Max-plus approximations:
from optimal control to template methods
[email protected]
INRIA and CMAP, École Polytechnique, CNRS
ENSEEIHT, Toulouse
May 12-14, 2014
Survey of work with Akian, Lakhoua, McEneaney, Qu, Sridharan
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
1 / 58
Templates
Introduced in static analysis by Manna, Sankaranarayanan and
Sipma (VMCAI’05)
A template is a collection of vectors w1 , . . . , wp ∈ Rd .
→ sets parametrized by λ ∈ Rp ,
T (λ) = {x ∈ Rd , wi · x 6 λi ,
1 6 i 6 p}
Nonlinear templates, wi are now non-linear functions (eg
quadratic forms), Adjé, SG, Goubault (ESOP’10), Adjé’s PhD
T (λ) = {x ∈ Rd , wi (x) 6 λi ,
1 6 i 6 p}
Template methods developed/applied by several authors:
Dang, Garoche, Gawlitza, Magron’s PhD (formal proof) . . .
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
2 / 58
A good idea often arises independently in different
contexts
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
3 / 58
Max-plus basis methods
Introduced by Fleming and McEneaney 00, developed by
McEneaney, Akian, Lakhoua, SG, Dower, Qu, Sridharan, . . .
A function f : Rd → R is approximated by
f (x) ' Fα (x) := sup (wi (x) + αi )
16i6p
maxplus basis functions
level sets
−→
templates:
S(Fα ) := {x | Fα (x) 6 0} = {x | wi (x) 6 −αi } = T (−α) .
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
4 / 58
This talk:
survey of the maxplus basis methods in optimal control
curse of dimensionality attenuation ↔ instance of
dynamic templates
pruning of maxplus bases ↔ compression of templates
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
5 / 58
Max-plus or tropical algebra
In an exotic country, children are taught that:
“a + b” = max(a, b)
“a × b” = a + b
So
“2 + 3” =
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
6 / 58
Max-plus or tropical algebra
In an exotic country, children are taught that:
“a + b” = max(a, b)
“a × b” = a + b
So
“2 + 3” = 3
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
6 / 58
Max-plus or tropical algebra
In an exotic country, children are taught that:
“a + b” = max(a, b)
“a × b” = a + b
So
“2 + 3” = 3
“2 × 3” =
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
6 / 58
Max-plus or tropical algebra
In an exotic country, children are taught that:
“a + b” = max(a, b)
“a × b” = a + b
So
“2 + 3” = 3
“2 × 3” =5
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
6 / 58
Max-plus or tropical algebra
In an exotic country, children are taught that:
“a + b” = max(a, b)
“a × b” = a + b
So
“2 + 3” = 3
“2 × 3” =5
“5/2” =
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
6 / 58
Max-plus or tropical algebra
In an exotic country, children are taught that:
“a + b” = max(a, b)
“a × b” = a + b
So
“2 + 3” = 3
“2 × 3” =5
“5/2” =3
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
6 / 58
Max-plus or tropical algebra
In an exotic country, children are taught that:
“a + b” = max(a, b)
“a × b” = a + b
So
“2 + 3” = 3
“2 × 3” =5
“5/2” =3
“23 ” =
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
6 / 58
Max-plus or tropical algebra
In an exotic country, children are taught that:
“a + b” = max(a, b)
“a × b” = a + b
So
“2 + 3” = 3
“2 × 3” =5
“5/2” =3
“23 ” =“2 × 2 × 2” = 6
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
6 / 58
Max-plus or tropical algebra
In an exotic country, children are taught that:
“a + b” = max(a, b)
“a × b” = a + b
So
“2 + 3” = 3
“2 × 3” =5
“5/2” =3
“23 ” =“2 × 2 × 2” = 6
√
“ −1” =
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
6 / 58
Max-plus or tropical algebra
In an exotic country, children are taught that:
“a + b” = max(a, b)
“a × b” = a + b
So
“2 + 3” = 3
7 0
2
“
”=
−∞ 3
1
“2 × 3” =5
“5/2” =3
“23 ” =“2 × 2 × 2” = 6
√
“ −1” =−0.5
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
6 / 58
Max-plus or tropical algebra
In an exotic country, children are taught that:
“a + b” = max(a, b)
“a × b” = a + b
So
“2 + 3” = 3
7 0
2
9
“
”=
−∞ 3
1
4
“2 × 3” =5
“5/2” =3
“23 ” =“2 × 2 × 2” = 6
√
“ −1” =−0.5
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
6 / 58
Lagrange problem / Lax-Oleinik semigroup
Z
v (t, x) =
t
L(x(s), ẋ(s))ds + φ(x(t))
sup
x(0)=x, x(·)
0
Lax-Oleinik semigroup: (St )t>0 , St φ := v (t, ·).
Superposition principle: ∀λ ∈ R, ∀φ, ψ,
St (sup(φ, ψ)) = sup(St φ, St ψ)
St (λ + φ)
= λ + St φ
So St is max-plus linear.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
7 / 58
Lagrange problem / Lax-Oleinik semigroup
Z
v (t, x) =
t
L(x(s), ẋ(s))ds + φ(x(t))
sup
x(0)=x, x(·)
0
Lax-Oleinik semigroup: (St )t>0 , St φ := v (t, ·).
Superposition principle: ∀λ ∈ R, ∀φ, ψ,
St (“φ + ψ”) = “St φ + St ψ”
St (“λφ”)
= “λSt φ”
So St is max-plus linear.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
7 / 58
The function v is solution of the Hamilton-Jacobi
equation
∂v
∂v
= H(x, )
∂t
∂x
v (0, ·) = φ
Max-plus linearity ⇔ Hamiltonian convex in p
H(x, p) = sup(L(x, u) + p · u)
u
Hopf formula, when L = L(u) concave:
v (t, x) = sup tL(
y ∈Rn
Stephane Gaubert (INRIA and CMAP)
x −y
) + φ(y ) .
t
Max-plus approximations
ENSEEIHT
8 / 58
The function v is solution of the Hamilton-Jacobi
equation
∂v
∂v
= H(x, )
∂t
∂x
v (0, ·) = φ
Max-plus linearity ⇔ Hamiltonian convex in p
H(x, p) = sup(L(x, u) + p · u)
u
Hopf formula, when L = L(u) concave:
Z
v (t, x) = “ G (x − y )φ(y )dy ” .
