Semialgebraic Relaxations using Moment-SOS Hierarchies

Transcription

1 Semialgebraic Relaxations using Moment-SOS Hierarchies Victor Magron, Postdoc LAAS-CNRS 17 September 2014 SIERRA Seminar Laboratoire d Informatique de l Ecole Normale Superieure 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 Semialgebraic Relaxations using Moment-SOS hierarchies 1 / 47

2 Personal Background : Master at Tokyo University HIERARCHICAL DOMAIN DECOMPOSITION METHODS (S. Yoshimura) : PhD at Inria Saclay LIX/CMAP FORMAL PROOFS FOR NONLINEAR OPTIMIZATION (S. Gaubert and B. Werner) 2014 now: Postdoc at LAAS-CNRS MOMENT-SOS APPLICATIONS (J.B. Lasserre and D. Henrion) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 2 / 47

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 Semialgebraic Relaxations using Moment-SOS hierarchies 3 / 47

4 Errors and Proofs Possible workaround: proof assistants COQ (Coquand, Huet 1984) HOL-LIGHT (Harrison, Gordon 1980) Built in top of OCAML V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 3 / 47

5 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 Semialgebraic Relaxations using Moment-SOS hierarchies 4 / 47

6 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 Semialgebraic Relaxations using Moment-SOS hierarchies 5 / 47

7 ...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 Semialgebraic Relaxations using Moment-SOS hierarchies 6 / 47

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 Project Completion on 10 August by the Flyspeck team!! V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 6 / 47

9 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 Semialgebraic Relaxations using Moment-SOS hierarchies 7 / 47

10 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 Semialgebraic Relaxations using Moment-SOS hierarchies 7 / 47

11 A Simple Example In the computational part: Transcendental functions T : composition of semialgebraic functions with arctan, exp, sin, +,,,... V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 7 / 47

12 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 Semialgebraic Relaxations using Moment-SOS hierarchies 7 / 47

13 Existing Formal Frameworks Formal proofs for Global Optimization: Bernstein polynomial methods [Zumkeller 08] restricted to polynomials Taylor + Interval arithmetic [Melquiond 12, Solovyev 13] robust but subject to the CURSE OF DIMENSIONALITY V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 8 / 47

14 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 Semialgebraic Relaxations using Moment-SOS hierarchies 8 / 47

15 Introduction Moment-SOS relaxations Another look at Nonnegativity New Applications of Moment-SOS Hierarchies Conclusion

16 Polynomial Optimization Problems 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] e.g. x 2 1 2x 1x 2 + x 2 2 = (x 1 x 2 ) 2 Q(K) := { } σ 0 (x) + m j=1 σ j(x)g j (x), with σ j Σ[x] V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 9 / 47

17 Polynomial Optimization Problems Archimedean module The set K is compact and the polynomial N x 2 2 Q(K) for some N > 0. belongs to Assume that K is a box: product of closed intervals 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 2 belongs to Q(K ) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 10 / 47

18 Convexification and the K Moment Problem Borel σ-algebra B (generated by the open sets of R n ) M + (K): set of probability measures supported on K. If µ M + (K) then 1 µ : B [0, 1], µ( ) = 0, µ(r n ) < 2 µ( i B i ) = i µ(b i ), for any countable (B i ) B 3 K µ(dx) = 1 supp(µ) is the smallest set K such that µ(r n \K) = 0 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 11 / 47

19 Convexification and the K Moment Problem p = inf p(x) = inf p dµ x K µ M + (K) K V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 11 / 47

20 Convexification and the K Moment Problem Let (x α ) α N n be the monomial basis Definition A sequence y has a representing measure on K if there exists a finite measure µ supported on K such that y α = K x α µ(dx), α N n. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 11 / 47

21 Convexification and the K Moment Problem L y (q) : q R[x] α q α y α Theorem [Putinar 93] Let K be compact and Q(K) be Archimedean. Then y has a representing measure on K iff L y (σ) 0, L y (g j σ) 0, σ Σ[x]. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 11 / 47

22 Lasserre s Hierarchy of SDP relaxations Moment matrix M(y) u,v := L y (u v), u, v monomials Localizing matrix M(g j y) associated with g j M(g j y) u,v := L y (u v g j ), u, v monomials V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 12 / 47

23 Lasserre s Hierarchy of SDP relaxations M k (y) contains ( n+2k ) variables, has size (n+k n n ) Truncated matrix of order k = 2 with variables x 1, x 2 : M 2 (y) = 1 x 1 x 2 x 2 1 x 1 x 2 x y 1,0 y 0,1 y 2,0 y 1,1 y 0,2 x 1 y 1,0 y 2,0 y 1,1 y 3,0 y 2,1 y 1,2 x 2 y 0,1 y 1,1 y 0,2 y 2,1 y 1,2 y 0,3 x 2 1 y 2,0 y 3,0 y 2,1 y 4,0 y 3,1 y 2,2 x 1 x 2 y 1,1 y 2,1 y 1,2 y 3,1 y 2,2 y 1,3 x 2 2 y 0,2 y 1,2 y 0,3 y 2,2 y 1,3 y 0,4 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 12 / 47

