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Semialgebraic Relaxations usingMoment-SOS Hierarchies
Victor Magron, Postdoc LAAS-CNRS
17 September 2014
SIERRA SeminarLaboratoire d’Informatique de l’Ecole Normale Superieure
b
y
b 7→ sin(√
b)
par−b1
par−b2
par−b3
par+b1
par+b2
par+b3
1 b1 b2 b3 = 500
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Personal Background
2008− 2010: Master at Tokyo UniversityHIERARCHICAL DOMAIN DECOMPOSITION METHODS
(S. Yoshimura)
2010− 2013: PhD at Inria Saclay LIX/CMAPFORMAL PROOFS FOR NONLINEAR OPTIMIZATION
(S. Gaubert and B. Werner)
2014−now: Postdoc at LAAS-CNRSMOMENT-SOS APPLICATIONS
(J.B. Lasserre and D. Henrion)
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Errors and Proofs
Mathematicians want to eliminate all the uncertainties ontheir results. Why?
M. Lecat, Erreurs des Mathématiciens des origines ànos jours, 1935.
130 pages of errors! (Euler, Fermat, Sylvester, . . . )
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Errors and Proofs
Possible workaround: proof assistants
COQ (Coquand, Huet 1984)
HOL-LIGHT (Harrison, Gordon 1980)
Built in top of OCAML
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Complex Proofs
Complex mathematical proofs / mandatory computation
K. Appel and W. Haken , Every Planar Map isFour-Colorable, 1989.
T. Hales, A Proof of the Kepler Conjecture, 1994.
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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
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...to Flyspeck Nonlinear Inequalities
The proof of T. Hales (1998) contains mathematical andcomputational parts
Computation: check thousands of nonlinear inequalities
Robert MacPherson, editor of The Annals of Mathematics:“[...] the mathematical community will have to get used tothis state of affairs.”
Flyspeck [Hales 06]: Formal Proof of Kepler Conjecture
Project Completion on 10 August by the Flyspeck team!!
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...to Flyspeck Nonlinear Inequalities
The proof of T. Hales (1998) contains mathematical andcomputational parts
Computation: check thousands of nonlinear inequalities
Robert MacPherson, editor of The Annals of Mathematics:“[...] the mathematical community will have to get used tothis state of affairs.”
Flyspeck [Hales 06]: Formal Proof of Kepler Conjecture
Project Completion on 10 August by the Flyspeck team!!
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A “Simple” Example
In the computational part:
Multivariate Polynomials:∆x := x1x4(−x1 + x2 + x3 − x4 + x5 + x6) + x2x5(x1 − x2 + x3 +
x4 − x5 + x6) + x3x6(x1 + x2 − x3 + x4 + x5 − x6)− x2(x3x4 +
x1x6)− x5(x1x3 + x4x6)
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A “Simple” Example
In the computational part:
Semialgebraic functions: composition of polynomials with| · |,√,+,−,×, /, sup, inf, . . .
p(x) := ∂4∆x q(x) := 4x1∆xr(x) := p(x)/
√q(x)
l(x) := −π
2+ 1.6294− 0.2213 (
√x2 +
√x3 +
√x5 +
√x6 − 8.0) +
0.913 (√
x4 − 2.52) + 0.728 (√
x1 − 2.0)
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A “Simple” Example
In the computational part:
Transcendental functions T : composition of semialgebraicfunctions with arctan, exp, sin, +,−,×, . . .
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A “Simple” Example
In the computational part:
Feasible set K := [4, 6.3504]3 × [6.3504, 8]× [4, 6.3504]2
Lemma9922699028 from Flyspeck:
∀x ∈ K, arctan( p(x)√
q(x)
)+ l(x) > 0
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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
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Existing Formal Frameworks
Lemma9922699028 from Flyspeck:
∀x ∈ K, arctan( ∂4∆x√
4x1∆x
)+ l(x) > 0
Dependency issue using Interval Calculus:One can bound ∂4∆x/
√4x1∆x and l(x) separately
Too coarse lower bound: −0.87
Subdivide K to prove the inequality
K=⇒
K0K1
K2
K3
K4
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Introduction
Moment-SOS relaxations
Another look at Nonnegativity
New Applications of Moment-SOS Hierarchies
Conclusion
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Polynomial Optimization Problems
Semialgebraic set K := {x ∈ Rn : g1(x) > 0, . . . , gm(x) > 0}
p∗ := minx∈K
p(x): NP hard
Sums of squares Σ[x]e.g. x2
1 − 2x1x2 + x22 = (x1 − x2)
2
Q(K) :={
σ0(x) + ∑mj=1 σj(x)gj(x), with σj ∈ Σ[x]
}
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Polynomial Optimization Problems
Archimedean module
The set K is compact and the polynomial N − ‖x‖22 belongs to
Q(K) for some N > 0.
