Asymptotic behavior of Multiscaled Gradient Dynamics. Applications - - PowerPoint PPT Presentation

Sixi` emes journ´ ees Franco-Chiliennes d’Optimisation Universit´ e de Toulon 19-21 mai 2008

Asymptotic behavior of Multiscaled Gradient Dynamics. Applications to Coupled systems, Games and PDE’s.

Hedy ATTOUCH (Joint work with M.-O. Czarnecki)

Institut de Math´ ematiques et de Mod´ elisation de Montpellier, UMR CNRS 5149, Universit´ e de Montpellier 2 Supported by ANR under grant ANR-05-BLAN-0248

SETTING

• H Hilbert space
• Φ : H → IR ∪ {+∞} closed convex proper function.
• Ψ : H → IR+ ∪ {+∞} closed convex proper function, C = argminΨ = Ψ−1(0) = ∅.
• β : IR+ → IR+ a function of t which tends to +∞ as t goes to +∞.

(MAG) ˙ z(t) + ∂Φ(z(t)) + β(t)∂Ψ(z(t)) ∋ 0. Φ + β(t)Ψ ↑ Φ + δC as t → +∞: (MAG)= “Multiscale Asymptotic Gradient” system. Claim: Under ad hoc conditions on β, Φ, Ψ, z(t) → z∞ ∈ argminCΦ as t → +∞. Motivation: Dynamic and Algorithmic approach to Optimization and Potential Games: min {f(x) + g(y) : Ax − By = 0} .

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COUPLED GRADIENT SYSTEMS

• H = X × Y the cartesian product of two Hilbert spaces, z = (x, y).
• Φ(z) = f(x) + g(y) , f ∈ Γ0(X), g ∈ Γ0(Y ).
• Ψ(z) = 1

2Ax − By2, A and B linear continuous operators.

(MAG) ˙ z(t) + ∂Φ(z(t)) + β(t)∂Ψ(z(t)) ∋ 0.

• ˙

x(t) + ∂f(x(t)) + β(t)At(Ax(t) − By(t)) ∋ 0 ˙ y(t) + ∂g(y(t)) + β(t)Bt(By(t) − Ax(t)) ∋ 0 Claim: z(t) = (x(t), y(t)) → z∞ = (x∞, y∞) where (x∞, y∞) is a solution of min {f(x) + g(y) : Ax − By = 0}.

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EXAMPLE 1: DECOMPOSITION OF DOMAINS IN PDE’s. Ω1 Ω2 Γ Dirichlet problem on Ω: h ∈ L2(Ω) given, find z : Ω → IR solution of −∆z = h on Ω z = 0 on ∂Ω Variational formulation: min

• 1

2

• Ω1 ∇z1|2 + 1

2

• Ω2 |∇z2|2 −
• Ω hz :

z1 ∈ X1, z2 ∈ X2, [z] = 0

• n Γ
• .
• Xi = {z ∈ H1(Ωi), z = 0 on ∂Ω ∩ ∂Ωi}, z = zi on Ωi, i = 1, 2.
• [z] = jump of z through the interface Γ.

min {f1(z1) + f2(z2) : z1 ∈ X1, z2 ∈ X2, A1(z1) − A2(z2) = 0} . Ai : H1(Ωi) → Z = L2(Γ) is the trace operator, i = 1, 2.

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EXAMPLE 2: POTENTIAL GAMES, BEST RESPONSE DYNAMICS Static loss functions of players 1 and 2:    F : (ξ, η) ∈ X × Y → F(ξ, η) = f(ξ) + βΨ(ξ, η) G : (ξ, η) ∈ X × Y → G(ξ, η) = g(η) + µΨ(ξ, η). Best reply dynamic with cost to change, (players 1 and 2 play alternatively): zk = (xk, yk) − → (xk+1, yk) − → zk+1 = (xk+1, yk+1) k = 0, 1, ... xk+1 = argmin{f(ξ) + βkΨ(ξ, yk) + α

