Certification of Inequalities involving Transcendental Functions: - PowerPoint PPT Presentation

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization Certification of Inequalities involving Transcendental Functions: combining SDP and Max-plus Approximation Joint Work withX. Allamigeon, S. Gaubert and B. Werner Third year PhD Victor MAGRON LIX/CMAP INRIA, ´ Ecole Polytechnique ECC 2013 Thursday July 18 th Third year PhD Victor MAGRON Maxplus SOS

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization The Kepler Conjecture Kepler Conjecture (1611): π √ The maximal density of sphere packings in 3D-space is 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 Third year PhD Victor MAGRON Maxplus SOS

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization Contents Flyspeck-Like Global Optimization 1 Classical Approach: Taylor + SOS 2 Max-Plus Estimators 3 Certified Global Optimization 4 Third year PhD Victor MAGRON Maxplus SOS

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization The Kepler Conjecture Inequalities issued from Flyspeck non-linear part involve: Multivariate Polynomials: 1 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 ) Semi-Algebraic functions algebra A : composition of 2 polynomials with | · | , √ , + , − , × , /, sup , inf , · · · Transcendental functions T : composition of semi-algebraic 3 functions with arctan , exp , sin , + , − , × , · · · Lemma from Flyspeck (inequality ID 6096597438 ) ∀ x ∈ [3 , 64] , 2 π − 2 x arcsin(cos(0 . 797) sin( π/x ))+0 . 0331 x − 2 . 097 ≥ 0 Third year PhD Victor MAGRON Maxplus SOS

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization The Kepler Conjecture Inequalities issued from Flyspeck non-linear part involve: Multivariate Polynomials: 1 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 ) Semi-Algebraic functions algebra A : composition of 2 polynomials with | · | , √ , + , − , × , /, sup , inf , · · · Transcendental functions T : composition of semi-algebraic 3 functions with arctan , exp , sin , + , − , × , · · · Lemma from Flyspeck (inequality ID 6096597438 ) ∀ x ∈ [3 , 64] , 2 π − 2 x arcsin(cos(0 . 797) sin( π/x ))+0 . 0331 x − 2 . 097 ≥ 0 Third year PhD Victor MAGRON Maxplus SOS

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization The Kepler Conjecture Inequalities issued from Flyspeck non-linear part involve: Multivariate Polynomials: 1 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 ) Semi-Algebraic functions algebra A : composition of 2 polynomials with | · | , √ , + , − , × , /, sup , inf , · · · Transcendental functions T : composition of semi-algebraic 3 functions with arctan , exp , sin , + , − , × , · · · Lemma from Flyspeck (inequality ID 6096597438 ) ∀ x ∈ [3 , 64] , 2 π − 2 x arcsin(cos(0 . 797) sin( π/x ))+0 . 0331 x − 2 . 097 ≥ 0 Third year PhD Victor MAGRON Maxplus SOS

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization The Kepler Conjecture Inequalities issued from Flyspeck non-linear part involve: Multivariate Polynomials: 1 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 ) Semi-Algebraic functions algebra A : composition of 2 polynomials with | · | , √ , + , − , × , /, sup , inf , · · · Transcendental functions T : composition of semi-algebraic 3 functions with arctan , exp , sin , + , − , × , · · · Lemma from Flyspeck (inequality ID 6096597438 ) ∀ x ∈ [3 , 64] , 2 π − 2 x arcsin(cos(0 . 797) sin( π/x ))+0 . 0331 x − 2 . 097 ≥ 0 Third year PhD Victor MAGRON Maxplus SOS

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization Global Optimization Problems: Examples from the Literature   4 3 � � a ij ( x j − p ij ) 2 H3 : x ∈ [0 , 1] 3 − min c i exp  −  i =1 j =1 sin( x 1 + x 2 ) + ( x 1 − x 2 ) 2 − 0 . 5 x 2 + 2 . 5 x 1 + 1 min MC : x 1 ∈ [ − 3 , 3] x 2 ∈ [ − 1 . 5 , 4] 5 n � � � � SBT : min j cos(( j + 1) x i + j ) x ∈ [ − 10 , 10] n i =1 j =1 n ( x i + ǫx i +1 ) sin( √ x i ) � x ∈ [1 , 500] n − min ( ǫ ∈ { 0 , 1 } ) SWF : i =1 Third year PhD Victor MAGRON Maxplus SOS

