New applications of moment-sos hierarchies

Transcription

1 New applications of moment-sos hierarchies Victor Magron, Postdoc LAAS-CNRS 13 August 2014 専攻談話会 ( セミナー ) Tokyo Institute of Technology Dept. of Math. & Comput. Sci. y b sin( par + b) b 1 1 b1 b2 par + b3 = par 500 b2 b3 par + b3 b par b2 par b 1 V. Magron New applications of moment-sos hierarchies 1 / 42

2 Personal Background : Master s thesis at Tokyo University Hierarchical Domain Decomposition methods Collaboration with S. Yoshimura Sensei and Sugimoto : PhD at Ecole Polytechnique 2014 now: Postdoc at LAAS-CNRS V. Magron New applications of moment-sos hierarchies 2 / 42

3 Errors and Proofs Mathematicians want to eliminate all the uncertainties on their results. Why? M. Lecat, Erreurs des Mathématiciens des origines à nos jours, pages of errors! (Euler, Fermat, Sylvester,... ) V. Magron New applications of moment-sos hierarchies 3 / 42

4 Errors and Proofs Possible workaround: proof assistants COQ (Coquand, Huet 1984) HOL-LIGHT (Harrison, Gordon 1980) Built in top of OCAML V. Magron New applications of moment-sos hierarchies 3 / 42

5 Computer Science and Mathematics PhD on Formal Proofs for Global Optimization: Templates and Sums of Squares Collaboration with: Benjamin Werner (LIX Polytechnique) Stéphane Gaubert (Maxplus Team CMAP/INRIA Polytechnique) Xavier Allamigeon (Maxplus Team) V. Magron New applications of moment-sos hierarchies 4 / 42

6 Complex Proofs Complex mathematical proofs / mandatory computation K. Appel and W. Haken, Every Planar Map is Four-Colorable, T. Hales, A Proof of the Kepler Conjecture, V. Magron New applications of moment-sos hierarchies 5 / 42

7 From Oranges Stack... Kepler Conjecture (1611): The maximal density of sphere packings in 3D-space is π 18 Face-centered cubic Packing Hexagonal Compact Packing V. Magron New applications of moment-sos hierarchies 6 / 42

8 ...to Flyspeck Nonlinear Inequalities The proof of T. Hales (1998) contains mathematical and computational parts Computation: check thousands of nonlinear inequalities Robert MacPherson, editor of The Annals of Mathematics: [...] the mathematical community will have to get used to this state of affairs. Flyspeck [Hales 06]: Formal Proof of Kepler Conjecture V. Magron New applications of moment-sos hierarchies 7 / 42

9 ...to Flyspeck Nonlinear Inequalities The proof of T. Hales (1998) contains mathematical and computational parts Computation: check thousands of nonlinear inequalities Robert MacPherson, editor of The Annals of Mathematics: [...] the mathematical community will have to get used to this state of affairs. Flyspeck [Hales 06]: Formal Proof of Kepler Conjecture Project Completion last Sunday by the Flyspeck team!! V. Magron New applications of moment-sos hierarchies 7 / 42

10 A Simple Example In the computational part: Multivariate Polynomials: 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 ) V. Magron New applications of moment-sos hierarchies 8 / 42

11 A Simple Example In the computational part: Semialgebraic functions: composition of polynomials with,, +,,, /, sup, inf,... p(x) := 4 x q(x) := 4x 1 x r(x) := p(x)/ q(x) l(x) := π ( x 2 + x 3 + x 5 + x 6 8.0) ( x ) ( x 1 2.0) V. Magron New applications of moment-sos hierarchies 8 / 42

12 A Simple Example In the computational part: Transcendental functions T : composition of semialgebraic functions with arctan, exp, sin, +,,,... V. Magron New applications of moment-sos hierarchies 8 / 42

13 A Simple Example In the computational part: Feasible set K := [4, ] 3 [6.3504, 8] [4, ] 2 Lemma from Flyspeck: ( p(x) ) x K, arctan + l(x) 0 q(x) V. Magron New applications of moment-sos hierarchies 8 / 42

14 Existing Formal Frameworks Formal proofs for Global Optimization: Bernstein polynomial methods [Zumkeller s PhD 08] SMT methods [Gao et al. 12] Interval analysis and Sums of squares V. Magron New applications of moment-sos hierarchies 9 / 42

