handles moderate size problems Restricted to polynomials V. Magron NLCertify : A Tool for Formal Nonlinear Optimization 9 / 44
Existing Formal Frameworks Approximation theory: Chebyshev/Taylor models mandatory for non-polynomial problems hard to combine with SOS techniques (degree of approximation) V. Magron NLCertify : A Tool for Formal Nonlinear Optimization 9 / 44
Existing Formal Frameworks Can we develop a new approach with both keeping the respective strength of interval and precision of SOS? Proving Flyspeck Inequalities is challenging: medium-size and tight V. Magron NLCertify : A Tool for Formal Nonlinear Optimization 9 / 44
New Framework (in my PhD thesis) Certificates for lower bounds of Nonlinear optimization using: Moment-SOS hierarchies Maxplus approximation (Optimal Control) Verification of these certificates inside C OQ V. Magron NLCertify : A Tool for Formal Nonlinear Optimization 10 / 44
New Framework (in my PhD thesis) Software Implementation NLCertify : https://forge.ocamlcore.org/projects/nl-certify/ 15 000 lines of OC AML code 4000 lines of C OQ code V. Magron NLCertify : A Tool for Formal Nonlinear Optimization 10 / 44
Introduction Moment-SOS relaxations Semialgebraic Maxplus Optimization Formal Nonlinear Optimization
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 NLCertify : A Tool for Formal Nonlinear Optimization 11 / 44
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 NLCertify : A Tool for Formal Nonlinear Optimization 12 / 44
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 NLCertify : A Tool for Formal Nonlinear Optimization 13 / 44
Convexification and the K Moment Problem � p ∗ = inf x ∈ K p ( x ) = K p d µ inf µ ∈M + ( K ) V. Magron NLCertify : A Tool for Formal Nonlinear Optimization 13 / 44
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 NLCertify : A Tool for Formal Nonlinear Optimization 13 / 44
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 NLCertify : A Tool for Formal Nonlinear Optimization 13 / 44
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 NLCertify : A Tool for Formal Nonlinear Optimization 14 / 44
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