Certification of inequalities involving transcendental functions - PowerPoint PPT Presentation

Certification of inequalities involving transcendental functions using Semi-Definite Programming Supervisor: Benjamin Werner (TypiCal) Co-Supervisor: St´ ephane Gaubert (Maxplus) 2 nd year PhD Victor MAGRON LIX, ´ Ecole Polytechnique Wednesday July 11 st 2012 2 nd 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 Basic Semi-Algebraic Relaxations Transcendental Functions Underestimators Multi-Relaxations Algorithm Local Solutions to Global Issues 3 Multivariate Taylor-Models Underestimators Branch and Bound Algorithm Decrease the SDP Problems Size 2 nd year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Flyspeck-Like Problems 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, arccos, 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 2 − arctan − ∂ 4 ∆ x √ 4 x 1 ∆ x + 1 . 6294 − 0 . 2213 ( √ x 2 + √ x 3 + √ x 5 + √ x 6 − 8 . 0) + 0 . 913 ( √ x 4 − 2 . 52) + 0 . 728 ( √ x 1 − 2 . 0) ≥ 0 . 2 nd year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Flyspeck-Like Problems Hales and Solovyev Method: Real numbers are represented by interval arithmetic Arithmetic is floating point with IEEE-754 directed rounding Analytic functions f are approximated with Taylor expansions with rigorously computed error terms: | f ( x ) − f ( x 0 ) − ▽ f ( x 0 ) ( x − x 0 ) | < � m ij ǫ i ǫ j , i,j ǫ i = | x i − x 0 i | The domain K is partitioned into smaller rectangles as needed until the Taylor approximations are accurate enough to yield the desired inequalities. The Taylor expansions are generated by symbolic differentiation using the chain rule, product rule, and so forth. A few primitive functions ( √· , 1 · , arctan and some common polynomials) are hand-coded. 2 nd 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, minor 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 We reduce the problem to compute x ∈ K sa f sa ( x ) to a inf 2 polynomial optimization problem in a lifted space K pop x ∈ K pop f pop ( x ) using a inf We classicaly solve the POP problem 3 hierarchy of SDP relaxations by Lasserre If the relaxations are accurate enough, f ∗ ≥ f ∗ sa ≥ f ∗ pop ≥ 0 . 2 nd 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: 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 ∈ R [ X ] The SOS assumption is always verified if there exists N ∈ N such n m � � X 2 that N − i = u 0 + u j g j . In our case, it as always i =1 j =1 verified since all the polynomial variables X i are bounded. 2 nd 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]: ∃ L : R [ X ] → R s.t. ( ∃ µ ∈ P ( K pop ) , ∀ p ∈ R [ X ] , L ( p ) = m � � p dµ ) ⇐ ⇒ ( L (1) = 1 and L ( s 0 + s j g j ) ≥ 0 for any SOS j =1 s 0 ,..., s m ∈ R [ X ]) . Equivalent formulation: f ∗ pop = min { L ( f ) : L : R [ X ] → R linear, L (1) = 1 and each L g j is psd } , 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 ) 2 nd year PhD Victor MAGRON Certification of transcendental inequalities using SDP

SOS and SDP Relaxations Let B the monomial basis ( X α ) α ∈ N n and set y b = L ( b ) for b ∈ B identifies L with the infinite series y = ( y b ) b ∈B . 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 ⌉} . By truncating the previous matrices by considering only rows and columns indexed by elements in B of degree at most k , consider the hierarchy Q k of semidefinite relaxations: inf y L ( f ) Q k : M k −⌈ deg g j / 2 ⌉ ( g j y ) 0 , 0 ≤ j ≤ m, � y 1 = 1 2 nd year PhD Victor MAGRON Certification of transcendental inequalities using SDP

SOS and SDP Relaxations Convergence Theorem [Lasserre]: Let the SOS assumption holds. Then the sequence inf( Q k ) k ≥ k 0 is monotically non-decreasing and converges to f ∗ pop SDP relaxations: Let B = |B| . Many solvers (Sedumi, SDPA) solve the following standard form semidefinite program and its dual:  B �  P : min c α y α    y   α =1   B    �  F α y α − F 0 � 0 subject to ( SDP ) α =1     D : max Trace ( F 0 Y )   Y     Trace ( F α Y ) = c α ( α = 1 , ..., B )  subject to  2 nd year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Basic Semi-Algebraic Relaxations Let A be a set of semi-algebraic functions and f sa ∈ A . We consider the problem f ∗ sa = x ∈ K sa f sa ( x ) with inf K sa := { x ∈ R n : g 1 ( x ) ≥ 0 , ..., g m ( x ) ≥ 0 } a basic semi-algebraic set Basic Semi-Algebraic Lifting: A function f sa ∈ A is said to have a basic semi-algebraic lifting (a b.s.a.l.), or f is basic semi-algebraic (b.s.a.) if ∃ p, s ∈ N , polynomi- als ( h k ) 1 ≤ k ≤ s ∈ R [ X, Z 1 , ..., Z p ] and a b.s.a. set K pop := { ( x, z ) ∈ R n + p : x ∈ K sa , h k ( x, z ) ≥ 0 , k = 1 , ..., s } such that the graph of f sa (denoted Ψ f sa ) satisfies: Ψ f sa := { ( x, f sa ( x )) : x ∈ K sa } = { ( x, z p ) : ( x, z ) ∈ K pop } 2 nd year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Basic Semi-Algebraic Relaxations b.s.a.l. lemma [Lasserre, Putinar] : Let A be the semi-algebraic functions algebra obtained by composi- 1 p ( p ∈ N 0 ) , + , − , × , /, sup , inf . Then tion of polynomials with | · | , ( · ) every well-defined f sa ∈ A has a basic semi-algebraic lifting. Example from Flyspeck: f sa := − ∂ 4 ∆ x √ 4 x 1 ∆ x , K sa := [4; 6 . 3504] 3 × [6 . 3504; 8] × [4; 6 . 3504] 2 . � Define z 1 := 4 x 1 ∆ x , m 1 = x ∈ K sa z 1 ( x ) , M 1 = sup inf z 1 ( x ) . x ∈ K sa Define h 1 := z 1 − m 1 , h 2 := M 1 − z 1 , h 3 := z 2 � 1 − 4 x 1 ∆ x , h 4 := − z 2 � 1 + 4 x 1 ∆ x , h 5 := z 1 , h 6 := z 2 z 1 + ∂ 4 ∆ x , h 7 := − z 2 z 1 − ∂ 4 ∆ x , s = 7 , p = 2 . K pop := { ( x, z ) ∈ R 6+2 : x ∈ K sa , h k ( x, z ) ≥ 0 , k = 1 , ..., 7 } . Ψ f sa := { ( x, f sa ( x )) : x ∈ K sa } = { ( x, z 2 ) : ( x, z ) ∈ K pop } . 2 nd year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Basic Semi-Algebraic Relaxations Example from Flyspeck: f sa := − ∂ 4 ∆ x √ 4 x 1 ∆ x , K sa := [4; 6 . 3504] 3 × [6 . 3504; 8] × [4; 6 . 3504] 2 . Define g 1 := x 1 − 4 , g 2 := 6 . 3504 − x 1 , ..., g 11 := x 6 − 4 , g 12 := 6 . 3504 − x 6 . Solve: inf y L ( f pop ) = inf y y 0 ... 01 Q k : M k −⌈ deg g j / 2 ⌉ ( g j y ) 0 , 1 ≤ j ≤ 12 , � M k −⌈ deg h k / 2 ⌉ ( h k y ) 0 , 1 ≤ k ≤ 7 , � y 0 ... 0 = 1 2 nd year PhD Victor MAGRON Certification of transcendental inequalities using SDP

Basic Semi-Algebraic Relaxations Example from Flyspeck: f sa := − ∂ 4 ∆ x √ 4 x 1 ∆ x , K sa := [4; 6 . 3504] 3 × [6 . 3504; 8] × [4; 6 . 3504] 2 . Define g 1 := x 1 − 4 , g 2 := 6 . 3504 − x 1 , ..., g 11 := x 6 − 4 , g 12 := 6 . 3504 − x 6 . Solve: inf y y 0 ... 01 Q k : M k − 1 ( g j y ) 0 , 1 ≤ j ≤ 12 , � M k −⌈ deg h k / 2 ⌉ ( h k y ) 0 , 1 ≤ k ≤ 7 , � y 0 ... 0 = 1 b.s.a.l. Convergence: Let k ≥ k 0 := max { f pop , 1 , ⌈ deg h 1 / 2 ⌉ , ..., ⌈ deg h 7 / 2 ⌉} . The sequence inf( Q k ) k ≥ k 0 is monotically non-decreasing and converges to f ∗ sa . 2 nd year PhD Victor MAGRON Certification of transcendental inequalities using SDP