Certification of inequalities involving transcendental functions - PowerPoint PPT Presentation

Certification of inequalities involving transcendental functions using SDP Joint Work with B. Werner, S. Gaubert and X. Allamigeon Second year PhD Victor MAGRON LIX/INRIA, ´ Ecole Polytechnique MAP 2012 Tuesday 18 th September Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Flyspeck-Like Problems The Kepler Conjecture Kepler Conjecture (1611): The maximal density of sphere packings in 3-space is π 18 It corresponds to the way people would intuitively stack oranges, as a pyramid shape The proof of T. Hales (1998) consists of thousands of non-linear inequalities Many recent efforts have been done to give a formal proof of these inequalities: Flyspeck Project Motivation: get positivity certificates and check them with Proof assistants like COQ Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Flyspeck-Like Problems Lemma Example Inequalities issued from Flyspeck non-linear part involve: Semi-Algebraic functions algebra A : composition of 1 1 p ( p ∈ N 0 ) , + , − , × , /, sup , inf polynomials with | · | , ( · ) Transcendental functions T : composition of semi-algebraic 2 functions with arctan , arcos , arcsin , exp , log , | · | , 1 p ( p ∈ N 0 ) , + , − , × , /, sup , inf ( · ) Lemma 9922699028 from Flyspeck K := [4; 6 . 3504] 3 × [6 . 3504; 8] × [4; 6 . 3504] 2 P, Q ∈ R [ X ] + 1 . 6294 − 0 . 2213 ( √ x 2 + √ x 3 + ∀ x ∈ K, − π 2 + arctan P ( x ) � Q ( x ) √ x 5 + √ x 6 − 8 . 0) + 0 . 913 ( √ x 4 − 2 . 52) + 0 . 728 ( √ x 1 − 2 . 0) ≥ 0 . Tight inequality: global optimum = 1 . 7 × 10 − 4 Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

General Framework Given K a compact set, and f a transcendental function, bound from below f ∗ = inf x ∈ K f ( x ) and prove f ∗ ≥ 0 f is underestimated by a semi-algebraic function f sa on a 1 compact set K sa ⊃ K x ∈ K sa f sa ( x ) to a polynomial optimization inf Reduce the problem 2 problem (POP) in a lifted space K pop Solve classicaly the POP problem x ∈ K pop f pop ( x ) using a inf 3 sparse variant hierarchy of SDP relaxations by Lasserre f ∗ ≥ f ∗ sa ≥ f ∗ pop ≥ 0 ���� If the relaxations are accurate enough Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

General Framework Given K a compact set, and f a transcendental function, bound from below f ∗ = inf x ∈ K f ( x ) and prove f ∗ ≥ 0 f is underestimated by a semi-algebraic function f sa on a 1 compact set K sa ⊃ K x ∈ K sa f sa ( x ) to a polynomial optimization inf Reduce the problem 2 problem (POP) in a lifted space K pop Solve classicaly the POP problem x ∈ K pop f pop ( x ) using a inf 3 sparse variant hierarchy of SDP relaxations by Lasserre f ∗ ≥ f ∗ sa ≥ f ∗ pop ≥ 0 ���� If the relaxations are accurate enough Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

General Framework Given K a compact set, and f a transcendental function, bound from below f ∗ = inf x ∈ K f ( x ) and prove f ∗ ≥ 0 f is underestimated by a semi-algebraic function f sa on a 1 compact set K sa ⊃ K x ∈ K sa f sa ( x ) to a polynomial optimization inf Reduce the problem 2 problem (POP) in a lifted space K pop Solve classicaly the POP problem x ∈ K pop f pop ( x ) using a inf 3 sparse variant hierarchy of SDP relaxations by Lasserre f ∗ ≥ f ∗ sa ≥ f ∗ pop ≥ 0 ���� If the relaxations are accurate enough Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