Assume that K is a box: product of closed intervals
Normalize the feasibility set to get K′ := [−1, 1]n
K′ := {x ∈ Rn : g1 := 1− x21 > 0, · · · , gn := 1− x2
n > 0}
n− ‖x‖22 belongs to Q(K′)
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Convexification and the K Moment Problem
Borel σ-algebra B (generated by the open sets of Rn)
M+(K): set of probability measures supported on K.If µ ∈ M+(K) then
1 µ : B → [0, 1], µ(∅) = 0, µ(Rn) < ∞
2 µ(⋃
i Bi) = ∑i µ(Bi), for any countable (Bi) ⊂ B
3∫
K µ(dx) = 1
supp(µ) is the smallest set K such that µ(Rn\K) = 0
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Convexification and the K Moment Problem
p∗ = infx∈K
p(x) = infµ∈M+(K)
∫K
p dµ
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Convexification and the K Moment Problem
Let (xα)α∈Nn be the monomial basis
DefinitionA sequence y has a representing measure on K if there exists afinite measure µ supported on K such that
yα =∫
Kxαµ(dx) , ∀ α ∈Nn .
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Convexification and the K Moment Problem
Ly(q) : q ∈ R[x] 7→ ∑α qαyα
Theorem [Putinar 93]
Let K be compact and Q(K) be Archimedean.Then y has a representing measure on K
iff
Ly(σ) > 0 , Ly(gj σ) > 0 , ∀σ ∈ Σ[x] .
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Lasserre’s Hierarchy of SDP relaxations
Moment matrixM(y)u,v := Ly(u · v), u, v monomials
Localizing matrix M(gj y) associated with gj
M(gj y)u,v := Ly(u · v · gj), u, v monomials
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Lasserre’s Hierarchy of SDP relaxations
Mk(y) contains (n+2kn ) variables, has size (n+k
n )
Truncated matrix of order k = 2 with variables x1, x2:
M2(y) =
1 | x1 x2 | x21 x1x2 x2
2
1 1 | y1,0 y0,1 | y2,0 y1,1 y0,2
− − − − − − − −x1 y1,0 | y2,0 y1,1 | y3,0 y2,1 y1,2
x2 y0,1 | y1,1 y0,2 | y2,1 y1,2 y0,3
− − − − − − − − −x2
1 y2,0 | y3,0 y2,1 | y4,0 y3,1 y2,2
x1x2 y1,1 | y2,1 y1,2 | y3,1 y2,2 y1,3
x22 y0,2 | y1,2 y0,3 | y2,2 y1,3 y0,4
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Lasserre’s Hierarchy of SDP relaxations
Consider g1(x) := 2− x21 − x2
2. Then v1 = ddeg g1/2e = 1.
M1(g1 y) =
1 x1 x2
1 2− y2,0 − y0,2 2y1,0 − y3,0 − y1,2 2y0,1 − y2,1 − y0,3x1 2y1,0 − y3,0 − y1,2 2y2,0 − y4,0 − y2,2 2y1,1 − y3,1 − y1,3x2 2y0,1 − y2,1 − y0,3 2y1,1 − y3,1 − y1,3 2y0,2 − y2,2 − y0,4
M1(g1 y)(3, 3) = L(g1(x) · x2 · x2) = L(2x22 − x2
1x22 − x4
2)
= 2y0,2 − y2,2 − y0,4
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Lasserre’s Hierarchy of SDP relaxations
Truncation with moments of order at most 2k
vj := ddeg gj/2e
Hierarchy of semidefinite relaxations:infy Ly(p) = ∑α
∫K pα xα µ(dx) = ∑α pα yα
Mk(y) < 0 ,Mk−vj(gj y) < 0 , 1 6 j 6 m,
y1 = 1 .
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Semidefinite Optimization
F0, Fα symmetric real matrices, cost vector c
Primal-dual pair of semidefinite programs:
(SDP)
P : infy ∑α cαyα
s.t. ∑α Fα yα − F0 < 0
D : supY Trace (F0 Y)s.t. Trace (Fα Y) = cα , Y < 0 .