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
8 / 58
Classical
Maxplus
Expectation
sup
Brownian motion
L(ẋ(s)) = (ẋ(s))2 /2
Heat equation:
Hamilton-Jacobi equation:
1
∂v
1 ∂v 2
∂v
=
−
∆v
=
∂t
2
∂t
2 ∂x
− 12 kxk2
exp(− 12 kxk2 )
Fenchel transform:
R Fourier transform:
exp(ihx, y i)f (x)dx
supx hx, y i − f (x)
inf or sup-convolution
convolution
See Akian, Quadrat, Viot 97 Duality & Opt. . . .
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
9 / 58
Max-plus basis / finite-element method
Fleming, McEneaney 00-;
Akian, Lakhoua, SG 04- Approximate the value function by a
“linear comb.” of “basis” functions with coeffs. λi (t) ∈ R:
v (t, ·) '“
X
λi (t)wi ”
i∈[p]
The wi are given finite elements, to be chosen depending
on the regularity of v (t, ·)
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
10 / 58
Max-plus basis / finite-element method
Fleming, McEneaney 00-;
Akian, Lakhoua, SG 04- Approximate the value function by a
“linear comb.” of “basis” functions with coeffs. λi (t) ∈ R:
v (t, ·) ' sup λi (t) + wi
i∈[p]
The wi are given finite elements, to be chosen depending
on the regularity of v (t, ·)
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
10 / 58
Best max-plus approximation
P(f ) := max{g 6 f | g “linear comb.” of wi }
linear forms wi : x 7→ hyi , xi
hyi , xi
adapted if v is convex
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
11 / 58
Best max-plus approximation
P(f ) := max{g 6 f | g “linear comb.” of wi }
wi : x 7→ − C2 kxk2 + hyi , xi
− C2 kxk2 + hyi , xi
adapted if v is C -semi-convex, i.e. v + C kxk2 /2 convex
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
11 / 58
Best max-plus approximation
P(f ) := max{g 6 f | g “linear comb.” of wi }
cone like functions wi : x 7→ −C kx − xi k
xi
adapted if v is C -Lip
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
11 / 58
Max-plus basis propagation
Max-plus linearity is essential in max-plus basis method:
Vt ' Ṽt = sup λti + wi
i
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
12 / 58
Max-plus basis propagation
Max-plus linearity is essential in max-plus basis method:
Vt ' Ṽt = sup λti + wi
i
Vt+τ ' Sτ [Ṽt ]
Stephane Gaubert (INRIA and CMAP)
dynamic programming principle
Max-plus approximations
ENSEEIHT
12 / 58
Max-plus basis propagation
Max-plus linearity is essential in max-plus basis method:
Vt ' Ṽt = sup λti + wi
i
Vt+τ ' Sτ [Ṽt ]
= sup Sτ [λti + wi ]
dynamic programming principle
i
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
12 / 58
Max-plus basis propagation
Max-plus linearity is essential in max-plus basis method:
Vt ' Ṽt = sup λti + wi
i
Vt+τ ' Sτ [Ṽt ]
= sup Sτ [λti + wi ]
dynamic programming principle
i
= sup λti + Sτ [wi ]
maxplus linearity
i
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
12 / 58
Max-plus basis propagation
Max-plus linearity is essential in max-plus basis method:
Vt ' Ṽt = sup λti + wi
i
Vt+τ ' Sτ [Ṽt ]
= sup Sτ [λti + wi ]
dynamic programming principle
i
= sup λti + Sτ [wi ]
maxplus linearity
' sup λti + S̃τ [wi ]
semigroup approximation step
i
i
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
12 / 58
Max-plus basis propagation
Max-plus linearity is essential in max-plus basis method:
Vt ' Ṽt = sup λti + wi
i
Vt+τ ' Sτ [Ṽt ]
= sup Sτ [λti + wi ]
dynamic programming principle
i
= sup λti + Sτ [wi ]
maxplus linearity
' sup λti + S̃τ [wi ]
semigroup approximation step
' sup λt+τ
+ wi
i
maxplus projection step
i
i
i
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
12 / 58
Max-plus basis propagation
Max-plus linearity is essential in max-plus basis method:
Vt ' Ṽt = sup λti + wi
i
Vt+τ ' Sτ [Ṽt ]
= sup Sτ [λti + wi ]
dynamic programming principle
i
= sup λti + Sτ [wi ]
maxplus linearity
' sup λti + S̃τ [wi ]
semigroup approximation step
' sup λt+τ
+ wi
i
maxplus projection step
i
i
i
In summary:
VT (x) = {Sτ }N [V0 ] '
{P ◦ S̃τ }N [V0 ] .
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
12 / 58
The Lax-Oleinik semigroup St plays the role of the
order-preserving functional of abstract interpretation.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
13 / 58
Max-plus basis methods
Several max-plus basis methods have been proposed:
[Fleming,McEneaney 00]:
A first development of max-plus basis method
[Akian,SG,Lakhoua 06]:
A finite element max-plus basis method
[McEneaney 07]:
A curse of dimensionality free method
[McEneaney,Deshpande,SG 08],
[Sridharan,James,McEneaney 10], [Dower,McEneaney 11], ......
