Formal Proofs of Inequalities and Semi-Definite Programming Supervisor: Benjamin Werner (TypiCal) Co-Supervisor: Stéphane Gaubert (Maxplus) 2 nd year PhD Victor MAGRON LIX, École Polytechnique Friday November 27 th 2011 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
Contents Background Difficulties Sums of Squares (SOS) and Semi-Definite Programming (SDP) Relaxations Formal Proofs of Non-linear Inequalities Certificates and Oracles 1 Flyspeck 2 Bernstein 3 SOS and Transcendental Functions 4 Possible Framework 5 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
Background Computational Proofs: Primality, Four colors theorem Autarcic approach: a program prime : nat → bool computes prime numbers with an algorithm proved sound and correct in Coq, no need of certificates to check the primality Sceptic approach: a program prime : nat ∗ cert → bool in Coq checks primality, helped with the certificate imported from an external tool Hales proof of the Kepler conjecture generated hundred of non-linear inequalities: need automatic proofs 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
Difficulties Multiple interests: A part of the mathematics is related to these technics The interface between the deductive « conventional » part and the computational part is particularly favorable to errors Opening new fields to proof systems while allowing some results automatization Improve the tools developed by Roland Zumkeller by using SDP tools (strong interest for the related applied mathematics) Limit the size of the certificate while using hybrid format for numbers, mixing classical numerical and symbolic representation 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
SOS and SDP Relaxations Polynomial Optimization Problem (POP): Let f k ∈ R [ x ] ( k = 0 , 1 , ..., m ) : minimize f 0 ( x ) subject to f k ( x ) � 0 ( k = 1 , 2 ..., m ) Generalized Lagrangian dual: k = 1 ϕ k ( x ) f k ( x ) ( ∀ x ∈ R n and ∀ ϕ ∈ Φ ) , L ( x , ϕ ) = f 0 ( x ) − � m Φ = { ϕ = ( ϕ 1 , ϕ 2 , ..., ϕ m ) : ∀ k ∈ { 1 , 2 ..., m } , ϕ k SOS } Lagrangian relaxation problem: L ∗ ( ϕ ) = inf { L ( x , ϕ ) : x ∈ R n } L ∗ ( ϕ ) � ζ ∗ ζ ∗ = inf { f 0 ( x ) : f k ( x ) � 0 ( k = 1 , 2 ..., m ) 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
SOS and SDP Relaxations Constrained optimization problems with semi-definite positive matrices: Find X ∈ S n , solution of the primal problem: inf � C , X � (P) A ( X ) = b X � 0 . Such formulations can be derived from the previous problem as primal SDP relaxations. 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
Formal Proofs of Non-linear Inequalities - Certificates and Oracles Proof systems like Coq have several ways to solve such problems: Without certificates, with pure functional computations (OCaml 1 fragment) : autarcic approach (Bernstein, TM) Coq checks certificates imported from external solvers (e.g. 2 Gloptipoly, SparsePOP , RAGlib, CSDP ,...): sceptical approach with formal computations Micromega: psatz tactic in Coq, developed by F. Besson, uses sceptical approach by verification of certificates imported from CSDP computations Such tactics can be developed with several computational tools: Bernstein, SOS, rational functions minimization, transcendental approximations,... 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
Formal Proofs of Non-linear Inequalities - Flyspeck Two types of inequalities issued from Flyspeck non-linear part: Pure polynomials 1 Transcendentals 2 2 + arctan − ∂ 4 ∆ x Example: dih x = π √ 4 x 1 ∆ x K = ([ 4 ; 6 . 3504 ] 3 , [ 6 . 3504 ; 6 . 3504 ] , [ 4 ; 6 . 3504 ] 2 ) � � 0 1 1 1 1 � � � � x 1 x 4 (− x 1 + x 2 + x 3 − x 4 + x 5 + x 6 ) � � x 3 x 2 x 1 � 1 0 � � � + x 2 x 5 ( x 1 − x 2 + x 3 + x 4 − x 5 + x 6 ) ∆ x = 1 � � = � 1 x 3 0 x 4 x 5 � 2 � � + x 3 x 6 ( x 1 + x 2 − x 3 + x 4 + x 5 − x 6 ) � � � x 2 x 4 x 6 � 1 0 � � − x 2 x 3 x 4 − x 1 x 3 x 5 − x 1 x 2 x 6 − x 4 x 5 x 6 � � � � x 1 x 5 x 6 1 0 � � Lemma 2570626711 : ∀ x ∈ K , dih x � 1 . 15 . 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
Formal Proofs of Non-linear Inequalities - Bernstein PhD thesis of Roland Zumkeller about Bernstein polynomials and Taylor models (TM): Global Optimization in Type Theory Software: sergei written in Haskell can provide bounds for multivariate polynomials Sufficent for the former example: ∀ x ∈ ([ 4 ; 6 . 3504 ] 3 , [ 6 . 3504 ; 6 . 3504 ] , [ 4 ; 6 . 3504 ] 2 ) , max (( ∂ 4 ∆ x ) 2 − 0 . 2 ( 4 x 1 ∆ x )) < 0 and √ 0 . 2 ) + π dih x = arctan (− 2 > 1 . 1502 > 1 . 15 Work in progress: a formal study of Bernstein coefficients and polynomials by Bertot, Guilhot and Mahboubi 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
Formal Proofs of Non-linear Inequalities - SOS and Transcendental Functions Need to deal with rational functions minimization or constrained POP: Taylor Models in Coq, Gloptipoly, SparsePOP , RAGlib Gloptipoly or RAGlib can solve the former example Not sufficent to solve many inequalities, e.g. with sums or multiplications of transcendental functions 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
Formal Proofs of Non-linear Inequalities - Possible Framework Build abstract syntax tree from an inequality, where leaves are polynomials and nodes are transcendental functions (arctan, √ , ...) or basic operations ( + , ∗ , − , / ), e.g. : + Use basic convexity properties and monotonicity of elementary functions to π arctan find lower and upper piecewise 2 polynomial bounds for each node , e.g.: y / √ tan 2 tan 1 − ∂ 4 ∆ x arctan x chord m i − 1 M i − 1 4 x 1 ∆ x 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
Formal Proofs of Non-linear Inequalities - Possible Framework Recursive algorithm solving successive constrained POP at unary or binary nodes i, e.g.: i , k tan k ( P − i − 1 ( x )) = P − � i i i chord ( P + � i − 1 ( x )) = P + i min z = m i max z = M i i − 1 � i − 1 P − � i − 1 P + z � P − z � P + i ( x ) i ( x ) i − 1 i − 1 m i − 1 M i − 1 x ∈ K x ∈ K Works out sometimes with a single tangent at each node and sergei but fails with several tangents and SOS solvers 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
Formal Proofs of Non-linear Inequalities - Possible Framework For the binary node of addition: min z max z z � z 1 + z 2 z � z 1 + z 2 k P − k P + z 1 � � z 1 � � k k l P − l P + z 2 � � z 2 � � l l 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
End Thank you for your attention! 2 nd year PhD Victor MAGRON Formal Proofs of Inequalities and Semi-Definite Programming
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