24 Lasserre s Hierarchy of SDP relaxations Consider g 1 (x) := 2 x 2 1 x2 2. Then v 1 = deg g 1 /2 = 1. 1 x 1 x y 2,0 y 0,2 2y 1,0 y 3,0 y 1,2 2y 0,1 y 2,1 y 0,3 M 1 (g 1 y) = x 1 2y 1,0 y 3,0 y 1,2 2y 2,0 y 4,0 y 2,2 2y 1,1 y 3,1 y 1,3 x 2 2y 0,1 y 2,1 y 0,3 2y 1,1 y 3,1 y 1,3 2y 0,2 y 2,2 y 0,4 M 1 (g 1 y)(3, 3) = L(g 1 (x) x 2 x 2 ) = L(2x 2 2 x 2 1 x2 2 x 4 2) = 2y 0,2 y 2,2 y 0,4 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 12 / 47

25 Lasserre s Hierarchy of SDP relaxations Truncation with moments of order at most 2k v j := deg g j /2 Hierarchy of semidefinite relaxations: inf y L y (p) = α K p α x α µ(dx) = α p α y α M k (y) 0, M k vj (g j y) 0, 1 j m, y 1 = 1. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 12 / 47

26 Semidefinite Optimization F 0, F α symmetric real matrices, cost vector c Primal-dual pair of semidefinite programs: P : inf y α c α y α s.t. α F α y α F 0 0 (SDP) D : sup Y Trace (F 0 Y) s.t. Trace (F α Y) = c α, Y 0. Freely available SDP solvers (CSDP, SDPA, SEDUMI) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 13 / 47

27 Primal-dual Moment-SOS 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 Semialgebraic Relaxations using Moment-SOS hierarchies 14 / 47

28 Primal-dual Moment-SOS Truncated quadratic module Q k (K) := Q(K) R 2k [x] For large enough k, zero duality gap [Lasserre 01]: Polynomial Optimization Problems (POP) (Moment) (SOS) inf p α y α = sup λ α s.t. M k vj (g j y) 0, 0 j m, s.t. λ R, y 1 = 1 p λ Q k (K) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 14 / 47

29 Practical Computation Hierarchy { of SOS relaxations: } λ k := sup λ : p λ Q k (K) λ Convergence guarantees λ k p [Lasserre 01] If p p Q k (K) for some k then: y := (1, x 1, x 2, (x 1) 2, x 1 x 2,..., (x 1) 2k,..., (x n) 2k ) is a global minimizer of the primal SDP [Lasserre 01]. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 15 / 47

30 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 on [0, 1] 4 SOS of degree at most 4 Redundant constraints x 2 1 1,..., x2 4 1 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 15 / 47

31 The No Free Lunch Rule Exponential dependency in 1 Relaxation order k (SOS degree) 2 number of variables n Computing λ k involves ( n+2k n ) variables At fixed k, O(n 2k ) variables V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 16 / 47

32 Introduction Moment-SOS relaxations Another look at Nonnegativity New Applications of Moment-SOS Hierarchies Conclusion

33 Another look at Nonnegativity Knowledge of K through µ M + (K) Independent of the representation of K Typical from INVERSE PROBLEMS reconstruct K from measuring moments of µ M + (K) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 17 / 47

34 Another look at Nonnegativity Borel σ-algebra B (generated by the open sets of R n ) Lemma A continuous function p : R n R is nonnegative on K iff the set function ν : B B K B p(x) dµ(x) belongs to M +. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 18 / 47

35 Another look at Nonnegativity ν : B B K B p(x) dµ(x) Proof 1 Only if part is straightforward 2 If part If ν M + then p(x) 0 for µ-almost all x K, i.e. there exists G B such that µ(g) = 0 and p(x) 0 on K\G. K = K\G (from the support definition). Let x K. There is a sequence (x l ) K\G such that x l x, as l. By continuity of p and p(x l ) 0, one has p(x) 0. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 19 / 47

36 The K Moment Problem Definition A sequence y has a representing measure on K if there exists a finite measure µ supported on K such that y α = K x α µ(dx), α N n. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 20 / 47

37 The K Moment Problem L y (q) : q R[x] α q α y α Theorem [Putinar 93] Let K be compact and Q(K) be Archimedean. Then y has a representing measure on K iff L y (σ) 0, L y (g j σ) 0, σ Σ[x]. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 20 / 47