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
14 / 58
Maxplus finite element projector
Define the max-plus scalar product
hg |f i := sup f (x) + g (x),
g ∈ G, f ∈ F.
x∈X
Two functions f and f˜
f 6 f˜ ⇔ hg |f i 6 hg |f˜i, ∀g ∈ G.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
15 / 58
Maxplus finite element projector
The Max-plus projector equals to
PB [f ] = sup{f˜ ∈ span B : f˜ 6 f }
= sup{f˜ ∈ span B : hg |f˜ i 6 hg |f i, g ∈ G}.
Analogous to Petrov-Galerkin Method
A finite set of basis functions B ⊂ F, a finite set of
test functions Z ⊂ G
˜
˜
ΠZ
B [f ] := sup{f ∈ Span B : hz|f i 6 hz|f i, ∀z ∈ Z}
Theorem ([Cohen,SG,Quadrat 96])
Z
ΠZ
B = PB ◦ P .
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
15 / 58
Example of max-plus finite element projector
B = {−|x|2 , −|x ± 1.5|2 , −|x ± 3|2 , −|x ± 4|2 }
π
−Z = {|x|, |x ± π|, |x ± |}
2
f (x) = cos(x)
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
16 / 58
Example of max-plus finite element projector
B = {−|x|2 , −|x ± 1.5|2 , −|x ± 3|2 , −|x ± 4|2 }
π
−Z = {|x|, |x ± π|, |x ± |}
2
f (x) = cos(x)
P Z [f ] :
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
16 / 58
Example of max-plus finite element projector
B = {−|x|2 , −|x ± 1.5|2 , −|x ± 3|2 , −|x ± 4|2 }
π
−Z = {|x|, |x ± π|, |x ± |}
2
f (x) = cos(x)
PB ◦ P Z [f ] :
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
16 / 58
Max-plus finite element method
Wh = span{w1 , . . . , wp } max-plus space of finite elements
Zh = span{z1 , . . . , zq } max-plus space of test functions
Substitute v t ∼ vht := sup16j6p λtj + wj in v t+τ = S τ v t ,
and take the scalar product with every zi ,
sup λjt+τ +hwj , zi i = sup λtj +hS τ wj , zi i
16j6q
∀1 6 i 6 q .
16j6q
This is of the form Mλt+τ = K λt , M and K analogues of
the mass and stiffness matrices, respectively.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
17 / 58
Mλt+τ = K λt
Need to compute λt+τ as a function of λt . The equation
Mµ = K λt may not have a solution, so we take the
greatest solution of Mµ 6 K λt ,
λt+τ = M ] K λt
where M ] is the adjoint (it is a min-plus linear operator):
(M ] ν)j = min −Mij + νi .
16i6q
In summary: Approx. v t by vht := sup16j6p λth + w t
where the λ0j are given, and
λt+τ
= min −hwj , zi i+ max hS τ wk , zi i+λtk
j
16i6q
16k6p
t = τ, 2τ, . .
Zero-sum two player game !
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
18 / 58
To compute
λt+τ
= min −hwj , zi i+ max hS τ wk , zi i+λtk t = τ, 2τ, . . .
j
16i6q
16k6p
we need to evaluate off line
Mij = hwj , zi i = supx∈X wj (x) + zi (x)
Kik = hzi , S τ wk i = supx∈X , u(·) zi (x) +
Rτ
0
`(x(s), u(s)) ds + wk (x)
1. Typically, wj or zi are concave, say x 7→ p · x − x ∗ Ax
with A positive semidefinite, so computing Mij is a convex
programming problem (sometimes explicit formulæ).
2. Computing Kik is an optimal control problem, horizon
τ is small, final and terminal rewards wj and zi nice, so
convexity propagates ⇒ global optimum accessible by
Pontryagin under suitable assumptions. Sometimes,
Riccati!
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
19 / 58
1
Error estimates
[Akian,SG,Lakhoua 06]
T
) max kS τ [wi ] − S̃ τ [wi ]k∞,X
τ i=1,...,p
+ max kVt − PB [Vt ]k∞,X
t=0,τ,...,T
+ max kVt − P Z [Vt ]k∞,X
kVT − ṼT k∞,X 6 (1 +
t=0,τ,...,T
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
20 / 58
Error estimates
[Akian,SG,Lakhoua 06]
T
) max kS τ [wi ] − S̃ τ [wi ]k∞,X
τ i=1,...,p
+ max kVt − PB [Vt ]k∞,X
t=0,τ,...,T
+ max kVt − P Z [Vt ]k∞,X
kVT − ṼT k∞,X 6 (1 +
t=0,τ,...,T
This also applies to [Fleming,McEneaney 00]:
T
kVT − ṼT k∞,X 6 (1 + ) max kS τ [wi ] − S̃ τ [wi ]k∞,X
τ i=1,...,p
+ max kPB ◦ S̃ τ [wi ] − S̃ τ [wi ]k∞,X
i=1,...,p
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
20 / 58
Error estimates
Approximation of the semigroup (Euler scheme)
S τ [wi ] ' S̃ τ [wi ] = wi + τ H(x, ∇wi ).
When X is bounded, under some technical
assumptions,
max kS τ [wi ] − S̃ τ [wi ]k∞,X ∼ O(τ 2 ).
i=1,...,p
Maxplus projection error of a C -semiconvex function f
kPB [f ] − f k∞,X
where B = {− 12 (x − xi )> C (x − xi )}i=1,...,p .