2 ξ − xk 2 X: ξ ∈ X}

yk+1 = argmin{g(η) + βkΨ(xk+1, η) + ν

2 η − yk 2 Y: η ∈ Y}

Corresponding continuous dynamical system (MAGS): z(t) = (x(t), y(t))

• ˙

x(t) + ∂f(x(t)) + β(t)∇xΨ(x(t), y(t)) ∋ 0 ˙ y(t) + ∂g(y(t)) + β(t)∇yΨ(x(t), y(t)) ∋ 0 β(t) → +∞ as t → +∞ = increasing weight of the cooperative behaviour aspects.

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CONTENTS

• 1. Multiscale features. Slow-Fast dynamics.
• 2. Ergodic convergence results:

2.1 β(t) → +∞. 2.2 ǫ(t) → 0. 2.3 Links with Passty theorem.

• 3. From ergodic convergence to convergence.

3.1 β(t) → +∞: the general case. 3.2 β(t) → +∞: the strongly monotone case. 3.3 β(t) → +∞: the finite dimensional case. 3.4 ǫ(t) → 0.

• 4. Rate of convergence results.
• 5. Applications to

4.1 domain decomposition for PDE’s. 4.2 potential games and best response dynamics.

• 6. Perspectives.
• 7. References.

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MULTISCALE FEATURES. SLOW-FAST DYNAMICS 1. (MAG) ˙ z(t) + ∂Φ(z(t)) + β(t)∂Ψ(z(t)) ∋ 0 is the combination of two dynamics:

• A slow dynamic: (1)

˙ z(t) + ∂Φ(z(t)) ∋ 0.

• A fast dynamic: (2)

˙ z(t) + β(t)∂Ψ(z(t)) ∋ 0. Change of time scaling in (2): take t = τ(s) and set z(τ(s)) = w(s). (2) ⇔

1 ˙ τ(s)β(τ(s)) ˙

w(s) + ∂Ψ(w(s)) ∋ 0. Take ˙ τ(s)β(τ(s)) = 1, i.e., τ(s) β(ξ)dξ = s. Assume +∞ β(ξ)dξ = +∞, then (2) ⇔ ˙ w(s) + ∂Ψ(w(s)) ∋ 0.

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MULTISCALE FEATURES. SLOW-FAST DYNAMICS 2. (MAG) ˙ z(t) + ∂Φ(z(t)) + β(t)∂Ψ(z(t)) ∋ 0. Change of time scaling: take t = τ(s) and set z(τ(s)) = w(s), ǫ(s) =

1 β(τ(s)).

Equivalent system with ǫ(s) → 0 as s → +∞, +∞ ǫ(s)ds = +∞. ˙ w(s) + ǫ(s)∂Φ(w(s)) + ∂Ψ(w(s)) ∋ 0. From ˙ τ(s)β(τ(s)) = 1 , ˙ τ(s) =

1 β(τ(s)) = ǫ(s), and

+∞ ǫ(s)ds = lims→+∞τ(s) = +∞. Classical situation: Φ(w) = 1

2w2 ⇒ Asymptotic Tikhonov selection property:

Att.-Cominetti, Att.-Czarnecki, Cabot, Combettes-Hirstoaga, Peypouquet.

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ERGODIC CONVERGENCE RESULTS: β(t) → +∞ (MAG) ˙ z(t) + A(z(t)) + β(t)∂Ψ(z(t)) ∋ 0.

• A : H → H general maximal monotone operator.
• Ψ : H → IR+ ∪ {+∞} closed convex proper, C = argminΨ = Ψ−1(0) = ∅.

Ψ∗= Fenchel conjugate of Ψ, σC= support function of C, NC= normal cone to C. Theorem 1 [A.-C.] Let us assume that,

• (H0)

A + NC is a maximal monotone operator and S := (A + NC)−1(0) = ∅ closed convex set.