Freely available SDP solvers (CSDP, SDPA, SEDUMI)
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Primal-dual Moment-SOS
M+(K): space of probability measures supported on K
Polynomial Optimization Problems (POP)
(Primal) (Dual)
inf∫
Kp dµ = sup λ
s.t. µ ∈ M+(K) s.t. λ ∈ R ,
p− λ ∈ Q(K)
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Primal-dual Moment-SOS
Truncated quadratic module Qk(K) := Q(K) ∩R2k[x]
For large enough k, zero duality gap [Lasserre 01]:
Polynomial Optimization Problems (POP)
(Moment) (SOS)
inf ∑α
pα yα = sup λ
s.t. Mk−vj(gj y) < 0 , 0 6 j 6 m, s.t. λ ∈ R ,
y1 = 1 p− λ ∈ Qk(K)
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Practical Computation
Hierarchy of SOS relaxations:
λk := supλ
{λ : p− λ ∈ Qk(K)
}Convergence guarantees λk ↑ p∗ [Lasserre 01]
If p− p∗ ∈ Qk(K) for some k then:
y∗ := (1, x∗1 , x∗2 , (x∗1)2, x∗1x∗2 , . . . , (x∗1)
2k, . . . , (x∗n)2k)
is a global minimizer of the primal SDP [Lasserre 01].
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Practical Computation
Caprasse Problem
∀x ∈ [−0.5, 0.5]4,−x1x33 + 4x2x2
3x4 + 4x1x3x24 + 2x2x3
4 +
4x1x3 + 4x23 − 10x2x4 − 10x2
4 + 5.1801 > 0.
Scale on [0, 1]4
SOS of degree at most 4
Redundant constraints x21 6 1, . . . , x2
4 6 1
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The “No Free Lunch” Rule
Exponential dependency in1 Relaxation order k (SOS degree)
2 number of variables n
Computing λk involves (n+2kn ) variables
At fixed k, O(n2k) variables
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Introduction
Moment-SOS relaxations
Another look at Nonnegativity
New Applications of Moment-SOS Hierarchies
Conclusion
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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)
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Another look at Nonnegativity
Borel σ-algebra B (generated by the open sets of Rn)
LemmaA continuous function p : Rn → R is nonnegative on K
iff
the set function ν : B ∈ B 7→∫
K∩B p(x) dµ(x) belongs toM+.
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Another look at Nonnegativity
ν : B ∈ B 7→∫
K∩B p(x) dµ(x)
Proof1 “Only if” part is straightforward
2 “If part”If ν ∈ M+ then p(x) > 0 for µ-almost all x ∈ K, i.e. thereexists 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 (xl) ⊂ K\G such that xl → x,as l→ ∞. By continuity of p and p(xl) > 0, one hasp(x) > 0.
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The K Moment Problem
DefinitionA sequence y has a representing measure on K if there exists afinite measure µ supported on K such that
yα =∫
Kxαµ(dx) , ∀ α ∈Nn .
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The K Moment Problem
Ly(q) : q ∈ R[x] 7→ ∑α qαyα
Theorem [Putinar 93]
Let K be compact and Q(K) be Archimedean.Then y has a representing measure on K
iff
Ly(σ) > 0 , Ly(gj σ) > 0 , ∀σ ∈ Σ[x] .
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The K Moment Problem
Theorem [Lasserre 11]Let K be compact and µ ∈ M+ be arbitrary fixed with moments
yα =∫
Kxαµ(dx) , ∀ α ∈Nn .
Then a polynomial p is nonnegative on K
iff
Mk(p y) < 0 , ∀k ∈N .
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Hierarchy of Outer Approximations
Cone of nonnegative polynomials C(K)d ⊂ Rd[x]
Each entry of the matrix Mk(p y) is linear in the coefficientsof p
The set ∆k := {p ∈ Rd[x] : Mk(p y) < 0} is closed andconvex, called spectrahedron
Nested outer approximations:∆0 ⊃ ∆1 · · · ⊃ ∆k · · · ⊃ C(K)d
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Hierarchy of Outer Approximations
Hierarchy of upper bounds for p∗ := infx∈K p(x)
Theorem [Lasserre 11]
Let K be closed and µ ∈ M+(K) be arbitrary fixed with mo-ments (yα), α ∈Nn. Consider the hierarchy of SDP:
uk := max{λ : Mk((p− λ) y) < 0}= max{λ : λ Mk(y) 4 Mk(p y)} .