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization Global Optimization Problems: a Framework Given K a compact set, and f a transcendental function, minor f ∗ = inf x ∈ K f ( x ) and prove f ∗ ≥ 0 f is underestimated by a semialgebraic function f sa 1 We reduce the problem f ∗ sa := inf x ∈ K f sa ( x ) to a polynomial 2 optimization problem in a lifted space K pop (with lifting variables z ) We solve the POP problem f ∗ pop := inf f pop ( x , z ) using 3 ( x , z ) ∈ K pop a hierarchy of SDP relaxations by Lasserre If the relaxations are accurate enough, f ∗ ≥ f ∗ sa ≥ f ∗ pop ≥ 0 . Third year PhD Victor MAGRON Maxplus SOS

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization Global Optimization Problems: a Framework Given K a compact set, and f a transcendental function, minor f ∗ = inf x ∈ K f ( x ) and prove f ∗ ≥ 0 f is underestimated by a semialgebraic function f sa 1 We reduce the problem f ∗ sa := inf x ∈ K f sa ( x ) to a polynomial 2 optimization problem in a lifted space K pop (with lifting variables z ) We solve the POP problem f ∗ pop := inf f pop ( x , z ) using 3 ( x , z ) ∈ K pop a hierarchy of SDP relaxations by Lasserre If the relaxations are accurate enough, f ∗ ≥ f ∗ sa ≥ f ∗ pop ≥ 0 . Third year PhD Victor MAGRON Maxplus SOS

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization Global Optimization Problems: a Framework Given K a compact set, and f a transcendental function, minor f ∗ = inf x ∈ K f ( x ) and prove f ∗ ≥ 0 f is underestimated by a semialgebraic function f sa 1 We reduce the problem f ∗ sa := inf x ∈ K f sa ( x ) to a polynomial 2 optimization problem in a lifted space K pop (with lifting variables z ) We solve the POP problem f ∗ pop := inf f pop ( x , z ) using 3 ( x , z ) ∈ K pop a hierarchy of SDP relaxations by Lasserre If the relaxations are accurate enough, f ∗ ≥ f ∗ sa ≥ f ∗ pop ≥ 0 . Third year PhD Victor MAGRON Maxplus SOS

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization Global Optimization Problems: a Framework Given K a compact set, and f a transcendental function, minor f ∗ = inf x ∈ K f ( x ) and prove f ∗ ≥ 0 f is underestimated by a semialgebraic function f sa 1 We reduce the problem f ∗ sa := inf x ∈ K f sa ( x ) to a polynomial 2 optimization problem in a lifted space K pop (with lifting variables z ) We solve the POP problem f ∗ pop := inf f pop ( x , z ) using 3 ( x , z ) ∈ K pop a hierarchy of SDP relaxations by Lasserre If the relaxations are accurate enough, f ∗ ≥ f ∗ sa ≥ f ∗ pop ≥ 0 . Third year PhD Victor MAGRON Maxplus SOS

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization Global Optimization Problems: a Framework Given K a compact set, and f a transcendental function, minor f ∗ = inf x ∈ K f ( x ) and prove f ∗ ≥ 0 f is underestimated by a semialgebraic function f sa 1 We reduce the problem f ∗ sa := inf x ∈ K f sa ( x ) to a polynomial 2 optimization problem in a lifted space K pop (with lifting variables z ) We solve the POP problem f ∗ pop := inf f pop ( x , z ) using 3 ( x , z ) ∈ K pop a hierarchy of SDP relaxations by Lasserre If the relaxations are accurate enough, f ∗ ≥ f ∗ sa ≥ f ∗ pop ≥ 0 . Third year PhD Victor MAGRON Maxplus SOS

Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Estimators Certified Global Optimization Contents Flyspeck-Like Global Optimization 1 Classical Approach: Taylor + SOS 2 Max-Plus Estimators 3 Certified Global Optimization 4 Third year PhD Victor MAGRON Maxplus SOS