Algorithmes de minimisation alternée avec frottement: H. Attouch

Transcription

1 Algorithmes de minimisation alternée avec frottement: Applications aux E.D.P., problèmes inverses et jeux dynamiques. H. Attouch Institut de Mathématiques et de Modélisation de Montpellier UMR CNRS 5149 programme ANR décisionprox (UMII-Paris 6) Séminaire du laboratoire de Mathématiques de l Université de Saint-Etienne (LaMUSE) 10 Mai 2007, Université Saint-Etienne.

2 Contents: 1. Classical alternating minimization algorithms 1.1 Von Neumann theorem. 1.2 Acker-Prestel theorem. 1.3 Application to the inverse Cauchy problem. 1.4 Passty Theorem. 2. Link with decision sciences and dynamical games. 2.1 Dynamical decision with costs to change. 2.2 Proximal algorithms in decision sciences. 2.3 Alternating proximal algorithms and Nash potential games. 3. Convergence results for proximal dynamics 3.1 The convex case. 3.2 The analytic and subanalytic cases. 3.3 Subdifferential calculus for nonsmooth subanalytic functions. 4. Alternating proximal minimization: the convex case. 4.1 Strong coupling. Attouch-Redont-Soubeyran (SIAM J. Opt.) 4.2 Weak coupling. 4.3 Splitting algorithms for P.D.E. 5. Alternating proximal minimization: the subanalytic case. Attouch-Bolte (Math. Programming 2007)

3 1 Alternating minimization algorithms. Some classical results. 1.1 Von Neumann theorem (1933), Annals of Math. (1950) H Hilbert C, D closed affine subspaces of H, C D x 0 H, x 1 = proj C x 0, x 2 = proj D x 1,... Then x k s H proj C D x 0 as k +. x 2 x 4 x 6 x 7 x 5 x 3 D x 0 x 1 C lim x 2k = lim x 2k+1 = proj C D x 0 Extension: Halperin (1962), N subspaces: C 1,..., C N x 0 H, x 1 = proj C1 x 0, x 2 = proj C2 x 1,..., x N = proj CN x N 1, x N+1 = proj C1 x N Solving numerically linear systems (Kaczmarz, 1937).

4 1.2 From linear convex analysis Theorem (Acker and Prestel, Ann. Toulouse, 190) H Hilbert f, g : H R {+ } convex, lower semicontinuous, + >< >: (x k, y k ) (x k+1, y k ) (x k+1, y k+1 ): x k+1 = argmin{f(ξ) + 1 2λ ξ y k 2 : ξ H} then y k+1 = argmin{g(η) + 1 2λ x k+1 η 2 : η H} Then (x k, y k ) w H H (x, y ) as k +, with (x, y ) argmin{f(ξ) + g(η) + 1 2λ ξ η 2 : ξ H, η H} Algorithm = alternating minimization of the bivariate function Application to inverse problems L(ξ, η) = f(ξ) + g(η) + 1 2λ ξ η 2 (Cheney and Goldstein 59; Gubin, Polyak and Raik 67) f = δ C, g = δ D, C, D closed convex sets in H x k w H x C, y k w H y D x y = min{ x y : x C, y D} C x 1 x x 3 x 2 y 3 y y 2 D y 1 y 0

5 1.3 Application to the inverse Cauchy problem Cimetière, Delvare, Jaoua, Pons, Inverse problems (2001) u = 0 Γ i Ω Γ d < : u = u 0 u n = u 1 Lacking data on Γ i superabundant measured data on Γ d. >< u = 0 on Ω u u = u 0, n >: = u 1 on Γ d u unknown on Γ i Ill-posed inverse problem: data may be incompatible (measurements, noise). Hadamard example of instability Ω = (0, 1) (0, 1), u 0 = 0, u 1 = 0, u n (x, y) = 1 n sin(nx)ch(ny). Alternating projection method in H = H 1 (Ω) with D, N closed affine subspaces of H = H 1 (Ω): D = {v H 1 (Ω) : v = 0 on Ω, v = u 0 on Γ d } N = {v H 1 (Ω) : v = 0 on Ω, v n = u 1 on Γ d }