15 Existing Formal Frameworks Interval analysis Certified interval arithmetic in COQ [Melquiond 12] Taylor methods in HOL Light [Solovyev thesis 13] Formal verification of floating-point operations robust but subject to the Curse of Dimensionality V. Magron New applications of moment-sos hierarchies 9 / 42

16 Existing Formal Frameworks Lemma from Flyspeck: ( 4 x ) x K, arctan + l(x) 0 4x1 x Dependency issue using Interval Calculus: One can bound 4 x/ 4x 1 x and l(x) separately Too coarse lower bound: 0.87 Subdivide K to prove the inequality K 3 K = K 1 K 0 K 4 K 2 V. Magron New applications of moment-sos hierarchies 9 / 42

17 Existing Formal Frameworks Sums of squares techniques Formalized in HOL-LIGHT [Harrison 07] COQ [Besson 07] Precise methods but scalibility and robustness issues (numerical) but powerful: global optimality certificates without branching not so robust: handles moderate size problems Restricted to polynomials V. Magron New applications of moment-sos hierarchies 9 / 42

18 Existing Formal Frameworks Approximation theory: Chebyshev/Taylor models mandatory for non-polynomial problems hard to combine with SOS techniques (degree of approximation) V. Magron New applications of moment-sos hierarchies 9 / 42

19 Existing Formal Frameworks Can we develop a new approach with both keeping the respective strength of interval and precision of SOS? Proving Flyspeck Inequalities is challenging: medium-size and tight V. Magron New applications of moment-sos hierarchies 9 / 42

20 New Framework (in my PhD thesis) Certificates for lower bounds of Global Optimization Problems using SOS and new ingredients in Global Optimization: Maxplus approximation (Optimal Control) Nonlinear templates (Static Analysis) Verification of these certificates inside COQ V. Magron New applications of moment-sos hierarchies 10 / 42

21 New Framework (in my PhD thesis) Software Implementation NLCertify: lines of OCAML code 4000 lines of COQ code V. Magron New applications of moment-sos hierarchies 10 / 42

22 Introduction Moment-SOS relaxations and Maxplus approximation Formal Nonlinear Optimization Approximating Pareto curves Polynomial image of semialgebraic sets Program Analysis with Polynomial Templates Conclusion

23 Moment-SOS relaxations Semialgebraic set K := {x R n : g 1 (x) 0,..., g m (x) 0} p := min p(x): NP hard x K Sums of squares Σ[x] Q(K) := { } σ 0 (x) + m j=1 σ j(x)g j (x), with σ j Σ[x] V. Magron New applications of moment-sos hierarchies 11 / 42

24 Moment-SOS relaxations M + (K): space of probability measures supported on K Polynomial Optimization Problems (POP) (Primal) (Dual) inf p dµ = sup λ K s.t. µ M + (K) s.t. λ R, p λ Q(K) V. Magron New applications of moment-sos hierarchies 11 / 42

25 Moment-SOS relaxations Truncated quadratic module Q k (K) := Q(K) R 2k [x] Polynomial Optimization Problems (POP) (Moment) (SOS) inf p dµ sup λ K s.t. µ M + (K) s.t. λ R, p λ Q k (K) V. Magron New applications of moment-sos hierarchies 11 / 42

26 Practical Computation Hierarchy { of SOS relaxations: } λ k := sup λ : p λ Q k (K) λ Convergence guarantees λ k p [Lasserre 01] Can be computed with SOS solvers (CSDP, SDPA) Extension to semialgebraic functions r(x) = p(x)/ q(x) [Lasserre-Putinar 10] V. Magron New applications of moment-sos hierarchies 12 / 42

27 Practical Computation Caprasse Problem x [ 0.5, 0.5] 4, x 1 x x 2x 2 3 x 4 + 4x 1 x 3 x x 2x x 1 x 3 + 4x x 2x 4 10x scale_pol = true: scaled on [0, 1] 4 relax_order = 2: SOS of degree at most 4 bound_squares_variables = true: rsedundant constraints x 2 1 1,..., x2 4 1 V. Magron New applications of moment-sos hierarchies 12 / 42

28 The General Informal Framework Given K a compact set and f a transcendental function, bound f = inf x K f (x) and prove f 0 f is underestimated by a semialgebraic function f sa Reduce the problem f sa := inf x K f sa (x) to a polynomial optimization problem (POP) V. Magron New applications of moment-sos hierarchies 13 / 42