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
21 / 58
Maxplus projection error: an upper bound
Theorem ([Akian,SG,Lakhoua 06])
Let X be a compact convex subset of Rd and f be a
C -semiconvex function and Lipschitz continuous of
Lipschitz constant L, then
kPB [f ] − f k∞,X 6 |C |ρ(X̂ ; x1 , . . . , xp ) diam X
where X̂ = X + B(0, |CL | ) and ρ(X̂ ; x1 , . . . , xp )
is the maximal radius of the Voronoi cells
of the space X̂ divided by the points {x1 , . . . , xp }.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
22 / 58
Maxplus projection error: an upper bound
Theorem ([Akian,SG,Lakhoua 06])
Let X be a compact convex subset of Rd and f be a
(C − α)-semiconvex function and Lipschitz continuous of
Lipschitz constant L, then
kPB [f ] − f k∞,X
Stephane Gaubert (INRIA and CMAP)
ρ(X̂ ; x1 , . . . , xp )2
diam X
6
α
Max-plus approximations
ENSEEIHT
22 / 58
Maxplus projection error: an upper bound
We have
kPB [f ] − f k∞,X 6 O(ρ(X̂ ; x1 , . . . , xp )2 )
It is known (covering surface with
discs [Hlawka 49, Rogers 64]) that:
min ρ(X̂ ; x1 , . . . , xp ) ∼ O(
1
1 ), as p → +∞
pd
Therefore, the minimal number of basis functions p()
needed to obtain an error of order O() is bounded by
1
p() 6 O( d )
2
.
x1 ,...,xp
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
23 / 58
Minimal number of basis functions p() needed to obtain
an error of order O() bounded by
p() 6 O(
1
d
2
)
First generation maxplus methods (Fleming-McEneaney,
Akian, SG, Lakhoua):
→ same complexity as Falcone’s grid based semilagrangean
schemes (state of the art competitor) + theory of
projection errors (explicit bounds).
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
24 / 58
Maxplus projection error: an asymptotic estimates
Strong analogy between
Covering a convex body by a
circumscribed polytope with
at most p faces
Asymptotic estimates for best
approximation of convex bodies
by P.M.Gruber .
Stephane Gaubert (INRIA and CMAP)
Projecting a convex function
into a max-plus linear
subspace generated by at
most p linear basis functions
Max-plus approximations
ENSEEIHT
25 / 58
Asymptotic estimates for best max-plus projection
Analogous result of [Gruber 93, Gruber 07]
Theorem ([SG,McEneaney,Qu 11])
Let X be a compact convex set and f ∈ C 2 (Rd : R) be a convex
function such that f 00 (x) > 0, ∀x ∈ Rd . Then,
min kPB [f ] − f k∞,X ∼ O(
x1 ,...,xp
min kPB [f ] − f k1,X ∼ O(
x1 ,...,xp
Stephane Gaubert (INRIA and CMAP)
1
2
),
as p → ∞
2
),
as p → ∞
pd
1
pd
Max-plus approximations
ENSEEIHT
26 / 58
A negative result
Corollary
Minimal number of linear forms p() to reach an approximation of
order O() is of order:
1
p() ∼ O( d )
2
A negative result for the max-plus basis method
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
27 / 58
A negative result
Corollary
Minimal number of linear forms p() to reach an approximation of
order O() is of order:
1
p() ∼ O( d )
2
A negative result for the max-plus basis method and for the dual
dynamic programming method. (if the value function is regular and
strictly convex)
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
27 / 58
Asymptotic estimates for best max-plus projection
Analogous result of [Gruber 93, Gruber 07]
Theorem ([SG,McEneaney,Qu 11])
Under the same assumptions, as p → ∞,
min kPB [f ] − f k∞,X ∼
x1 ,...,xp
C1
p
2
d
, min kPB [f ] − f k1,X ∼
x1 ,...,xp
C2
2
pd
,
where
C1 = α1
Z
1
(det(f 00 (x))) d+2 dx
d+2
d
X
C2 = α2
Z
00
1
2
(det(f (x))) dx
d2
X
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
28 / 58
Second generation maxplus methods
McEneaney’s curse of dimensionality attenuation
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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Switched optimal control problem
Infinite horizon switched optimal control problem
[McEneaney 07]:
Z ∞
1
γ2
0 µ(t)
V (x) = sup sup
x(t) D x(t) − |u(t)|2 dt,
2
2
µ
u
0
where
.
D∞ = {µ : [0, ∞) → {1, . . . , M} : measurable} ,
.
k
W = Lloc
2 ([0, ∞); R ) ,
and x(·) satisfies:
ẋ(t) = Aµ(t) x(t) + σ µ(t) u(t), x(0) = x ∈ Rd ,
arising from H∞ robust control, nonconvex (D 1 , . . . , D M < 0).
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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McEneaney’s curse of dimensionality free method
Semigroup approximation:
Sτ ' S̃τ = sup Sτm
m
Stm
is the semigroup associated to the control problem by letting
the switching control µ equal to m ∈ {1, . . . , M}:
Z t
1
γ2
m
St [φ](x) = sup
x(t)0 D m x(t) − |u(t)|2 dt + φ(x(t)).
2
u
0 2
ẋ(s) = Am x(s) + σ m u(s); x(0) = x ∈ Rd .
Stm [φ] is a quadratic function if φ is. (Riccati)
V ' VT = {Sτ }N [V0 ] ' {S̃τ }N [V0 ] =
sup SτiN ◦ . . . Sτi1 [V0 ] .
iN ,...,i1
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Max-plus approximations
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Arborescent propagation
V ' VT = {Sτ }N [V0 ] ' {S̃τ }N [V0 ] =
sup SτiN ◦ . . . Sτi1 [V0 ] .
iN ,...,i1
V0
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Max-plus approximations
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Arborescent propagation
V ' VT = {Sτ }N [V0 ] ' {S̃τ }N [V0 ] =
sup SτiN ◦ . . . Sτi1 [V0 ] .
iN ,...,i1
V0
Sτ1 [V0 ]
Stephane Gaubert (INRIA and CMAP)
......
Max-plus approximations
SτM [V0 ]
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Arborescent propagation
V ' VT = {Sτ }N [V0 ] ' {S̃τ }N [V0 ] =
sup SτiN ◦ . . . Sτi1 [V0 ] .
iN ,...,i1
V0
......