6 1.4 Passty theorem Theorem (Passty, J. Math Analysis Appl., 1979) H Hilbert f, g : H R {+ } convex, lower semicontinuous, + (f + g) = f + g. (x 0, y 0 ) H H given. (x k, y k ) (x k+1, y k ) (x k+1, y k+1 ): >< >: x k+1 = argmin{f(ξ) + 1 2λ k ξ y k 2 : ξ H} then y k+1 = argmin{g(η) + 1 2λ k x k+1 η 2 : η H} λ k 0 with P λ 2 k and P λ k =. Define s k = P k 1 λ p, X k = 1 s k P k 1 λ px p, Y k = 1 s k P k 1 λ py p Then, X k w H z, Y k w H z as k +, with z argmin{f(ξ) + g(ξ) : ξ H} Remarks on Passty theorem: 1. Valid for A, B maximal monotone operators, A+B max. monotone. Extension of Brezis-Lions thm. (Israel J. Math., 197). 2. Link with semigroups of contractions (Trotter-Lie-Kato formula). Questions: 1. Still valid without (f + g) = f + g? relaxation phenomena: Lehdili and Lemaire (1999), weak sum (Lapidus), variational sum (Attouch-Baillon-Thera, JCA., 1994), extended sum (Revalski-Thera). 2. Weak ergodic weak convergence? (Bruck, JFA., 1975). 3. P λ n = is enough? (Güler, SIAM J. Control Opt., 1991).

7 2 Link with decision sciences and dynamical games 2.1 Dynamical decision with costs to change Goal oriented human cognition (motivated agent): Vroom (1964), Locke and Lathan (1990). Agent = problem solver. Solving problem = reducing (gradually) unsatisfied needs. x 0 x k x k+1 goal Satisficing, bounded rationality: (Carnegie school), H. Simon, Nobel prize of economics (197), artificial intelligence. Landscape theory: The environment (landscape) is largely unknown, exploration aspects: Levinthal and Warglien (1999), learning aspects: Sobel (2000), Berthoz (2003). Worthwhile to move principle: Satisficing with not too much sacrificing. Attouch and Soubeyran (2005, J. Math. Psy.) >< >: X state, performance, strategy space g : X R gain function c : X X R + cost to change g(x k+1 ) g(x k ) θ k c(x k+1, x k ) Marginal gain Cost to change Cost to change Local features of the dynamics: c(x k, x k+1 ) 1 θ k (sup X g g(x 0 ))

8 2.2 Proximal algorithms in decision sciences. Worthwhile to move principle + Optimization Proximal Dynamics. The vision the agent has of his environment and of his gain function depend on his current position (local aspects). Difficulty to change, inertia Anchoring effect: at stage k, his gain function is given by ξ g(ξ) c(ξ, x k ). Proximal dynamics: x k x k+1 x k+1 argmax{g(ξ) c(ξ, x k ) : ξ X} Take ξ = x k, sum w.r. to k, Finite ressource assumption sup X g + g(x k+1 ) g(x k ) c(x k+1, x k ) P k c(x k+1, x k ) sup X g g(x 0 ) + c(x k+1, x k ) 0 as k + Classical prox. dynamics: H Hilbert, f = g, c(ξ, x) = ξ x 2, λ k 0 x k+1 argmin{f(ξ) + 1 2λ k ξ x k 2 : ξ H} General proximal dynamics: x k+1 ǫ k argmax{g(ξ) θ k c(ξ, x k ) : ξ E(x k, r k )} ǫ k : psychological features (motivation, degree of resolution) θ k : cognitive features (speed, learning, reactivity). E(x k, r k ) : exploration set.

9 2.3 Alternating algorithms and Nash potential games 2 agents (players) f: X R payoff of player 1, individual behaviour g: Y R payoff of player 2, individual behaviour c: X Y R joint payoff of the two players, coupling term with either attractive features (team game) or repulsive features (congestion games) Static normal form of the potential game (Monderer-Shapley, 1996) < : F(ξ, η) = f(ξ) + βc(ξ, η) G(ξ, η) = g(ξ) + µc(ξ, η) Dynamical game: Inertial non autonomous Nash equilibration process h: X X R + cost to move in X for player 1 g: Y Y R + cost to move in Y for player 2 non autonomous aspects: α k 0, ν k 0 >< >: x k+1 argmin{f(ξ) + βc(ξ, y k ) + α k h(x k, ξ) : ξ X } then y k+1 argmin{g(η) + µc(x k+1, η) + ν k k(y k, η) : η Y} Nash equilibrium (x, y) is a Nash equilibrium of the normal form game if < : x argmin{f(ξ) + βc(ξ, y) : ξ X } y argmin{g(η) + µc(x, η) : η Y}