29 Maxplus Approximation Initially introduced to solve Optimal Control Problems [Fleming-McEneaney 00] Curse of dimensionality reduction [McEaneney Kluberg, Gaubert-McEneaney-Qu 11, Qu 13]. Allowed to solve instances of dim up to 15 (inaccessible by grid methods) In our context: approximate transcendental functions V. Magron New applications of moment-sos hierarchies 14 / 42

30 Maxplus Approximation Definition Let γ 0. A function φ : R n R is said to be γ-semiconvex if the function x φ(x) + γ 2 x 2 2 is convex. y par + a 1 par + a 2 arctan par a 2 m a 1 a 2 M a par a 1 V. Magron New applications of moment-sos hierarchies 14 / 42

31 Nonlinear Function Representation Exact parsimonious maxplus representations y a V. Magron New applications of moment-sos hierarchies 15 / 42

32 Nonlinear Function Representation Exact parsimonious maxplus representations y a V. Magron New applications of moment-sos hierarchies 15 / 42

33 Nonlinear Function Representation Abstract syntax tree representations of multivariate transcendental functions: leaves are semialgebraic functions of A nodes are univariate functions of D or binary operations V. Magron New applications of moment-sos hierarchies 15 / 42

34 Nonlinear Function Representation For the Simple Example from Flyspeck: + l(x) arctan r(x) V. Magron New applications of moment-sos hierarchies 15 / 42

35 Maxplus Optimization Algorithm First iteration: + y arctan l(x) arctan r(x) m a 1 par a 1 M a 1 1 control point {a 1 } SOS Computation: m 1 = V. Magron New applications of moment-sos hierarchies 16 / 42

36 Maxplus Optimization Algorithm Second iteration: + y arctan l(x) arctan r(x) m a 2 par a 2 a 1 par a 1 M a 2 2 control points {a 1, a 2 }: m 2 = V. Magron New applications of moment-sos hierarchies 16 / 42

37 Maxplus Optimization Algorithm Third iteration: y + arctan l(x) arctan r(x) m a 2 a 3 par a 2 par par a 3 a 1 a a 1 M 3 3 control points {a 1, a 2, a 3 }: m 3 = 0.04 V. Magron New applications of moment-sos hierarchies 16 / 42

38 Maxplus Optimization Algorithm Input: tree t, box K, SOS relaxation order k, precision p Output: bounds m and M, approximations t 2 and t+ 2 1: if t A then t := t, t + := t 2: else if u := root(t) D with child c then 3: m c, M c, c, c + := samp_approx(c, K, k, p) 4: I := [m c, M c ] 5: u, u + := unary_approx(u, I, c, p) 6: t, t + := compose_approx(u, u, u +, I, c, c + ) 7: else if bop := root(t) with children c 1 and c 2 then 8: m i, M i, c i, c + i := samp_approx(c i, K, k, p) for i {1, 2} 9: t, t + := compose_bop(c 1, c+ 1, c 2, c+ 2, bop, [m 2, M 2 ]) 10: end 11: return min_sa(t, K, k), max_sa(t +, K, k), t, t + V. Magron New applications of moment-sos hierarchies 17 / 42

39 Contributions Published: X. Allamigeon, S. Gaubert, V. Magron, and B. Werner. Certification of inequalities involving transcendental functions: combining sdp and max-plus approximation, ECC Conference X. Allamigeon, S. Gaubert, V. Magron, and B. Werner. Certification of bounds of non-linear functions: the templates method, CICM Conference, In revision: X. Allamigeon, S. Gaubert, V. Magron, and B. Werner. Certification of Real Inequalities Templates and Sums of Squares, arxiv: , V. Magron New applications of moment-sos hierarchies 18 / 42

40 Introduction Moment-SOS relaxations and Maxplus approximation Formal Nonlinear Optimization Approximating Pareto curves Polynomial image of semialgebraic sets Program Analysis with Polynomial Templates Conclusion

41 The General Formal Framework We check the correctness of SOS certificates for POP We build certificates to prove interval bounds for semialgebraic functions We bound formally transcendental functions with semialgebraic approximations V. Magron New applications of moment-sos hierarchies 19 / 42

42 Formal SOS bounds When q Q(K), σ 0,..., σ m is a positivity certificate for q Check symbolic polynomial equalities q = q in COQ Existing tactic ring [Grégoire-Mahboubi 05] Polynomials coefficients: arbitrary-size rationals bigq [Grégoire-Théry 06] Much simpler to verify certificates using sceptical approach Extends also to semialgebraic functions V. Magron New applications of moment-sos hierarchies 20 / 42