Sτ1 [V0 ]
Sτ1 Sτ1 [V0 ]
...
Stephane Gaubert (INRIA and CMAP)
SτM Sτ1 [V0 ]
SτM [V0 ]
Sτ1 SτM [V0 ] . . . SτM SτM [V0 ]
Max-plus approximations
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Arborescent propagation
V ' VT = {Sτ }N [V0 ] ' {S̃τ }N [V0 ] =
sup SτiN ◦ . . . Sτi1 [V0 ] .
iN ,...,i1
V0
......
Sτ1 [V0 ]
Sτ1 Sτ1 [V0 ]
...
Sτ1 . . . Sτ1 [V0 ]
Stephane Gaubert (INRIA and CMAP)
SτM Sτ1 [V0 ]
SτM [V0 ]
Sτ1 SτM [V0 ] . . . SτM SτM [V0 ]
......
Max-plus approximations
SτM . . . SτM [V0 ]
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Arborescent propagation
V ' VT = {Sτ }N [V0 ] ' {S̃τ }N [V0 ] =
sup SτiN ◦ . . . Sτi1 [V0 ] .
iN ,...,i1
V0
......
Sτ1 [V0 ]
Sτ1 Sτ1 [V0 ]
...
Sτ1 . . . Sτ1 [V0 ]
SτM Sτ1 [V0 ]
SτM [V0 ]
Sτ1 SτM [V0 ] . . . SτM SτM [V0 ]
......
SτM . . . SτM [V0 ]
Comput. complexity: O(M N d 3 ) ⇒curse of dimensionality free
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Max-plus approximations
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Tree approximation + pruning
Computational complexity: O(M N d 3 )
The method has been applied to solve approximately problems of
dimension d = 4, number of switches M = 3, in [McEneaney 07]
dimension d = 6, number of switches M = 6,
in [McEneaney,Deshpande,SG 08] (with a semidefinite
programming pruning technique)
dimension d = 15, number of switches M = 6,
in [Sridharan,James,McEneaney 10] (quantum optimal gate
synthesis, SU(4))
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Max-plus approximations
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Introduction
Switched infinite horizon optimal control problem
Static HJ equation:
H(x, ∇V ) = 0, ∀x ∈ Rd ; V (0) = 0 .
where H(x, p) =
sup
m∈{1,...,M}
Assumption
(existence)
Assumption Σ:
Assumption
contraction:
1 0 m
xD x
2
+ 12 p 0 Σm p + (Am x)0 p.
0 ≺ D m 4 cD Id , 0 ≺ Σm 4 cΣ Id , ∀m
x 0 Am x 6 −cA |x|2 , ∀x ∈ Rd , ∀m. cA2 > cD cΣ .
Σm = Σ, m = 1, . . . , M .
D m > mD Id , mD cΣ > (cA −
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
p
cA2 − cD cΣ )2 .
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Introduction
Error bound
Theorem (Zheng Qu, PhD 2013)
Under Assumption existence and Assumption contraction,
the computational complexity to reach an error of order is
O(M − log()/ d 3 ) .
Compare with O(1/d/r ) for a grid scheme with an error
of order (∆x)r .
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Max-plus approximations
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Introduction
Take home message :
invariant Finsler metrics on the cone of positive definite
matrices
the standard Riccati flow is a contraction in these metrics
Ṗ = A0 P + PA + D − PΣP ,
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
D, Σ > 0 .
ENSEEIHT
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Introduction
Thompson’s part metric
C closed convex pointed cone in a Banach space C ,
x 6 y iff y − x ∈ C
dT (x, y ) = inf{log α | α−1 x 6 y 6 αx}
C = Rn+ , dT (x, y ) = k log x − log y k∞ .
C = Sn+ , dT (A, B) = k spec log A−1/2 BA−1/2 k∞ =
k log spec A−1 Bk∞ .
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Max-plus approximations
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Introduction
Invariant metrics on the cone of positive matrices
Thompson’s part metric:
dT (A, B) = k log spec A−1 Bk∞ , A, B 0
dT (UAU 0 , UBU 0 ) = dT (A, B) , U ∈ GL(n)
Z 1
dT (A, B) = inf
kγ̇(t)γ(t)−1 k∞ dt.
γ
0
Riemannian metric:
Z
d2 (A, B) = inf
γ
1
kγ̇(t)γ(t)−1 k2 dt
0
= k log spec A−1 Bk2
Invariant Finsler metric, ν convex positively homogeneous
Z 1
dν (A, B) = inf
ν(γ̇(t)γ(t)−1 )dt = ν(log spec A−1 B)
γ
Stephane Gaubert (INRIA and CMAP)
0
Max-plus approximations
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Introduction
Theorem ([Bougerol 93])
The standard Riccati operator (flow) is a strict contraction mapping
in the invariant Riemannian metric.
Theorem ([Liverani and Wojtkowski.94, Lawson and Lim 07])
The standard Riccati operator (flow) is a strict contraction mapping
in Thompson’s part metric.
Theorem ([Lee and Lim 07])
The standard Riccati operator (flow) is a strict contraction mapping
in any invariant Finsler metric.
The proof relies on the symplectic structure of the standard Riccati
flow
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Max-plus approximations
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Introduction
Ingredient: contraction property of Riccati flow
For all m ∈ {1, . . . , M}, the semigroup {Stm }t corresponds to the
flow of an indefinite Riccati equation:
Ṗ = (Am )0 P + PAm + D m + PΣm P .
(1)
sign changed, −PΣm P in Bougerol, Liverani, Wojtwoski. . . !