• (H1)

∀p ∈ NC +∞ β(t)

• Ψ∗( p

β(t)) − σC( p β(t))

• dt < +∞.

Then,

• w − limt→+∞

1 t

t

0 z(s)ds = z∞ exists with z∞ ∈ S.

• ∀a ∈ S limt→+∞z(t) − a2 exists.
• +∞

β(t)Ψ(z(t))dt < +∞.

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Interpretation of the condition (H1) ∀p ∈ NC +∞ β(t)

• Ψ∗( p

β(t)) − σC( p β(t))

• dt < +∞.
• Model situation: Ψ(z) = 1

2dist2(z, C) = 1 2.2 ∇ δC.

Ψ∗(z) = 1

2z2 + σC(z) and Ψ∗(z) − σC(z) = 1 2z2.

(H1) ⇔ +∞

1 β(t)dt < +∞.

• Ψ = 0. Then (H1) is automatically satisfied (NC = 0) and

Theorem 1 ⇔ Baillon-Brezis ergodic convergence theorem for ˙ z(t) + A(z(t)) ∋ 0 with A maximal monotone operator.

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ERGODIC CONVERGENCE RESULTS: ǫ(t) → 0 Equivalent system with ǫ(s) → 0 as s → +∞, +∞ ǫ(s)ds = +∞. ˙ w(s) + ǫ(s)A(w(s)) + ∂Ψ(w(s)) ∋ 0.

• A : H → H general maximal monotone operator.
• Ψ : H → IR+ ∪ {+∞} closed convex proper, C = argminΨ = Ψ−1(0) = ∅.

Theorem 2 [A.-C.] Let us assume that,

• (H0)

A + NC is a maximal monotone operator and S := (A + NC)−1(0) = ∅.

• (H1)

∀p ∈ NC +∞ [Ψ∗(ǫ(s)p) − σC(ǫ(s)p)] ds < +∞. Then, w − lims→+∞ 1 s

0 ǫ(τ)dτ

s w(τ)ǫ(τ)dτ = w∞ exists with w∞ ∈ S.

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LINKS WITH PASSTY THEOREM

• H = X × Y the cartesian product of two Hilbert spaces, X = Y , z = (x, y).
• A and B two maximal monotone operators, M(z) = M(x, y) = (Ax, By).
• Ψ(z) = 1

2x − y2 (strong coupling).

˙ w(s) + ǫ(s)M(w(s)) + ∂Ψ(w(s)) ∋ 0.

• ˙

x(s) + ǫ(s)A(x(s)) + x(s) − y(s) ∋ 0 ˙ y(t) + ǫ(s)B(y(s)) + y(s) − x(s) ∋ 0 Discrete version:

• xk+1 − xk + ǫ(sk)A(xk+1) + xk − yk ∋ 0

yk+1 − yk + ǫ(sk)B(yk+1) + yk − xk+1 ∋ 0 yk+1 = (I + ǫkB)−1(I + ǫkA)−1yk Theorem [Passty, JMMA, 1979]: Suppose (ǫk)k∈II N ∈ l2(II N) \ l1(II N), then zn =

1 n

1 ǫk

n

1 ǫkyk → z∞ weakly in X with Az∞ + Bz∞ ∋ 0.

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FROM ERGODIC CONVERGENCE TO CONVERGENCE: β(t) → +∞ Take A = ∂Φ a subdifferential operator, and use energy estimates. (MAG) ˙ z(t) + ∂Φ(z(t)) + β(t)∂Ψ(z(t)) ∋ 0. Theorem 3 [A.-C.] Let us assume

• (H0), (H1).
• (H2)

β : IR+ → IR+ is a smooth (C1) increasing function and there exists some positive constant k > 0 such that for t large enough: ˙ β(t) ≤ kβ(t). Then,

• w − limt→+∞ z(t) = z∞

exists with z∞ ∈ S.

• limt→+∞ Ψ(z(t)) = 0.
• limt→+∞ Φ(z(t)) = infCΦ.