Then uk ↓ p∗, as k→ ∞.
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Hierarchy of Outer Approximations
Hierarchy of upper bounds for p∗ := infx∈K p(x)
Computing uk: GENERALIZED EIGENVALUE PROBLEM forthe pair [Mk(p y), Mk(y)].
Index the matrices in the basis of orthonormal polynomialsw.r.t. µ:
uk = max{λ : λ I 4 Mk(p y)}= λmin(Mk(p y)) ,
a standard MINIMAL EIGENVALUE PROBLEM.
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Hierarchy of Outer Approximations
Primal-Dual Moment-SOS
(Primal) (Dual)
u′k := min∫
Kp(x) σ(x) dµ(x) > uk := max λ
s.t.∫
Kσ(x) dµ(x) = 1 , s.t. λ ∈ R ,
σ ∈ Σk[x] λ Mk(y) 4 Mk(p y) .
Then u′k, uk → p∗, as k→ ∞.
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Interpretation of the Primal Problem
u∗k := minν
{ ∫K
p σ dµ︸︷︷︸dν
: ν(K) = 1 , ν(Rn\K) = 0 , σ ∈ Σk[x]}
The measure ν approximates the DIRAC measure δx=x∗ at aglobal minimizer x∗ ∈ K and
1 ν is absolutely continuous w.r.t. µ
2 the density of ν is σ ∈ Σk[x]
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Example with uniform probability measure
Polynomial p := 0.375− 5x + 21x2 − 32x3 + 16x4 on K := [0, 1]
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Example with uniform probability measure
Probability Density σ ∈ Σ10[x]
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Extension to the non-compact case
In this case, the measure µ
needs to satisfy CARLEMAN-type sufficient condition tolimit the growth of its moments (yα):
∞
∑t=1
Ly(x2ti )−1/(2t) = +∞ , i = 1, . . . , n .
e.g. dµ(x) := exp(−‖x‖22/2) dµ0, with µ0 ∈ M+(K) finite
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Extension to the non-compact case
dµ(x) := exp(−‖x‖22/2) dx, for K = Rn
dµ(x) := exp(−∑ni=1 xi) dx, for K = Rn
+
dµ(x) := dx, when K is a box, simplex
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Introduction
Moment-SOS relaxations
Another look at Nonnegativity
New Applications of Moment-SOS Hierarchies
Conclusion
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Introduction
Moment-SOS relaxations
Another look at Nonnegativity
New Applications of Moment-SOS HierarchiesSemialgebraic Maxplus OptimizationFormal Nonlinear OptimizationPareto CurvesPolynomial Images of Semialgebraic SetsProgram Analysis with Polynomial Templates
Conclusion
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General informal Framework
Given K a compact set and f a transcendental function, boundf ∗ = inf
x∈Kf (x) and prove f ∗ > 0
f is underestimated by a semialgebraic function fsa
Reduce the problem f ∗sa := infx∈K fsa(x) to a polynomialoptimization problem (POP)
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General informal Framework
Approximation theory: Chebyshev/Taylor models
mandatory for non-polynomial problems
hard to combine with Sum-of-Squares techniques (degreeof approximation)
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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 bygrid methods)
In our context: approximate transcendental functions
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Maxplus Approximation
DefinitionLet γ > 0. A function φ : Rn → R is said to be γ-semiconvex ifthe function x 7→ φ(x) + γ
2 ‖x‖22 is convex.
a
y
par+a1
par+a2
par−a2
par−a1
a2a1
arctan
m M
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Nonlinear Function Representation
Exact parsimonious maxplus representations
a
y
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Nonlinear Function Representation
Exact parsimonious maxplus representations
a
y
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Nonlinear Function Representation
Abstract syntax tree representations of multivariatetranscendental functions:
leaves are semialgebraic functions of Anodes are univariate functions of D or binary operations
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Nonlinear Function Representation
For the “Simple” Example from Flyspeck:
+
l(x) arctan
r(x)
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Maxplus Optimization Algorithm
First iteration:
+
l(x) arctan
r(x)
a
y
par−a1
arctan
m Ma1
1 control point {a1}: m1 = −4.7× 10−3 < 0
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Maxplus Optimization Algorithm
Second iteration:
+
l(x) arctan
r(x)
a
y
par−a1
par−a2
arctan
m Ma1a2
2 control points {a1, a2}: m2 = −6.1× 10−5 < 0
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Maxplus Optimization Algorithm
Third iteration:
+
l(x) arctan
r(x)
a
y
par−a1
par−a2
par−a3
arctan
m Ma1a2 a3
3 control points {a1, a2, a3}: m3 = 4.1× 10−6 > 0
OK!
