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 = argmin Ψ = Ψ − 1 (0) � = ∅ . • β : IR + → IR + a function of t which tends to + ∞ as t goes to + ∞ . z ( t ) + ∂ Φ( z ( t )) + β ( t ) ∂ Ψ( z ( t )) ∋ 0 . ( MAG ) ˙ 1 Φ + β ( t )Ψ ↑ Φ + δ C as t → + ∞ : (MAG)= “Multiscale Asymptotic Gradient” system. Claim : Under ad hoc conditions on β, Φ , Ψ , z ( t ) → z ∞ ∈ argmin C Φ as t → + ∞ . Motivation : Dynamic and Algorithmic approach to Optimization and Potential Games: min { f ( x ) + g ( y ) : Ax − By = 0 } .

COUPLED GRADIENT SYSTEMS • H = X × Y the cartesian product of two Hilbert spaces, z = ( x, y ) . • Φ( z ) = f ( x ) + g ( y ) , f ∈ Γ 0 ( X ) , g ∈ Γ 0 ( Y ) . • Ψ( z ) = 1 2 � Ax − By � 2 , A and B linear continuous operators. ( MAG ) z ( t ) + ∂ Φ( z ( t )) + β ( t ) ∂ Ψ( z ( t )) ∋ 0 . ˙ � 2 � x ( t ) + ∂f ( x ( t )) + β ( t ) A t ( Ax ( t ) − By ( t )) ∋ 0 ˙ y ( t ) + ∂g ( y ( t )) + β ( t ) B t ( 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 } .

EXAMPLE 1: DECOMPOSITION OF DOMAINS IN PDE’s. Ω 1 Γ Ω 2 Dirichlet problem on Ω : h ∈ L 2 (Ω) given, find z : Ω → IR solution of � − ∆ z = h on Ω z = 0 on ∂ Ω Variational formulation : 3 � � Ω 1 ∇ z 1 | 2 + 1 Ω 2 |∇ z 2 | 2 − 1 � � � z 1 ∈ X 1 , z 2 ∈ X 2 , [ z ] = 0 min Ω hz : on Γ . 2 2 • X i = { z ∈ H 1 (Ω i ) , z = 0 on ∂ Ω ∩ ∂ Ω i } , z = z i on Ω i , i = 1 , 2 . • [ z ] = jump of z through the interface Γ . min { f 1 ( z 1 ) + f 2 ( z 2 ) : z 1 ∈ X 1 , z 2 ∈ X 2 , A 1 ( z 1 ) − A 2 ( z 2 ) = 0 } . A i : H 1 (Ω i ) → Z = L 2 (Γ) is the trace operator , i = 1 , 2 .

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 ) + β Ψ( ξ, η )  G : ( ξ, η ) ∈ X × Y → G ( ξ, η ) = g ( η ) + µ Ψ( ξ, η ) .  Best reply dynamic with cost to change, (players 1 and 2 play alternatively): z k = ( x k , y k ) − → ( x k +1 , y k ) − → z k +1 = ( x k +1 , y k +1 ) k = 0 , 1 , ... 4 � x k +1 = argmin { f ( ξ ) + β k Ψ( ξ, y k ) + α 2 � ξ − x k � 2 X : ξ ∈ X} y k +1 = argmin { g ( η ) + β k Ψ( x k +1 , η ) + ν 2 � η − y k � 2 Y : η ∈ Y} Corresponding continuous dynamical system (MAGS): z ( t ) = ( x ( t ) , y ( t )) � x ( t ) + ∂f ( x ( t )) + β ( t ) ∇ x Ψ( x ( t ) , y ( t )) ∋ 0 ˙ y ( t ) + ∂g ( y ( t )) + β ( t ) ∇ y Ψ( x ( t ) , y ( t )) ∋ 0 ˙ β ( t ) → + ∞ as t → + ∞ = increasing weight of the cooperative behaviour aspects.

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. 5 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.

MULTISCALE FEATURES. SLOW-FAST DYNAMICS 1. z ( t ) + ∂ Φ( z ( t )) + β ( t ) ∂ Ψ( z ( t )) ∋ 0 ( MAG ) ˙ 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 . ˙ 6 Change of time scaling in (2): take t = τ ( s ) and set z ( τ ( s )) = w ( s ) . 1 (2) ⇔ w ( s ) + ∂ Ψ( w ( s )) ∋ 0 . τ ( s ) β ( τ ( s )) ˙ ˙ � τ ( s ) Take ˙ τ ( s ) β ( τ ( s )) = 1 , i.e., β ( ξ ) dξ = s. 0 � + ∞ β ( ξ ) dξ = + ∞ , then Assume 0 (2) ⇔ ˙ w ( s ) + ∂ Ψ( w ( s )) ∋ 0 .

MULTISCALE FEATURES. SLOW-FAST DYNAMICS 2. z ( t ) + ∂ Φ( z ( t )) + β ( t ) ∂ Ψ( z ( t )) ∋ 0 . ( MAG ) ˙ 1 Change of time scaling : take t = τ ( s ) and set z ( τ ( s )) = w ( s ) , ǫ ( s ) = β ( τ ( s )) . � + ∞ Equivalent system with ǫ ( s ) → 0 as s → + ∞ , ǫ ( s ) ds = + ∞ . 0 7 w ( s ) + ǫ ( s ) ∂ Φ( w ( s )) + ∂ Ψ( w ( s )) ∋ 0 . ˙ � + ∞ 1 From ˙ τ ( s ) β ( τ ( s )) = 1 , ˙ τ ( s ) = β ( τ ( s )) = ǫ ( s ) , and ǫ ( s ) ds = lim s → + ∞ τ ( s ) = + ∞ . 0 2 � w � 2 ⇒ Asymptotic Tikhonov selection property: Classical situation : Φ( w ) = 1 Att.-Cominetti, Att.-Czarnecki, Cabot, Combettes-Hirstoaga, Peypouquet.