General Framework Given K a compact set, and f a transcendental function, bound from below f ∗ = inf x ∈ K f ( x ) and prove f ∗ ≥ 0 f is underestimated by a semi-algebraic function f sa on a 1 compact set K sa ⊃ K x ∈ K sa f sa ( x ) to a polynomial optimization inf Reduce the problem 2 problem (POP) in a lifted space K pop Solve classicaly the POP problem x ∈ K pop f pop ( x ) using a inf 3 sparse variant hierarchy of SDP relaxations by Lasserre f ∗ ≥ f ∗ sa ≥ f ∗ pop ≥ 0 ���� If the relaxations are accurate enough Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Transcendental Functions Underestimators Let f ∈ T be a transcendental univariate elementary function such as arctan , exp , ..., defined on a real interval I . Basic convexity/semiconvexity properties and monotonicity of f are used to find lower and upper semi-algebraic bounds. Example with arctan : arctan is semiconvex on I : ∃ c < 0 such that arctan − c 2( · ) 2 is convex on I i ∈C { par − ∀ a ∈ I = [ m ; M ] , arctan ( a ) ≥ max a i ( a ) } where C define an index collection of parabola tangent to the function curve and underestimating f . Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Transcendental Functions Underestimators Let f ∈ T be a transcendental univariate elementary function such as arctan , exp , ..., defined on a real interval I . Basic convexity/semiconvexity properties and monotonicity of f are used to find lower and upper semi-algebraic bounds. Example with arctan : arctan is semiconvex on I : ∃ c < 0 such that arctan − c 2( · ) 2 is convex on I i ∈C { par − ∀ a ∈ I = [ m ; M ] , arctan ( a ) ≥ max a i ( a ) } where C define an index collection of parabola tangent to the function curve and underestimating f . Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Transcendental Functions Underestimators Let f ∈ T be a transcendental univariate elementary function such as arctan , exp , ..., defined on a real interval I . Basic convexity/semiconvexity properties and monotonicity of f are used to find lower and upper semi-algebraic bounds. Example with arctan : arctan is semiconvex on I : ∃ c < 0 such that arctan − c 2( · ) 2 is convex on I i ∈C { par − ∀ a ∈ I = [ m ; M ] , arctan ( a ) ≥ max a i ( a ) } where C define an index collection of parabola tangent to the function curve and underestimating f . Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Transcendental Functions Underestimators Example with arctan: y par + 2 par + arctan 1 par − 2 a m M par − 1 Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Adaptative Semi-algebraic Approximations Algorithm The first step is to build the abstract syntax tree from an inequality, where leaves are semi-algebraic functions and nodes are univariate transcendental functions (arctan, exp, ...) or basic operations ( + , × , − , / ). 2 + 1 . 6294 − 0 . 2213 ( √ x 2 + √ x 3 + √ x 5 + √ x 6 − With l := − π 8 . 0) + 0 . 913 ( √ x 4 − 2 . 52) + 0 . 728 ( √ x 1 − 2 . 0) , the tree is: + l ( x ) arctan P ( x ) � Q ( x ) Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Adaptative Semi-algebraic Approximations algo iter First iteration: + y arctan l ( x ) arctan par − 1 a a 1 m M P ( x ) � Q ( x ) Evaluate f with randeval and obtain a minimizer guess x 1 . 1 P ( x 1 ) Compute a 1 := = 0 . 84460 � Q ( x 1 ) Get the equation of par − 2 1 1 ( P ( x ) x ∈ K { l ( x ) + par − Compute m 1 ≤ min ) } 3 � Q ( x ) Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Adaptative Semi-algebraic Approximations algo iter Second iteration: + y arctan l ( x ) arctan par − 1 a a 2 a 1 m M P ( x ) par − � 2 Q ( x ) m 1 = − 0 . 746 < 0 , obtain a new minimizer x 2 . 1 P ( x 2 ) = − 0 . 374 and par − Compute a 2 := 2 � 2 Q ( x 2 ) i ( P ( x ) i ∈{ 1 , 2 } { par − Compute m 2 ≤ min x ∈ K { l ( x ) + max ) }} 3 � Q ( x ) Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Adaptative Semi-algebraic Approximations algo iter Third iteration: + y arctan par − l ( x ) 3 arctan par − 1 a m a 2 a 3 a 1 M P ( x ) par − � 2 Q ( x ) m 2 = − 0 . 112 < 0 , obtain a new minimizer x 3 . 1 P ( x 3 ) = 0 . 357 and par − Compute a 3 := 2 � 3 Q ( x 3 ) i ( P ( x ) i ∈{ 1 , 2 , 3 } { par − Compute m 3 ≤ min x ∈ K { l ( x ) + max ) }} 3 � Q ( x ) Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Adaptative Semi-algebraic Approximations m 3 = − 0 . 0333 < 0 , obtain a new minimizer x 4 and iterate again... Theorem: Convergence of Semi-algebraic underestimators Let f ∈ T and ( x opt p ) p ∈ N be a sequence of control points obtained to define the hierarchy of f -underestimators in the previous algorithm algo iter and x ∗ be an accumulation point of ( x opt p ) p ∈ N . Then, x ∗ is a global minimizer of f on K . Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Adaptative Semi-algebraic Approximations m 3 = − 0 . 0333 < 0 , obtain a new minimizer x 4 and iterate again... Theorem: Convergence of Semi-algebraic underestimators Let f ∈ T and ( x opt p ) p ∈ N be a sequence of control points obtained to define the hierarchy of f -underestimators in the previous algorithm algo iter and x ∗ be an accumulation point of ( x opt p ) p ∈ N . Then, x ∗ is a global minimizer of f on K . Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP