Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with Certification of Bounds of Non-linear Functions : the Templates Method Joint Work with B. Werner, S. Gaubert and X. Allamigeon Third year PhD Victor MAGRON LIX/CMAP INRIA, ´ Ecole Polytechnique CICM 2013 Monday July 8 th Third year PhD Victor MAGRON Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with 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 tetrahedron 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 Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with Contents Flyspeck-Like Global Optimization 1 Classical Approach: Taylor + SOS 2 Max-Plus Based Templates 3 Certified Global Optimization with Coq 4 Third year PhD Victor MAGRON Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with 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 Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with 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 Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with 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 Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with Contents Flyspeck-Like Global Optimization 1 Classical Approach: Taylor + SOS 2 Max-Plus Based Templates 3 Certified Global Optimization with Coq 4 Third year PhD Victor MAGRON Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with Semialgebraic Optimization Problems Polynomial Optimization Problem (POP): p ∗ := min x ∈ K p ( x ) with K the compact set of constraints: K = { x ∈ R n : g 1 ( x ) ≥ 0 , · · · , g m ( x ) ≥ 0 } Let Σ d [ x ] be the cone of Sum-of-Squares (SOS) of degree at most 2 d : � � � q i ( x ) 2 , with q i ∈ R d [ x ] Σ d [ x ] = i Let g 0 := 1 and M d ( g ) be the quadratic module: � m � � M d ( g ) = σ j ( x ) g j ( x ) , with σ j ∈ Σ[ x ] , ( σ j g j ) ∈ R 2 d [ x ] j =0 Certificates for positive polynomials: Sum-of-Squares Third year PhD Victor MAGRON Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with Semialgebraic Optimization Problems � M ( g ) := M d ( g ) d ∈ N Proposition (Putinar) Suppose x ∈ [ a , b ] . p ( x ) − p ∗ > 0 on K = ⇒ ( p ( x ) − p ∗ ) ∈ M ( g ) M 0 ( g ) ⊂ M 1 ( g ) ⊂ M 2 ( g ) ⊂ · · · ⊂ M ( g ) Hence, we consider the hierarchy of SOS relaxation � � programs: µ k := sup µ : ( p ( x ) − µ ) ∈ M k ( g ) µ,σ 0 , ··· ,σ m µ k ↑ p ∗ (Lasserre Hierarchy Convergence) Third year PhD Victor MAGRON Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with Semialgebraic Optimization Problems Example from Flyspeck: Also works for Semialgebraic functions via lifting variables: ∆( 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 6 ) − x 5 ( x 1 x 3 + x 4 x 6 ) ∂ 4 ∆ x = x 1 ( − x 1 + x 2 + x 3 − x 4 + x 5 + x 6 )+ x 2 x 5 + x 3 x 6 − x 2 x 3 − x 5 x 6 ∂ 4 ∆ x f ∗ √ 4 x 1 ∆ x sa := min x ∈ [4 , 6 . 3504] 6 Third year PhD Victor MAGRON Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with Semialgebraic Optimization Problems Example from Flyspeck: � z 1 := 4 x 1 ∆ x , m 1 = x ∈ [4 , 6 . 3504] 6 z 1 ( x ) , M 1 = inf x ∈ [4 , 6 . 3504] 6 z 1 ( x ) . sup K := { ( x , z ) ∈ R 8 : x ∈ [4 , 6 . 3504] 6 , h 1 ( x , z ) ≥ 0 , · · · , h 7 ( x , z ) ≥ 0 } h 4 := − z 2 h 1 := z 1 − m 1 1 + 4 x 1 ∆ x h 2 := M 1 − z 1 h 5 := z 2 z 1 − ∂ 4 ∆ x h 3 := z 2 1 − 4 x 1 ∆ x h 6 := − z 2 z 1 + ∂ 4 ∆ x p ∗ := ( x , z ) ∈ K z 2 = f ∗ inf sa . We obtain µ 2 = − 0 . 618 and µ 3 = − 0 . 445 . Third year PhD Victor MAGRON Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with Taylor Approximation of Transcendental Functions n ( x i + x i +1 ) sin( √ x i ) � x ∈ [1 , 500] n f ( x ) = − min SWF : Classical idea: approximate sin( √· ) by a degree- d Taylor i =1 n � Polynomial f d , solve x ∈ [1 , 500] n − min ( x i + x i +1 ) f d ( x i ) (POP) i =1 Issues: Lack of accuracy if d is not large enough = ⇒ expensive Branch and Bound POP may involve many lifting variables : depends on semialgebraic and univariate transcendental components of f No free lunch: solving POP with Sum-of-Squares of degree 2 k involves O ( n 2 k ) variables SWF with n = 10 , d = 4 : takes already 38 min to certify a lower bound of − 430 n Third year PhD Victor MAGRON Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with Contents Flyspeck-Like Global Optimization 1 Classical Approach: Taylor + SOS 2 Max-Plus Based Templates 3 Certified Global Optimization with Coq 4 Third year PhD Victor MAGRON Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with Max-Plus Estimators Goals: Reduce the O ( n 2 k ) polynomial dependency: decrease the number of lifting variables Reduce the O ( n 2 k ) exponential dependency: use low degree approximations Reduce the Branch and Bound iterations: refine the approximations with an adaptive iterative algorithm Third year PhD Victor MAGRON Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with Max-Plus Estimators Let ˆ f ∈ T be a transcendental univariate function ( arctan , exp ) defined on an interval I . ˆ f is semi-convex: there exists a constant c j > 0 s.t. f ( a ) + c j / 2( a − a j ) 2 is convex a �→ ˆ By convexity: f ( a ) ≥ − c j / 2( a − a j ) 2 + ˆ ∀ a ∈ I, ˆ f ′ ( a j )( a − a j ) + ˆ f ( a j ) = par − a j ( a ) ∀ j, ˆ f ≥ par − ⇒ ˆ j { par − a j = f ≥ max a j } Max-Plus underestimator Example with arctan : 1 f ′ ( a j ) = ˆ {− ˆ f ′′ ( a ) } (always work) , c j = sup 1 + a 2 a ∈ I j c j depends on a j and the curvature variations of arctan on the considered interval I Third year PhD Victor MAGRON Templates SOS
Flyspeck-Like Global Optimization Classical Approach: Taylor + SOS Max-Plus Based Templates Certified Global Optimization with Max-Plus Estimators Example with arctan : y par + a 2 par + arctan a 1 par − a 2 a a 1 a 2 m M par − a 1 Third year PhD Victor MAGRON Templates SOS
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