38 The K Moment Problem Theorem [Lasserre 11] Let K be compact and µ M + be arbitrary fixed with moments y α = K x α µ(dx), α N n. Then a polynomial p is nonnegative on K iff M k (p y) 0, k N. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 20 / 47

39 Hierarchy of Outer Approximations Cone of nonnegative polynomials C(K) d R d [x] Each entry of the matrix M k (p y) is linear in the coefficients of p The set k := {p R d [x] : M k (p y) 0} is closed and convex, called spectrahedron Nested outer approximations: 0 1 k C(K) d V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 21 / 47

40 Hierarchy of Outer Approximations Hierarchy of upper bounds for p := inf x K p(x) Theorem [Lasserre 11] Let K be closed and µ M + (K) be arbitrary fixed with moments (y α ), α N n. Consider the hierarchy of SDP: Then u k p, as k. u k := max{λ : M k ((p λ) y) 0} = max{λ : λ M k (y) M k (p y)}. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 22 / 47

41 Hierarchy of Outer Approximations Hierarchy of upper bounds for p := inf x K p(x) Computing u k : GENERALIZED EIGENVALUE PROBLEM for the pair [M k (p y), M k (y)]. Index the matrices in the basis of orthonormal polynomials w.r.t. µ: u k = max{λ : λ I M k (p y)} = λ min (M k (p y)), a standard MINIMAL EIGENVALUE PROBLEM. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 22 / 47

42 Hierarchy of Outer Approximations Primal-Dual Moment-SOS (Primal) (Dual) u k := min p(x) σ(x) dµ(x) K u k := max λ s.t. σ(x) dµ(x) = 1, s.t. λ R, K σ Σ k [x] λ M k (y) M k (p y). Then u k, u k p, as k. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 23 / 47

43 Interpretation of the Primal Problem u k { := min p σ dµ : ν(k) = 1, ν(r n \K) = 0, σ Σ ν K }{{} k [x] } dν The measure ν approximates the DIRAC measure δ x=x at a global minimizer x K and 1 ν is absolutely continuous w.r.t. µ 2 the density of ν is σ Σ k [x] V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 24 / 47

44 Example with uniform probability measure Polynomial p := x + 21x 2 32x x 4 on K := [0, 1] V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 25 / 47

45 Example with uniform probability measure Probability Density σ Σ 10 [x] V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 25 / 47

46 Extension to the non-compact case In this case, the measure µ needs to satisfy CARLEMAN-type sufficient condition to limit the growth of its moments (y α ): L y (x 2t i ) 1/(2t) = +, i = 1,..., n. t=1 e.g. dµ(x) := exp( x 2 2 /2) dµ 0, with µ 0 M + (K) finite V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 26 / 47

47 Extension to the non-compact case dµ(x) := exp( x 2 2 /2) dx, for K = Rn dµ(x) := exp( n i=1 x i) dx, for K = R n + dµ(x) := dx, when K is a box, simplex V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 26 / 47

48 Introduction Moment-SOS relaxations Another look at Nonnegativity New Applications of Moment-SOS Hierarchies Conclusion

49 Introduction Moment-SOS relaxations Another look at Nonnegativity New Applications of Moment-SOS Hierarchies Semialgebraic Maxplus Optimization Formal Nonlinear Optimization Pareto Curves Polynomial Images of Semialgebraic Sets Program Analysis with Polynomial Templates Conclusion

50 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 Semialgebraic Relaxations using Moment-SOS hierarchies 27 / 47

51 General informal Framework Approximation theory: Chebyshev/Taylor models mandatory for non-polynomial problems hard to combine with Sum-of-Squares techniques (degree of approximation) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 27 / 47

52 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 Semialgebraic Relaxations using Moment-SOS hierarchies 28 / 47

53 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 Semialgebraic Relaxations using Moment-SOS hierarchies 28 / 47

54 Nonlinear Function Representation Exact parsimonious maxplus representations y a V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 29 / 47

55 Nonlinear Function Representation Exact parsimonious maxplus representations y a V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 29 / 47

56 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 Semialgebraic Relaxations using Moment-SOS hierarchies 29 / 47

57 Nonlinear Function Representation For the Simple Example from Flyspeck: + l(x) arctan r(x) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 29 / 47

58 Maxplus Optimization Algorithm First iteration: + y arctan l(x) arctan r(x) m a 1 par a 1 M a 1 control point {a 1 }: m 1 = < 0 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 30 / 47

59 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 control points {a 1, a 2 }: m 2 = < 0 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 30 / 47

60 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 control points {a 1, a 2, a 3 }: m 3 = > 0 OK! V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 30 / 47