ERGODIC CONVERGENCE RESULTS: β ( t ) → + ∞ z ( t ) + A ( z ( t )) + β ( t ) ∂ Ψ( z ( t )) ∋ 0 . ( MAG ) ˙ • A : H → H general maximal monotone operator. • Ψ : H → IR + ∪ { + ∞} closed convex proper, C = argmin Ψ = Ψ − 1 (0) � = ∅ . Ψ ∗ = Fenchel conjugate of Ψ , σ C = support function of C , N C = normal cone to C . Theorem 1 [A.-C.] Let us assume that, 8 A + N C is a maximal monotone operator and S := ( A + N C ) − 1 (0) � = ∅ closed • ( H 0 ) convex set. � + ∞ � � Ψ ∗ ( p β ( t ) ) − σ C ( p • ( H 1 ) ∀ p ∈ N C dt < + ∞ . β ( t ) β ( t ) ) 0 Then, � t 1 • w − lim t → + ∞ 0 z ( s ) ds = z ∞ exists with z ∞ ∈ S. t • ∀ a ∈ S lim t → + ∞ � z ( t ) − a � 2 exists. � + ∞ • β ( t )Ψ( z ( t )) dt < + ∞ . 0

Interpretation of the condition � + ∞ � � Ψ ∗ ( p β ( t ) ) − σ C ( p ( H 1 ) ∀ p ∈ N C dt < + ∞ . β ( t ) β ( t ) ) 0 2 � . � 2 ∇ δ C . • Model situation: Ψ( z ) = 1 2 dist 2 ( z, C ) = 1 2 � z � 2 + σ C ( z ) and Ψ ∗ ( z ) − σ C ( z ) = 1 Ψ ∗ ( z ) = 1 2 � z � 2 . 9 � + ∞ 1 ( H 1 ) ⇔ β ( t ) dt < + ∞ . 0 • Ψ = 0 . Then ( H 1 ) is automatically satisfied ( N C = 0 ) and Theorem 1 ⇔ Baillon-Brezis ergodic convergence theorem for z ( t ) + A ( z ( t )) ∋ 0 ˙ with A maximal monotone operator.

ERGODIC CONVERGENCE RESULTS: ǫ ( t ) → 0 � + ∞ Equivalent system with ǫ ( s ) → 0 as s → + ∞ , ǫ ( s ) ds = + ∞ . 0 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 = argmin Ψ = Ψ − 1 (0) � = ∅ . 10 Theorem 2 [A.-C.] Let us assume that, A + N C is a maximal monotone operator and S := ( A + N C ) − 1 (0) � = ∅ . • ( H 0 ) � + ∞ [Ψ ∗ ( ǫ ( s ) p ) − σ C ( ǫ ( s ) p )] ds < + ∞ . • ( H 1 ) ∀ p ∈ N C 0 Then, � s 1 w − lim s → + ∞ w ( τ ) ǫ ( τ ) dτ = w ∞ exists with w ∞ ∈ S. � s 0 ǫ ( τ ) dτ 0

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 ) . 2 � x − y � 2 (strong coupling). • Ψ( z ) = 1 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 ˙ 11 Discrete version : � x k +1 − x k + ǫ ( s k ) A ( x k +1 ) + x k − y k ∋ 0 y k +1 − y k + ǫ ( s k ) B ( y k +1 ) + y k − x k +1 ∋ 0 y k +1 = ( I + ǫ k B ) − 1 ( I + ǫ k A ) − 1 y k N ∈ l 2 (II N) \ l 1 (II Theorem [Passty, JMMA, 1979] : Suppose ( ǫ k ) k ∈ II N) , then � n 1 1 ǫ k y k → z ∞ weakly in X with Az ∞ + Bz ∞ ∋ 0 . z n = � n 1 ǫ k

FROM ERGODIC CONVERGENCE TO CONVERGENCE: β ( t ) → + ∞ Take A = ∂ Φ a subdifferential operator, and use energy estimates. z ( t ) + ∂ Φ( z ( t )) + β ( t ) ∂ Ψ( z ( t )) ∋ 0 . ( MAG ) ˙ Theorem 3 [A.-C.] Let us assume • ( H 0 ) , ( H 1 ) . β : IR + → IR + is a smooth ( C 1 ) increasing function and there exists some positive • ( H 2 ) constant k > 0 such that for t large enough: 12 ˙ β ( t ) ≤ kβ ( t ) . Then, • w − lim t → + ∞ z ( t ) = z ∞ exists with z ∞ ∈ S . • lim t → + ∞ Ψ( z ( t )) = 0 . • lim t → + ∞ Φ( z ( t )) = inf C Φ .

STRONG CONVERGENCE RESULTS z ( t ) + A ( z ( t )) + β ( t ) ∂ Ψ( z ( t )) ∋ 0 . ( MAG ) ˙ • A : H → H maximal monotone operator which is strongly monotone, i.e., ∃ α > 0 such that � Au − Av, u − v � ≥ α � u − v � 2 ∀ u, v ∈ H. • Ψ : H → IR + ∪ { + ∞} closed convex proper, C = argmin Ψ = Ψ − 1 (0) � = ∅ . 13 Theorem 4 [A.-C.] Let us assume that A is a strongly monotone operator and • ( H 0 ) A + N C is a maximal monotone operator. • β ( t ) → + ∞ as t → + ∞ . Then, • S = ( A + N C ) − 1 0 is reduced to a single element z. • s − lim t → + ∞ z ( t ) = z .