43 Benchmarks for Flyspeck Inequalities Inequality #boxes Time Time s 2218 s s s Comparable with Taylor interval methods in HOL-LIGHT [Hales-Solovyev 13] Bottleneck of informal optimizer is SOS solver 22 times slower! = Current bottleneck is to check polynomial equalities V. Magron New applications of moment-sos hierarchies 21 / 42

44 Contribution For more details on the formal side: X. Allamigeon, S. Gaubert, V. Magron and B. Werner. Formal Proofs for Nonlinear Optimization. Submitted for publication, arxiv: V. Magron New applications of moment-sos hierarchies 22 / 42

45 Introduction Moment-SOS relaxations and Maxplus approximation Formal Nonlinear Optimization Approximating Pareto curves Polynomial image of semialgebraic sets Program Analysis with Polynomial Templates Conclusion

46 Bicriteria Optimization Problems Let f 1, f 2 R d [x] two conflicting criteria Let S := {x R n : g 1 (x) 0,..., g m (x) 0} a semialgebraic set { } (P) min (f 1(x) f 2 (x)) x S Assumption The image space R 2 is partially ordered in a natural way (R 2 + is the ordering cone). V. Magron New applications of moment-sos hierarchies 23 / 42

47 Bicriteria Optimization Problems g 1 := (x 1 2) 3 /2 x , g 2 := x 1 x 2 + 8( x 1 + x ) , S := {x R 2 : g 1 (x) 0, g 2 (x) 0}. f 1 := (x 1 + x 2 7.5) 2 /4 + ( x 1 + x 2 + 3) 2, f 2 := (x 1 1) 2 /4 + (x 2 4) 2 /4. V. Magron New applications of moment-sos hierarchies 23 / 42

48 Parametric sublevel set approximation Inspired by previous research on multiobjective linear optimization [Gorissen-den Hertog 12] Workaround: reduce P to a parametric POP (P λ ) : f (λ) := min x S { f 2(x) : f 1 (x) λ }, V. Magron New applications of moment-sos hierarchies 24 / 42

49 A Hierarchy of Polynomial underestimators Moment-SOS approach [Lasserre 10]: (D d ) max q R 2d [λ] 2d q k /(1 + k) k=0 s.t. f 2 (x) q(λ) Q 2d (K). The hierarchy (D d ) provides a sequence (q d ) of polynomial underestimators of f (λ). lim d 1 0 (f (λ) q d (λ))dλ = 0 V. Magron New applications of moment-sos hierarchies 25 / 42

50 A Hierarchy of Polynomial underestimators Degree 4 V. Magron New applications of moment-sos hierarchies 26 / 42

51 A Hierarchy of Polynomial underestimators Degree 6 V. Magron New applications of moment-sos hierarchies 26 / 42

52 A Hierarchy of Polynomial underestimators Degree 8 V. Magron New applications of moment-sos hierarchies 26 / 42

53 Contributions Numerical schemes that avoid computing finitely many points. Pareto curve approximation with polynomials, convergence guarantees in L 1 -norm V. Magron, D. Henrion, J.B. Lasserre. Approximating Pareto Curves using Semidefinite Relaxations. Operations Research Letters. arxiv: , April V. Magron New applications of moment-sos hierarchies 27 / 42

54 Introduction Moment-SOS relaxations and Maxplus approximation Formal Nonlinear Optimization Approximating Pareto curves Polynomial image of semialgebraic sets Program Analysis with Polynomial Templates Conclusion

55 Approximation of sets defined with Let B R m be the unit ball and assume that f (S) B. Another point of view: f (S) = {y B : x S s.t. h(x, y) 0}, with h(x, y) := y f (x) 2 2. Approximate f (S) as closely as desired by a sequence of sets of the form : F d f := {y B : q d (y) 0}, for some polynomials q d R 2d [y]. V. Magron New applications of moment-sos hierarchies 28 / 42

56 A hierarchy of outer approximations of f (S) Let K = S B, g 0 := 1 and Q d (K) be the d-truncated quadratic module generated by g 0,..., g m : Q d (K) = { m } σ l (x, y)g l (x), with σ l Σ d vl [x, y] l=0 Define H(y) := min x S h(x, y) Hierarchy of Semidefinite programs: ρ d := } min (H q)dy : h q Q d (K) q R 2d [y],σ l { B Yet another SOS program with an optimal solution q d R 2d [y]!. V. Magron New applications of moment-sos hierarchies 29 / 42