61 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 Semialgebraic Relaxations using Moment-SOS hierarchies 31 / 47

62 Introduction Moment-SOS relaxations Another look at Nonnegativity New Applications of Moment-SOS Hierarchies Semialgebraic Maxplus Optimization Formal Nonlinear Optimization Pareto Curves Polynomial Images of Semialgebraic Sets Program Analysis with Polynomial Templates Conclusion

63 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 Semialgebraic Relaxations using Moment-SOS hierarchies 32 / 47

64 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 Semialgebraic Relaxations using Moment-SOS hierarchies 33 / 47

65 Formal Nonlinear Optimization 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 Semialgebraic Relaxations using Moment-SOS hierarchies 34 / 47

66 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 Semialgebraic Relaxations using Moment-SOS hierarchies 35 / 47

67 Introduction Moment-SOS relaxations Another look at Nonnegativity New Applications of Moment-SOS Hierarchies Semialgebraic Maxplus Optimization Formal Nonlinear Optimization Pareto Curves Polynomial Images of Semialgebraic Sets Program Analysis with Polynomial Templates Conclusion

68 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 Semialgebraic Relaxations using Moment-SOS hierarchies 36 / 47

69 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 Semialgebraic Relaxations using Moment-SOS hierarchies 36 / 47

70 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 Semialgebraic Relaxations using Moment-SOS hierarchies 37 / 47

71 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 Semialgebraic Relaxations using Moment-SOS hierarchies 38 / 47

72 A Hierarchy of Polynomial underestimators Degree 4 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 39 / 47

73 A Hierarchy of Polynomial underestimators Degree 6 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 39 / 47

74 A Hierarchy of Polynomial underestimators Degree 8 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 39 / 47

75 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 Semialgebraic Relaxations using Moment-SOS hierarchies 40 / 47

76 Introduction Moment-SOS relaxations Another look at Nonnegativity New Applications of Moment-SOS Hierarchies Semialgebraic Maxplus Optimization Formal Nonlinear Optimization Pareto Curves Polynomial Images of Semialgebraic Sets Program Analysis with Polynomial Templates Conclusion

77 Approximation of sets defined with Let B R 2 be the unit ball and assume that f (S) B. Another point of view: with f (S) = {y B : x S s.t. h(x, y) 0}, h(x, y) := y f (x) 2 2 = (y 1 f 1 (x)) 2 + (y 2 f 2 (x)) 2. Approximate f (S) as closely as desired by a sequence of sets of the form : Θ d := {y B : q d (y) 0}, for some polynomials q d R 2d [y]. V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 41 / 47

78 A Hierarchy of Outer approximations for f (S) f (x) := (x 1 + x 1 x 2, x 2 x 3 1 )/2 Degree 2 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 42 / 47

79 A Hierarchy of Outer approximations for f (S) f (x) := (x 1 + x 1 x 2, x 2 x 3 1 )/2 Degree 4 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 42 / 47

80 A Hierarchy of Outer approximations for f (S) f (x) := (x 1 + x 1 x 2, x 2 x 3 1 )/2 Degree 6 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 42 / 47

81 A Hierarchy of Outer approximations for f (S) f (x) := (x 1 + x 1 x 2, x 2 x 3 1 )/2 Degree 8 V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 42 / 47

82 Introduction Moment-SOS relaxations Another look at Nonnegativity New Applications of Moment-SOS Hierarchies Semialgebraic Maxplus Optimization Formal Nonlinear Optimization Pareto Curves Polynomial Images of Semialgebraic Sets Program Analysis with Polynomial Templates Conclusion

83 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 Semialgebraic Relaxations using Moment-SOS hierarchies 43 / 47

84 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 Semialgebraic Relaxations using Moment-SOS hierarchies 44 / 47

85 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 Semialgebraic Relaxations using Moment-SOS hierarchies 45 / 47

86 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 Semialgebraic Relaxations using Moment-SOS hierarchies 45 / 47

87 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 Semialgebraic Relaxations using Moment-SOS hierarchies 45 / 47

88 Contribution For more details: A. Adjé and V. Magron. Polynomial Template Generation using Sum-of-Squares Programming. Submitted for publication, arxiv: V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 46 / 47

89 Introduction Moment-SOS relaxations Another look at Nonnegativity New Applications of Moment-SOS Hierarchies Conclusion

90 Conclusion With MOMENT-SOS HIERARCHIES, you can Optimize nonlinear (transcendental) functions Approximate Pareto Curves, images and projections of semialgebraic sets Analyze programs V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 47 / 47

91 Conclusion Further research: Alternative polynomial bounds using geometric programming (T. de Wolff, S. Iliman) Mixed LP/SOS certificates (trade-off CPU/precision) V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 47 / 47

92 End Thank you for your attention!