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Contributions
Published:
X. Allamigeon, S. Gaubert, V. Magron, and B. Werner.Certification of inequalities involving transcendental functions:combining sdp and max-plus approximation, ECC Conference2013.
X. Allamigeon, S. Gaubert, V. Magron, and B. Werner.Certification of bounds of non-linear functions: the templatesmethod, CICM Conference, 2013.
In revision:
X. Allamigeon, S. Gaubert, V. Magron, and B. Werner.Certification of Real Inequalities – Templates and Sums ofSquares, arxiv:1403.5899, 2014.
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Introduction
Moment-SOS relaxations
Another look at Nonnegativity
New Applications of Moment-SOS HierarchiesSemialgebraic Maxplus OptimizationFormal Nonlinear OptimizationPareto CurvesPolynomial Images of Semialgebraic SetsProgram Analysis with Polynomial Templates
Conclusion
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The General “Formal Framework”
We check the correctness of SOS certificates for POP
We build certificates to prove interval bounds forsemialgebraic functions
We bound formally transcendental functions withsemialgebraic approximations
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Formal SOS bounds
When q ∈ Q(K), σ0, . . . , σm is a positivity certificate for qCheck 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
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Formal Nonlinear Optimization
Inequality #boxesTime Time
9922699028 39 190 s 2218 s3318775219 338 1560 s 19136 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 checkpolynomial equalities
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Contribution
For more details on the formal side:
X. Allamigeon, S. Gaubert, V. Magron and B. Werner. FormalProofs for Nonlinear Optimization. Submitted for publication,arxiv:1404.7282
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Introduction
Moment-SOS relaxations
Another look at Nonnegativity
New Applications of Moment-SOS HierarchiesSemialgebraic Maxplus OptimizationFormal Nonlinear OptimizationPareto CurvesPolynomial Images of Semialgebraic SetsProgram Analysis with Polynomial Templates
Conclusion
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Bicriteria Optimization Problems
Let f1, f2 ∈ Rd[x] two conflicting criteria
Let S := {x ∈ Rn : g1(x) > 0, . . . , gm(x) > 0} asemialgebraic set
(P){
minx∈S
(f1(x) f2(x))>}
Assumption
The image space R2 is partially ordered in a natural way (R2+ is
the ordering cone).
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Bicriteria Optimization Problems
g1 := −(x1 − 2)3/2− x2 + 2.5 ,
g2 := −x1 − x2 + 8(−x1 + x2 + 0.65)2 + 3.85 ,
S := {x ∈ R2 : g1(x) > 0, g2(x) > 0} .
f1 := (x1 + x2 − 7.5)2/4 + (−x1 + x2 + 3)2 ,
f2 := (x1 − 1)2/4 + (x2 − 4)2/4 .
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 36 / 47
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Parametric sublevel set approximation
Inspired by previous research on multiobjective linearoptimization [Gorissen-den Hertog 12]
Workaround: reduce P to a parametric POP
(Pλ) : f ∗(λ) := minx∈S{ f2(x) : f1(x) 6 λ } ,
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 37 / 47
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A Hierarchy of Polynomial underestimators
Moment-SOS approach [Lasserre 10]:
(Dd)
maxq∈R2d[λ]
2d
∑k=0
qk/(1 + k)
s.t. f2(x)− q(λ) ∈ Q2d(K) .
The hierarchy (Dd) provides a sequence (qd) ofpolynomial underestimators of f ∗(λ).
limd→∞∫ 1
0 (f∗(λ)− qd(λ))dλ = 0
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 38 / 47
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A Hierarchy of Polynomial underestimators
Degree 4
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 39 / 47
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A Hierarchy of Polynomial underestimators
Degree 6
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 39 / 47
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A Hierarchy of Polynomial underestimators
Degree 8
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 39 / 47
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Contributions
Numerical schemes that avoid computing finitely manypoints.
Pareto curve approximation with polynomials,convergence guarantees in L1-norm
V. Magron, D. Henrion, J.B. Lasserre. Approximating ParetoCurves using Semidefinite Relaxations. Operations ResearchLetters. arxiv:1404.4772, April 2014.