57 A hierarchy of outer approximations of f (S) From the definition of q d, the sublevel sets F d f := {y B : q d (y) 0} f (S), provide a sequence of certified outer approximations of f (S). It comes from the following: (x, y) K, q d (y) h(x, y) y, q d (y) H(y). V. Magron New applications of moment-sos hierarchies 30 / 42

58 Strong convergence property Theorem 1 The sequence of underestimators (q d ) d d0 converges to H w.r.t the L 1 (B)-norm: lim H q d dy = 0. d B V. Magron New applications of moment-sos hierarchies 31 / 42

59 Strong convergence property Theorem 1 The sequence of underestimators (q d ) d d0 converges to H w.r.t the L 1 (B)-norm: lim H q d dy = 0. d B 2 lim d vol(fd f \f (S)) = 0. V. Magron New applications of moment-sos hierarchies 31 / 42

60 Approximation for polynomial image of semialgebraic sets Image of the unit ball S := {x R 2 : x 2 2 1} by f (x) := (x 1 + x 1 x 2, x 2 x 3 1 )/2 Degree 4 V. Magron New applications of moment-sos hierarchies 32 / 42

61 Approximation for polynomial image of semialgebraic sets Image of the unit ball S := {x R 2 : x 2 2 1} by f (x) := (x 1 + x 1 x 2, x 2 x 3 1 )/2 Degree 4 V. Magron New applications of moment-sos hierarchies 32 / 42

62 Approximation for polynomial image of semialgebraic sets Image of the unit ball S := {x R 2 : x 2 2 1} by f (x) := (x 1 + x 1 x 2, x 2 x 3 1 )/2 Degree 6 V. Magron New applications of moment-sos hierarchies 32 / 42

63 Approximation for polynomial image of semialgebraic sets Image of the unit ball S := {x R 2 : x 2 2 1} by f (x) := (x 1 + x 1 x 2, x 2 x 3 1 )/2 Degree 8 V. Magron New applications of moment-sos hierarchies 32 / 42

64 Semialgebraic set projections f (S) = (x 1, x 2 ): projection on R 2 of the semialgebraic set S := {x R 3 : x 2 2 1, 1/4 (x 1 + 1/2) 2 x 2 2 0, 1/9 (x 1 1/2) 4 x 4 2 0} Degree 4 V. Magron New applications of moment-sos hierarchies 33 / 42

65 Semialgebraic set projections f (S) = (x 1, x 2 ): projection on R 2 of the semialgebraic set S := {x R 3 : x 2 2 1, 1/4 (x 1 + 1/2) 2 x 2 2 0, 1/9 (x 1 1/2) 4 x 4 2 0} Degree 6 V. Magron New applications of moment-sos hierarchies 33 / 42

66 Semialgebraic set projections f (S) = (x 1, x 2 ): projection on R 2 of the semialgebraic set S := {x R 3 : x 2 2 1, 1/4 (x 1 + 1/2) 2 x 2 2 0, 1/9 (x 1 1/2) 4 x 4 2 0} Degree 8 V. Magron New applications of moment-sos hierarchies 33 / 42

67 Support function of a closed convex set Let S be the unit ball and q R[x] being convex on S. f := q q is the support function of the convex set f (S) q(x) := x x x 2 1 x /2(x x 2 2) (x 1 x 2 + x 1 + x 2 ) V. Magron New applications of moment-sos hierarchies 34 / 42

68 Support function of a closed convex set q(x) := x x x 2 1 x /2(x x 2 2) (x 1 x 2 + x 1 + x 2 ) Degree 4 V. Magron New applications of moment-sos hierarchies 34 / 42

69 Support function of a closed convex set q(x) := x x x 2 1 x /2(x x 2 2) (x 1 x 2 + x 1 + x 2 ) Degree 8 V. Magron New applications of moment-sos hierarchies 34 / 42

70 Approximating Pareto curves Back our previous nonconvex example: Degree 2 V. Magron New applications of moment-sos hierarchies 35 / 42

71 Approximating Pareto curves Back our previous nonconvex example: Degree 4 V. Magron New applications of moment-sos hierarchies 35 / 42

72 Approximating Pareto curves Back our previous nonconvex example: Degree 8 V. Magron New applications of moment-sos hierarchies 35 / 42

73 Approximating Pareto curves Zoom on the region which is hard to approximate: Degree 8 V. Magron New applications of moment-sos hierarchies 36 / 42