Theorem ([SG, Qu JDE14])
Under Assumption existence and Assumption contraction, there is
P0 0 and α > 0 such that for all solutions
P1 (·), P2 (·) : [0, T ] → (0, P0 ) of the indefinite Riccati flow (1) we
have:
dT (P1 (t), P2 (t)) 6 e −αt dT (P1 (0), P2 (0)), ∀t ∈ [0, T ] .
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Max-plus approximations
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Introduction
Contraction rate in Thompson’s part metric
Denote by Mst (·) the flow associated to the ODE
ẋ(t) = φ(t, x(t))
φ is a continuous function, C 1 in the second variable with bounded
differential. Let U ⊆ int C be an open set satisfying λU ⊆ U for all
λ ∈ (0, 1].
Theorem ([SG, Qu JDE14])
If the flow is order-preserving, then the best constant α such that
dT (Mst (x1 ), Mst (x2 )) 6 e −α(t−s) dT (x1 , x2 ), s 6 t < tU (s, xi )
holds for all xi ∈ U, i = 1, 2, is given by
−1 α := − sup λmax Dφs (x)x − φ(s, x) x
.
s∈J, x∈U
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Max-plus approximations
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Introduction
Arborescent propagation
V ' VT = {Sτ }N [V0 ] ' {S̃τ }N [V0 ] =
sup SτiN ◦ . . . Sτi1 [V0 ] .
iN ,...,i1
V0
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Max-plus approximations
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Introduction
Arborescent propagation
V ' VT = {Sτ }N [V0 ] ' {S̃τ }N [V0 ] =
sup SτiN ◦ . . . Sτi1 [V0 ] .
iN ,...,i1
V0
Sτ1 [V0 ]
Stephane Gaubert (INRIA and CMAP)
......
Max-plus approximations
SτM [V0 ]
ENSEEIHT
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Introduction
Arborescent propagation
V ' VT = {Sτ }N [V0 ] ' {S̃τ }N [V0 ] =
sup SτiN ◦ . . . Sτi1 [V0 ] .
iN ,...,i1
V0
......
Sτ1 [V0 ]
Sτ1 Sτ1 [V0 ]
...
Stephane Gaubert (INRIA and CMAP)
SτM Sτ1 [V0 ]
SτM [V0 ]
Sτ1 SτM [V0 ] . . . SτM SτM [V0 ]
Max-plus approximations
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Introduction
Arborescent propagation
V ' VT = {Sτ }N [V0 ] ' {S̃τ }N [V0 ] =
sup SτiN ◦ . . . Sτi1 [V0 ] .
iN ,...,i1
V0
......
Sτ1 [V0 ]
Sτ1 Sτ1 [V0 ]
...
Sτ1 . . . Sτ1 [V0 ]
Stephane Gaubert (INRIA and CMAP)
SτM Sτ1 [V0 ]
SτM [V0 ]
Sτ1 SτM [V0 ] . . . SτM SτM [V0 ]
......
Max-plus approximations
SτM . . . SτM [V0 ]
ENSEEIHT
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Introduction
Arborescent propagation
V ' VT = {Sτ }N [V0 ] ' {S̃τ }N [V0 ] =
sup SτiN ◦ . . . Sτi1 [V0 ] .
iN ,...,i1
V0
......
Sτ1 [V0 ]
Sτ1 Sτ1 [V0 ]
...
Sτ1 . . . Sτ1 [V0 ]
SτM Sτ1 [V0 ]
SτM [V0 ]
Sτ1 SτM [V0 ] . . . SτM SτM [V0 ]
......
SτM . . . SτM [V0 ]
Comput. complexity: O(M N d 3 ) ⇒curse of dimensionality free
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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Introduction
The Pruning Problem
Given
f = sup φi ,
φi quadratic Rd → R
i∈[p]
and k p, find I ⊂ [p], |I | = k, with a best
approximation of f by
sup φi .
i∈I
Equivalent to template compression
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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Pruning algorithms
Pruning operation:
y
x
φ = sup(φgreen , φred , φviolet ,
φyellow , φblack , φblue )
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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Pruning algorithms
Pruning operation:
y
y
x
φ = sup(φgreen , φred , φviolet ,
φyellow , φblack , φblue )
Stephane Gaubert (INRIA and CMAP)
x
φ = sup(φgreen , φblack , φblue )
Max-plus approximations
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Pruning algorithms
Pruning algorithms
Let Q1 , . . . , Qn be (d + 1) × (d + 1) symmetric matrices
such that the quadratic functions are given by
x
T
φi (x) = (x 1)Qi
, i = 1, . . . , n.
1
1 Pairwise pruning
φi > φj ⇔ Qi > Qj ⇒ Remove the function φj
2 Global pruning (nonconvex quadratic programming,
NP hard)
sup φi > φj ⇒ Remove the function φj
i6=j
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Max-plus approximations
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Pruning algorithms
Pruning algorithms
Importance metric
sup φi > φj ⇔ νj := sup
x
i6=j
φj (x) − supi6=j φi (x)
60
1 + |x|2
νj is called the importance metric of the function φj .
Optimisation form
νj = max ν
ν 6 z > (Qj − Qi )z, ∀i 6= j
z T z = 1.
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Max-plus approximations
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Pruning algorithms
Pruning algorithms
Optimisation problem
SDP relaxation
νj = max ν
ν 6 z > (Qj − Qi )z
z T z = 1.
ν j = max ν
ν 6 tr((Qj − Qi )Z )
Z > 0, tr(Z ) = 1.
Conservative pruning[McEneaney,Deshpande,SG 08]:
ν j 6 0 ⇒ νj 6 0 ⇒: Remove the function φj .
Over-pruning[McEneaney,Deshpande,SG 08]:
sort ν j , keep at most k functions and prune the rest.
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Max-plus approximations
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Pruning algorithms
Pruning = facility location for Bregman dist.