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 40 / 47
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Introduction
Moment-SOS relaxations
Another look at Nonnegativity
New Applications of Moment-SOS HierarchiesSemialgebraic Maxplus OptimizationFormal Nonlinear OptimizationPareto CurvesPolynomial Images of Semialgebraic SetsProgram Analysis with Polynomial Templates
Conclusion
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Approximation of sets defined with “∃”
Let B ⊂ R2 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) 6 0} ,
with
h(x, y) := ‖y− f (x)‖22 = (y1 − f1(x))2 + (y2 − f2(x))2 .
Approximate f (S) as closely as desired by a sequence ofsets of the form :
Θd := {y ∈ B : qd(y) 6 0} ,
for some polynomials qd ∈ R2d[y].
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 41 / 47
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A Hierarchy of Outer approximations for f (S)
f (x) := (x1 + x1x2, x2 − x31)/2
Degree 2
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 42 / 47
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A Hierarchy of Outer approximations for f (S)
f (x) := (x1 + x1x2, x2 − x31)/2
Degree 4
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 42 / 47
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A Hierarchy of Outer approximations for f (S)
f (x) := (x1 + x1x2, x2 − x31)/2
Degree 6
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 42 / 47
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A Hierarchy of Outer approximations for f (S)
f (x) := (x1 + x1x2, x2 − x31)/2
Degree 8
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 42 / 47
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Introduction
Moment-SOS relaxations
Another look at Nonnegativity
New Applications of Moment-SOS HierarchiesSemialgebraic Maxplus OptimizationFormal Nonlinear OptimizationPareto CurvesPolynomial Images of Semialgebraic SetsProgram Analysis with Polynomial Templates
Conclusion
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One-loop with Conditional Branching
r, s, Ti, Te ∈ R[x]
x0 ∈ X0, with X0 semialgebraic set
x = x0;while (r(x) 6 0){
if (s(x) 6 0){x = Ti(x);}
else{x = Te(x);}
}
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 43 / 47
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Bounding Template using SOS
Sufficient condition to get bounding inductive invariant:
α := minq∈R[x]
supx∈X0
q(x)
s.t. q− q ◦ Ti > 0 ,
q− q ◦ Te > 0 ,
q− ‖ · ‖22 > 0 .
Nontrivial correlations via polynomial templates q(x)
{x : q(x) 6 α} ⊃ ⋃k∈N
Xk
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 44 / 47
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Bounds for⋃
k∈N Xk
X0 := [0.9, 1.1]× [0, 0.2] r(x) := 1 s(x) := 1− x21 − x2
2
Ti(x) := (x21 + x3
2, x31 + x2
2) Te(x) := (12
x21 +
25
x32,−3
5x3
1 +310
x22)
Degree 6
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 45 / 47
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Bounds for⋃
k∈N Xk
X0 := [0.9, 1.1]× [0, 0.2] r(x) := 1 s(x) := 1− x21 − x2
2
Ti(x) := (x21 + x3
2, x31 + x2
2) Te(x) := (12
x21 +
25
x32,−3
5x3
1 +310
x22)
Degree 8
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 45 / 47
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Bounds for⋃
k∈N Xk
X0 := [0.9, 1.1]× [0, 0.2] r(x) := 1 s(x) := 1− x21 − x2
2
Ti(x) := (x21 + x3
2, x31 + x2
2) Te(x) := (12
x21 +
25
x32,−3
5x3
1 +310
x22)
Degree 10
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 45 / 47
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Contribution
For more details:
A. Adjé and V. Magron. Polynomial Template Generation usingSum-of-Squares Programming. Submitted for publication,arxiv:1409.3941
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 46 / 47
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Introduction
Moment-SOS relaxations
Another look at Nonnegativity
New Applications of Moment-SOS Hierarchies
Conclusion
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Conclusion
With MOMENT-SOS HIERARCHIES, you can
Optimize nonlinear (transcendental) functions
Approximate Pareto Curves, images and projections ofsemialgebraic sets
Analyze programs
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 47 / 47
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Conclusion
Further research:
Alternative polynomial bounds using geometricprogramming (T. de Wolff, S. Iliman)
Mixed LP/SOS certificates (trade-off CPU/precision)
V. Magron Semialgebraic Relaxations using Moment-SOS hierarchies 47 / 47
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End
Thank you for your attention!
http://homepages.laas.fr/vmagron/