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STRONG CONVERGENCE RESULTS (MAG) ˙ z(t) + A(z(t)) + β(t)∂Ψ(z(t)) ∋ 0.

• A : H → H maximal monotone operator which is strongly monotone, i.e.,

∃α > 0 such that Au − Av, u − v ≥ αu − v2 ∀u, v ∈ H.

• Ψ : H → IR+ ∪ {+∞} closed convex proper, C = argminΨ = Ψ−1(0) = ∅.

Theorem 4 [A.-C.] Let us assume that A is a strongly monotone operator and

• (H0)

A + NC is a maximal monotone operator.

• β(t) → +∞ as t → +∞.

Then,

• S = (A + NC)−10 is reduced to a single element z.
• s − limt→+∞ z(t) = z .

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CONVERGENCE RESULTS: THE FINITE DIMENSIONAL CASE (MAG) ˙ z(t) + ∂Φ(z(t)) + β(t)∂Ψ(z(t)) ∋ 0. Equivalent system with ǫ(s) → 0 as s → +∞, +∞ ǫ(s)ds = +∞ : ˙ w(s) + ǫ(s)∂

• Φ +

1 ǫ(s)Ψ

• (w(s)) ∋ 0.
• Φ +

1 ǫ(s)Ψ Mosco epi-converges to Φ + δC as s → +∞.

• Φ +

1 ǫ(s)Ψ converges uniformly to Φ + δC on S = argmin (Φ + δC).

From Baillon and Cominetti, J. Funct. Analysis (2001) dist(w(s), S) tends to 0 as s → +∞. Combining with the Fejer monotonicity property (valid under (H1) ) + Opial’s lemma ⇒:

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(MAG) ˙ z(t) + ∂Φ(z(t)) + β(t)∂Ψ(z(t)) ∋ 0. Theorem 5 [A.-C.] Let us assume that

• H is a finite dimensional space and S = argminCΦ is a bounded set.
• β(t) → +∞ as t → +∞.
• (H1)

∀p ∈ NC +∞ β(t)

• Ψ∗( p

β(t)) − σC( p β(t))

• dt < +∞.

Then,

• limt→+∞ z(t) = z∞

exists with z∞ ∈ S.

• limt→+∞ Ψ(z(t)) = 0.
• limt→+∞ Φ(z(t)) = infCΦ.

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CONVERGENCE RESULTS: ǫ(t) → 0 Equivalent system with ǫ(s) → 0 as s → +∞, +∞ ǫ(s)ds = +∞ : ˙ w(s) + ǫ(s)∂Φ(w(s)) + ∂Ψ(w(s)) ∋ 0.

• Φ : H → IR ∪ {+∞} closed convex proper function.
• Ψ : H → IR+ ∪ {+∞} closed convex proper, C = argminΨ = Ψ−1(0) = ∅.

Theorem 6 [A.-C.] Let us assume that,

• (H0)

∂Φ + NC is a maximal monotone operator and S := (∂Φ + NC)−1(0) = ∅.

• (H1)

∀p ∈ NC +∞ [Ψ∗(ǫ(s)p) − σC(ǫ(s)p)] ds < +∞.

• (H2)

There exists some k > 0 such that for s large enough − ˙

ǫ(s) ǫ2(s) ≤ k.

Then, weak − limt→+∞ w(t) = w∞ exists with w∞ ∈ S. limt→+∞ Ψ(w(t)) = 0, limt→+∞ Φ(w(t)) = infCΦ.