Discrete points X̃ = {x1 , . . . , xN } ⊂ Rd . The loss at
point xk if we remove the function φj is:
c(j, k) := sup φi (xk ) − sup φi (xk ) > 0
i
i6=j
Combinatorial optimisation problem
Minimizing the average lost → discrete k-median problem:
N
X
min
[min c(j, k)] .
|S|=k
k=1
j∈S
Minimizing the maximal lost → discrete k-center problem:
min
max [min c(j, k)] .
|S|=k k=1,...,N j∈S
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Max-plus approximations
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Pruning algorithms
Pruning algorithms: distribution of witness points
y
x
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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Pruning algorithms
Pruning algorithms: generation of witness points
[SG,McEneaney,Qu 11]
Optimisation problem
SDP relaxation
νj = max ν
ν 6 z > (Qj − Qi )z
z T z = 1.
ν j = max ν
ν 6 tr((Qj − Qi )Z )
Z > 0, tr(Z ) = 1.
Randomization technique [Aspremont,Boyd 03]: For each
function j, let Zj be the optimal solution of the SDP,
generate random points of distribution N (0, Zj ).
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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Experimental results
0.7
0.35
← sort upper bound
tau=0.1
tau=0.05
← J−V facility location
0.6
0.3
L(|H|)
L(|H|)
← sort upper bound
0.5
← J−V facility location
0.4
0.25
← greedy facility location
0.2
← sort lower bound
0.3
0.2
5
10
15
Iteration number
20
← sort lower bound
0.15
← greedy facility location
25
0.1
20
(a) τ = 0.1
25
30
35
40
Iteration number
45
50
(b) τ = 0.05
Figure: Comparison of the four pruning techniques by the evolution of the
discrete L1 norm of backsubstitution error on the rectangle
[−2, 2] × [−2, 2] of the x1 − x2 plane, with respect to the number of
iterations.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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Experimental results
Table: CPU time
τ =0.2, K =25
sort lower
sort upper
J-V p-d
greedy
Stephane Gaubert (INRIA and CMAP)
Total time
1.04h
1.34h
1.38h
1.43h
Propagation
1.85%
1.52%
1.45%
1.63%
Max-plus approximations
SDP
98.15%
98.43%
89.47%
97.84%
Pruning
0.00%
0.05%
9.08%
0.53%
ENSEEIHT
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Experimental results
Experimental results
Instance : d = 6, M = 6 Backsubstitution error at point
x : H(x, ∇V (x)).
backsubstitution error at iteration 1, time step h=0.1
backsubstitution error at iteration 25, time step h=0.1
0.15
backsubstitution error H
backsubstitution error H
2
1.5
1
0.5
0
−0.5
2
1
2
0
−0.05
−0.1
2
1
1
0
2
−2
Stephane Gaubert (INRIA and CMAP)
x1−axis
0
−1
−1
−2
1
0
0
−1
x2−axis
0.1
0.05
x2−axis
Max-plus approximations
−1
−2
−2
x1−axis
ENSEEIHT
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Experimental results
The method has been applied to a quantum optimal
control of dimension 15 where the state space is the
unitary group SU(4), dim 15, see
[Sridharan,James,McEneaney 10].
M
X
√
H(U, p) = sup hp, −i{
vk Hk }Ui − v T Rv ,
|v |=1
k=1
for all p, U ∈ Mn (C). Here {H1 , . . . , HM } ⊂ Mn (C)
generate the Lie algebra of the special unitary group
SU(n).
wi (X ) = hPi , X i + ci , ∀X ∈ Mn (C) , i = 0, . . . , m ,
P0 , P1 , . . . , Pm ⊂ U(n) unitary, c0 , c1 , . . . , cm ∈ R.
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Max-plus approximations
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Experimental results
SDP relaxation is exact
δ̄0 = max min hPi − P0 , X i + ci − c0 .
X ∈U(n) 16i6m
(2)
The following is a SDP relaxation of the importance
metric
δ̄1 = max min hPi − P0 , X i + ci − c0 .
X ∈B(n) 16i6m
(3)
Theorem ([SG,Qu,Sridharan MTNS14])
δ̄0 = δ̄1 .
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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Experimental results
800
computation time (second)
700
bundle method
cvx(sdpt3)
600
500
400
300
200
100
0
0
1
2
3
number of basis functions m
4
4
x 10
Figure: Computation time in seconds V.S. the number of basis functions
m, obtained by the bundle method (blue curve) and by the interior point
algorithm (green curve) for solving (3)
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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Experimental results
Recent maxplus randomized method by Zh. Qu, almost
sure convergence, experimentally much faster than
McEneaney’s curse of dim scheme (103 speedup on
quantum gate synthesis problem), but no cod estimate.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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Experimental results
Concluding remarks
Max plus basis method = nonlinear templates
Reason success: easy propagation of basis functions
(eg Riccati)
Generalizes to non-switched systems (approximate a
general Hamiltonian by sup of LQ)
Max-plus linearity of the Lax-Oleinik semigroup is
used, extensions to second order PDE is Isaacs/game
PDE are less practicable.
Max-plus approx. → coarse but sometimes cod free
estimates.
In verification, happy with coarse bound (exact is
undecidable) 6= In optimal control, error → 0.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
58 / 58
Akian,Gaubert,Lakhoua 06.
The max-plus finite element method for solving
deterministic optimal control problems: basic
properties and convergence analysis.
SICON. 47(2): 817-848, 2006
Garrett Birkhoff 57.
Extensions of Jentzsch’s theorem
Trans.Amer.Math.Soc.. 85: 219-227, 1957
Philippe Bougerol 93
Kalman filtering with random coefficients and
contractions.
J.Control Optim.. 31(4): 942-959, 1993
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
58 / 58
W. H. Fleming and W. M. McEneaney.
A max-plus-based algorithm for a
Hamilton-Jacobi-Bellman equation of nonlinear
filtering
SICON, 38(3):683-710, 2000.
Falcone,M.
