Certification of inequalities involving transcendental functions using Semi-Definite Programming Joint Work with B. Werner, S. Gaubert and X. Allamigeon Second year PhD Victor MAGRON LIX/INRIA, ´ Ecole Polytechnique ISMP 2012 Friday August 24 th 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 ∆ 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 3 x 5 − x 1 x 2 x 6 − x 4 x 5 x 6 ∀ x ∈ K, − π √ 4 x 1 ∆ x + 1 . 6294 − 0 . 2213 ( √ x 2 + √ x 3 + ∂ 4 ∆ x 2 + arctan √ 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
Contents Flyspeck-Like Problems 1 General Framework 2 Sums of Squares (SOS) and Semi-Definite Programming (SDP) Relaxations Transcendental Functions Underestimators Adaptative Semi-algebraic Approximations Local Solutions to Global Issues 3 Compute λ min by Robust-SDP Branch and Bound Algorithm Preliminary Results Conclusions and Further Work 4 Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP
Flyspeck-Like Problems [Hales and Solovyev Method] Real numbers are represented by interval arithmetic Analytic functions f (e.g. √· , 1 · , arctan ) are approximated with Taylor expansions and the error terms are bounded: � | f ( x ) − f ( x 0 ) − D f ( x 0 ) ( x − x 0 ) | < m ij ǫ i ǫ j i,j ǫ i := | x i − x i 0 | To satisfy the inequalities, the initial box K is partitioned into smaller boxes until the Taylor approximations are accurate enough (the error terms become small enough) The Taylor expansions are generated by symbolic differentiation using the chain rule, product rule Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP
General Framework We consider the same problem: 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 Reduce the problem x ∈ K sa f sa ( x ) to a polynomial optimization inf 2 problem (POP) in a lifted space K pop x ∈ K pop f pop ( x ) using a inf Solve classicaly the POP problem 3 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 We consider the same problem: 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 Reduce the problem x ∈ K sa f sa ( x ) to a polynomial optimization inf 2 problem (POP) in a lifted space K pop x ∈ K pop f pop ( x ) using a inf Solve classicaly the POP problem 3 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 We consider the same problem: 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 Reduce the problem x ∈ K sa f sa ( x ) to a polynomial optimization inf 2 problem (POP) in a lifted space K pop x ∈ K pop f pop ( x ) using a inf Solve classicaly the POP problem 3 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 We consider the same problem: 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 Reduce the problem x ∈ K sa f sa ( x ) to a polynomial optimization inf 2 problem (POP) in a lifted space K pop x ∈ K pop f pop ( x ) using a inf Solve classicaly the POP problem 3 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
SOS and SDP Relaxations Polynomial Optimization Problem (POP): Let f, g 1 , · · · , g m ∈ R [ X 1 , · · · , X n ] K pop := { x ∈ R n : g 1 ( x ) ≥ 0 , · · · , g m ( x ) ≥ 0 } is the feasible set General POP: compute f ∗ pop = x ∈ K pop f ( x ) inf SOS Assumption: [e.g. Lasserre] K is compact, ∃ u ∈ R [ X ] s.t. the level set { x ∈ R n : u ( x ) ≥ 0 } m � is compact and u = u 0 + u j g j for some sum of squares (SOS) j =1 u 0 , u 1 , · · · , u m ∈ Σ[ X ] 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 The polynomial u ( x ) := n − j satisfies the assumption j =1 Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP
SOS and SDP Relaxations To convexify the problem, use the equivalent formulation: � f ∗ pop = x ∈ K pop f pop ( x ) = inf inf f pop dµ , where P ( K pop ) is the µ ∈P ( K pop ) set of all probability measures µ supported on the set K pop . Theorem [Putinar]: Given L : R [ X ] → R , the following are equivalent: � ∃ µ ∈ P ( K pop ) , ∀ p ∈ R [ X ] , L ( p ) = p dµ 1 m � L (1) = 1 , L ( s 0 + s j g j ) ≥ 0 for any s 0 , · · · , s m ∈ Σ[ X ] 2 j =1 Equivalent formulation: f ∗ pop = min { L ( f ) : L : R [ X ] → R linear, L (1) = 1 and each L g j is SDP } , with g 0 = 1 , L g 0 , · · · , L g m defined by: L g j : R [ X ] × R [ X ] → R ( p, q ) �→ L ( p · q · g j ) Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP
SOS and SDP Relaxations: Lasserre Hierarchy Let B := ( X α ) α ∈ N n denote the monomial basis and set y α = L ( X α ) , this identifies L with the infinite series y = ( y α ) α ∈ N n . The infinite moment matrix M associated to y indexed by B is: M ( y ) u,v := L ( u · v ) , u, v ∈ B . The localizing matrix M ( g j y ) is: M ( g j y ) u,v := L ( u · v · g j ) , u, v ∈ B . Let k ≥ k 0 := max {⌈ deg f pop ⌉ / 2 , ⌈ deg g 0 / 2 ⌉ , · · · , ⌈ deg g m / 2 ⌉} . Truncate the previous matrices by considering only rows and columns indexed by elements in B of degree at most k , and consider the hierarchy Q k of semidefinite relaxations: � � f α x α dµ ( x ) = inf y L ( f ) = f α y α α Q k : M k −⌈ deg g j / 2 ⌉ ( g j y ) 0 , 0 ≤ j ≤ m, � y 1 = 1 Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP
SOS and SDP Relaxations Convergence Theorem [Lasserre]: The sequence inf( Q k ) k ≥ k 0 is non-decreasing and under the SOS assumption converges to f ∗ pop . SDP relaxations: Many solvers (e.g. Sedumi [ ? ], SDPA) solve the pair of (standard form) semidefinite programs: � P : min c α y α y α � subject to F α y α − F 0 � 0 ( SDP ) α D : max Trace ( F 0 Y ) Y Trace ( F α Y ) = c α subject to Second year PhD Victor MAGRON Certification of transcendental inequalities using SDP
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