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DECOMPOSITION OF DOMAINS IN PDE’s. Ω1 Ω2 Γ Dirichlet problem on Ω: h ∈ L2(Ω) given, find u : Ω → IR solution of −∆u = h on Ω u = 0 on ∂Ω Variational formulation: min

• 1

2

• Ω1 ∇v1|2 + 1

2

• Ω2 |∇v2|2 −
• Ω hv :

v1 ∈ X1, v2 ∈ X2, [v] = 0

• n Γ
• .
• Xi = {v ∈ H1(Ωi), v = 0 on ∂Ω ∩ ∂Ωi}, v = vi on Ωi, i = 1, 2.
• [v] = jump of v through the interface Γ.

min {f1(v1) + f2(v2) : v1 ∈ X1, v2 ∈ X2, A1(v1) − A2(v2) = 0} . Ai : H1(Ωi) → Z = L2(Γ) is the trace operator, i = 1, 2.

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Continuous dynamical system:                −∆∂u1

∂t − ∆u1 = h1 on Ω1

− ∆∂u2

∂t − ∆u2 = h2 on Ω2 ∂ ˙ u1(t) ∂ν1 + ∂u1 ∂ν1(t) − β(t) [u(t)] = 0 on Γ ∂ ˙ u2(t) ∂ν2 + ∂u2 ∂ν2(t) + β(t) [u(t)] = 0 on Γ

Discrete version: Alternating Algorithm with Dirichlet-Neumann transmission conditions: (u1,k, u2,k) → (u1,k+1, u2,k) → (u1,k+1, u2,k+1) with βk → +∞.        −(1 + α)∆u1,k+1 = h1 − α∆u1,k on Ω1 (1 + α)

∂u1,k+1 ∂ν1

+ βku1,k+1 = βku2,k + α

∂u1,k ∂ν1

• n Γ

u1,k+1 = 0 on ∂Ω1 ∩ ∂Ω          −(1 + α)∆u2,k+1 = h2 − α∆u2,k on Ω2 (1 + α)

∂u2,k+1 ∂ν2

+ βku2,k+1 = βku1,k+1 + α

∂u2,k ∂ν2

• n Γ

u2,k+1 = 0 on ∂Ω2 ∩ ∂Ω

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POTENTIAL GAMES AND BEST RESPONSE DYNAMICS Static loss functions of players 1 and 2:    F : (ξ, η) ∈ X × Y → F(ξ, η) = f(ξ) + βΨ(ξ, η) G : (ξ, η) ∈ X × Y → G(ξ, η) = g(η) + µΨ(ξ, η). Best reply dynamic with cost to change, (players 1 and 2 play alternatively): (xk, yk) − → (xk+1, yk) − → (xk+1, yk+1) k = 0, 1, ... xk+1 = argmin{f(ξ) + βkΨ(ξ, yk) + α

2 ξ − xk 2 X: ξ ∈ X}

yk+1 = argmin{g(η) + βkΨ(xk+1, η) + ν

2 η − yk 2 Y: η ∈ Y}

Corresponding continuous dynamical system (MAGS):

• ˙

x(t) + ∂f(x(t)) + β(t)∇xΨ(x(t), y(t)) ∋ 0 ˙ y(t) + ∂g(y(t)) + β(t)∇yΨ(x(t), y(t)) ∋ 0 β(t) → +∞ as t → +∞ = increasing weight of the collective behaviour aspects.

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PERSPECTIVES

• ˙

β(t) ≤ kβ(t) optimal? Examples, counterexamples.

• Is (MAG) asymptotically almost-equivalent to an autonomous gradient-like system (gradient-

projection...)?

• Can one drop the qualification assumption? (variational sum of monotone operators...)
• From Penalization to Lagrangian and augmented Lagrangian methods.
• From first to second order time derivatives differential systems: control of oscillations,

synchronization.

• From continuous to discrete dynamics. Passty theorem with general (weak) coupling .
• From convex to non-convex analysis. Kurdyka-Lojasiewicz inequality for gradient systems.
• Parallel computing via forward-backward algorithm, Att-Bruceno-Combettes.
• Numerical developments for domain decomposition problems.
• Potential games, coordination games, learning cooperative behaviours.

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with non-isolated equilibria, J. Differential Equations, 2002.

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coupled convex minimization problems, Journal of Convex Analysis, 2008.

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equations, J. Funct. Analysis, 2001.

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