A numerical approach to the infinite horizon problem
of deterministic control theory
Appl. Math. Optim. 1987: 1, 1-13
Falcone, M. and Ferretti, R.
Discrete time high-order schemes for viscosity
solutions of Hamilton-Jacobi-Bellman equations.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
58 / 58
Numerische Mathematik 1994: 3, 315-344
Carlini, Elisabeth and Falcone, Maurizio and Ferretti,
Roberto
DAn efficient algorithm for Hamilton-Jacobi equations
in high dimension.
Computing and Visualization in Science 2004: 1, 15-29
F. Camilli, M. Falcone, P. Lanucara, and A. Seghini.
A domain decomposition method for Bellman
equations
Contemp.Math. 1994: 1, 477-483
Maurizio Falcone, Piero Lanucara, and Alessandra
Seghini.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
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A splitting algorithm for Hamilton-Jacobi-Bellman
equations.
Appl.Numer.Math. 1994: 15(2), 207-218
Simone Cacace, Emiliano Cristiani, Maurizio Falcone,
and Athena Picarelli.
A patchy dynamic programming scheme for a class of
Hamilton-Jacobi-Bellman equations
SIAM J.Sci.Comput.: 34(5): A2615-A2649, 2012.
Jimmie Lawson and Yongdo Lim.
A Birkhoff contraction formula with applications to
Riccati equations.
SIAM J.Control Optim.: 46(3): 930-951, 2007.
Carlangelo Liverani and Maciej P. Wojtkowski.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
58 / 58
Generalization of the Hilbert metric to the space of
positive definite matrices.
Pacific J.Math.: 166(2): 339-355, 1994.
Hosoo Lee and Yongdo Lim.
Invariant metrics, contractions and nonlinear matrix
equations.
Nonlinearity: 21(4): 857-878, 2008.
Carmeliza Navasca and Arthur J. Krener.
Patchy solutions of Hamilton-Jacobi-Bellman partial
differential equations.
Modeling, estimation and control 2007: 364, 251-270
W. M. McEneaney, A. Deshpande and S. Gaubert
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
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Curse-of-Complexity Attenuation in the
Curse-of-Dimensionality-Free Method for HJB PDEs
Proc. of the 2008 ACC :4684-4690, 2008.
S. Gaubert, W. M. McEneaney and Z. QU
Curse of dimensionality reduction in max-plus based
approximation methods: Theoretical estimates and
improved pruning algorithms
Proc. of the 2011 CDC :1054-1061, 2011.
W. M. McEneaney
A curse-of-dimensionality-free numerical method for
solution of certain HJB PDEs
SICON 46(4):1239-1076, 2007.
W. M. McEneaney
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
58 / 58
Idempotent algorithms for discrete-time stochastic
control through distributed dynamic programming
Proc. of the 2009 CDC :1569-1574, 2009.
J-M. Bony
Principe du maximum, ingalit de harnack et unicit du
problme de cauchy pour les oprateurs elliptiques
dgnrs.
Anna : 19:277-304, 1969.
Haim Brezis
On a characterization of flow-invariant sets
Comm. Pure Appl. Math : 23:261-263, 1970.
R.H.Martin
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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Differential equations on closed subsets of a Banach
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Trans. Amer. Math. Soc. : 179: 399-414, 1973.
H. Kaise , W. M. McEneaney
Idempotent expansions for continuous-time stochastic
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Proc. of the 2010 CDC :7015-7020, 2010.
P.M.Gruber
Asymptotic estimates for best and stepwise
approximation of convex bodies.i
Forum Math. 5(5):281-297, 1993.
P.M.Gruber
Convex and discrete geometry
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
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Springer, Berlin. 2007.
E. Hlawka
Ausfüllung und Überdeckung konvexer Körper durch
konvexe Körper
Monatsh. Math. 53(1):81-131, 1949.
C. A. Rogers
Packing and covering
Cambridge University Press. 1964.
A.Aspremont, S.Boyd
Relaxations and Randomized Methods for Nonconvex
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Stanford University. 2003.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
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H. J. Kushner and P. Dupuis.
Numerical methods for for stochastic control problems
in continuous time.
Springer-Verlag, New York 2001.
Crandall, M. G. and Lions, P.-L.
Two approximations of solutions of Hamilton-Jacobi
equations.
Mathematics of Computation 1984.
Dower, P.M. and McEneaney, W.M.
A max-plus based fundamental solution for a class of
infinite dimensional Riccati equations
Proc. of the 2011 CDC :615 -620, 2011.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
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Sridharan, Srinivas and Gu, Mile and James, Matthew
R. and McEneaney, William M.
Reduced-complexity numerical method for optimal
gate synthesis
Phys. Rev. A 82(4):042319, 2010.
G.Cohen, S.Gaubert and J-P. Quadrat
Kernels,images and projections in dioids
Proc. of WODES, 1996
W.M.McEneaney and J.Kluberg
Convergence Rate for a Curse-of-Dimensionality-Free
Method for a Class of HJB PDEs
SIAM J. Control Optim., 48(5),3052-3079,2009.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
ENSEEIHT
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S.Gaubert, Z.Qu
The contraction rate in Thompson metric of
order-preserving flows on a cone - application to
generalized Riccati equations.
J. Diff. Eq., 2014.
Z.Qu
Contraction of Riccati flows applied to the
convergence analysis of a max-plus curse of
dimensionality free method
submitted to SIAM J. Control and Opt., 2012
Z.Qu
Numerical methods for Hamilton-Jacobi equations and
nonlinear Perron-Frobenius theory
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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PhD. thesis, 2013
S.Gaubert, Z. Qu, S. Sridharan.
Bundle-based pruning in the max-plus curse of
dimensionality free method
MTNS’2014, to appear.
Stephane Gaubert (INRIA and CMAP)
Max-plus approximations
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