74 Approximating Pareto curves Zoom on the region which is hard to approximate: Degree 10 V. Magron New applications of moment-sos hierarchies 36 / 42

75 Semialgebraic image of semialgebraic sets Image of the unit ball S := {x R 2 : x 2 2 1} by f (x) := (min(x 1 + x 1 x 2, x 2 1), x 2 x 3 1 )/3 Degree 2 V. Magron New applications of moment-sos hierarchies 37 / 42

76 Semialgebraic image of semialgebraic sets Image of the unit ball S := {x R 2 : x 2 2 1} by f (x) := (min(x 1 + x 1 x 2, x 2 1), x 2 x 3 1 )/3 Degree 4 V. Magron New applications of moment-sos hierarchies 37 / 42

77 Semialgebraic image of semialgebraic sets Image of the unit ball S := {x R 2 : x 2 2 1} by f (x) := (min(x 1 + x 1 x 2, x 2 1), x 2 x 3 1 )/3 Degree 6 V. Magron New applications of moment-sos hierarchies 37 / 42

78 Semialgebraic image of semialgebraic sets Image of the unit ball S := {x R 2 : x 2 2 1} by f (x) := (min(x 1 + x 1 x 2, x 2 1), x 2 x 3 1 )/3 Degree 8 V. Magron New applications of moment-sos hierarchies 37 / 42

79 Introduction Moment-SOS relaxations and Maxplus approximation Formal Nonlinear Optimization Approximating Pareto curves Polynomial image of semialgebraic sets Program Analysis with Polynomial Templates Conclusion

80 One-loop with Conditional Branching r, s, T i, T e R[x] x 0 X 0, with X 0 semialgebraic set x = x 0 ; while (r(x) 0){ if (s(x) 0){ x = T i (x); } else { x = T e (x); } } V. Magron New applications of moment-sos hierarchies 38 / 42

81 Bounding Template using SOS Sufficient condition to get bounding inductive invariant: α := min q R[x] sup q(x) x X 0 s.t. q q T i 0, q q T e 0, q Nontrivial correlations via polynomial templates q(x) {x : q(x) α} X k k N V. Magron New applications of moment-sos hierarchies 39 / 42

82 Bounds for k N X k X 0 := [0.9, 1.1] [0, 0.2] r(x) := 1 s(x) := 1 x 2 1 x 2 2 T i (x) := (x x 3 2, x3 1 + x2 2) T e (x) := ( 1 2 x x3 2, 3 5 x x2 2) Degree 6 V. Magron New applications of moment-sos hierarchies 40 / 42

83 Bounds for k N X k X 0 := [0.9, 1.1] [0, 0.2] r(x) := 1 s(x) := 1 x 2 1 x 2 2 T i (x) := (x x 3 2, x3 1 + x2 2) T e (x) := ( 1 2 x x3 2, 3 5 x x2 2) Degree 8 V. Magron New applications of moment-sos hierarchies 40 / 42

84 Bounds for k N X k X 0 := [0.9, 1.1] [0, 0.2] r(x) := 1 s(x) := 1 x 2 1 x 2 2 T i (x) := (x x 3 2, x3 1 + x2 2) T e (x) := ( 1 2 x x3 2, 3 5 x x2 2) Degree 10 V. Magron New applications of moment-sos hierarchies 40 / 42

85 Introduction Moment-SOS relaxations and Maxplus approximation Formal Nonlinear Optimization Approximating Pareto curves Polynomial image of semialgebraic sets Program Analysis with Polynomial Templates Conclusion

86 Conclusion Formal nonlinear optimization: NLCertify Safe solutions for challenging problems, e.g. Flyspeck Approximation of Pareto Curves, images and projections of semialgebraic sets Program Analysis with polynomial templates V. Magron New applications of moment-sos hierarchies 41 / 42

87 Conclusion Further research: OCAML API Alternative Polynomials bounds using geometric programming (T. de Wolff, S. Iliman) COQ tactic Improve formal polynomial checker Programs analysis with transcendental assignments/conditions V. Magron New applications of moment-sos hierarchies 41 / 42

88 Conclusion Further research: Generalized problem of moments (Moment) (SOS) inf p 0 dµ sup λ 0 + λ i b i K i s.t. p i dµ b i s.t. λ 0, λ i 0, K µ M + (K) p 0 λ 0 i λ i p i Q k (K) Formal bounds using SDP and ring V. Magron New applications of moment-sos hierarchies 41 / 42

89 End Thank you for your attention!