10 3 Convergence of proximal dynamics 3.1 The convex case Theorem (Martinet 70, Rockafellar 76, Güler 91) H Hilbert space f : H R {+ } convex, lower semicontinuous, inf X f P λ k 0, λk = + x k+1 = argmin{f(ξ) + 1 2λ k ξ x k 2 : ξ H} Then, a) (x k ) k N is a minimizing sequence for f. b) If argminf x k w H x as k +, with x argminf. Remarks 1. Proximal algorithm = implicit discretization of the generalized steepest descent differential inclusion (x(t k ) = x k ) 0 ẋ(t) + f(x(t)) 0 1 λ k (x(t k+1 ) x(t k )) + f (x(t k+1 )) λ k = t k+1 t k, X λ k = + t k + k Hence, asymptotic behaviour of proximal algorithms asymptotic behaviour of the steepest descent (Bruck theorem, Opial lemma). 2. Weak convergence of (x k ) k N and not strong convergence: see Hundal counterexample (Nonlinear Analysis, 2004).

11 3.2 The analytic and subanalytic case (Attouch and Bolte, Math. Programming, 2007) f : R n R {+ } lower semicontinuous, inf X f. The restriction of f to its domain is a continuous function. f satisfies the Lojasiewicz property : (L) For any limiting-critical point ˆx, that is f(ˆx) 0, there exist C, ǫ > 0 and θ [0, 1) s.t. f(x) f(ˆx) θ C x, x B(ˆx, ǫ), x f(x). Theorem (convergence) Let (x k ) k N be a bounded sequence generated by the proximal algorithm, then + X k=0 x k+1 x k < +, and the whole sequence (x k ) converges to some critical point of f. The rate of convergence is intimately related to the Lojasiewicz exponent which measures some flatness (curvature) property of f. Theorem (rate of convergence) Let x k x be a convergent sequence generated by the proximal algorithm. Let us denote by θ a Lojasiewicz exponent of x crit f. Then, (i) If θ = 0, the sequence (x k ) converges in a finite number of steps, (ii) If θ (0, 1 ] then there exist c > 0 and Q [0, 1) such that 2 x k x c Q k, (iii) If θ ( 1, 1) then there exists c > 0 such that 2 x k x c k 1 θ 2θ 1.

12 3.3 Subdifferential calculus for nonsmooth subanalytic functions Definition: Let f : R n R {+ } be a proper lower semicontinuous function. For each x domf, the Fréchet subdifferential of f at x, written b f(x), is the set of vectors x R n which satisfy lim inf y x y x 1 x y [f(y) f(x) x, y x ] 0. If x / domf, then b f(x)) =. The limiting-subdifferential of f at x R n, written f, is defined as follows f(x) := {x R n : x n x, f(x n ) f(x), b f(xn ) x }. Remarks: The above definition implies that b f(x) f(x), where the first set is convex while the second one is closed. Clearly a necessary condition for x R n to be a minimizer of f is f(x) 0. Unless f is convex the above is not a sufficient condition. A point x R n that satisfies f(x) 0 is called limiting-critical or critical. The set of critical points of f is denoted by critf.

13 On the class of nonsmooth real valued functions involving analytic features 1. Real-analytic functions have the Lojasiewicz property, see Lojasiewicz (Editions du CNRS, Paris, 1963). 2. If f is subanalytic and is continuous on its domain with dom f closed in R n, in particular if f is continuous and subanalytic, it has the Lojasiewicz property, see Bolte-Daniilidis-Lewis (JMAA., 2005), Kurdyka-Parusinski (CRAS., 1994). 3. An interesting class of functions satisfying the Lojasiewicz property is given by semialgebraic functions. These are functions whose graphs can be expressed as p i=1 q j=1 {x Rn : P ij (x) = 0, Q ij (x) 0} where for all 1 i p, 1 j q the P ij, Q ij : R n R are polynomial functions. Due to the Tarski-Seidenberg principle, which asserts that the linear projection of a set of the above type remains of this type, semialgebraic objects enjoy remarkable stability properties. 4. Convex functions satisfying the following growth condition near the infimal set do satisfy the Lojasiewicz property: bx argminf, C 0, r 1, ǫ 0, x B(bx, ǫ), f(x) f(bx) + Cd(x,argminf) r 5. Infinite-dimensional versions of the Lojasiewicz property have been developed in view of the asymptotic analysis of dissipative evolution equations, see Simon (Ann. Math., 193) and Haraux. 6. Kurdyka has recently established a Lojasiewicz-like inequality for functions definable in an arbitrary o-minimal structure (Ann. Inst. Fourier, 199).

14 4 Alternating proximal minimization: the convex case 4.1 Strong coupling Setting: X = Y = H Hilbert spaces. f : H R {+ } convex, lower semicontinuous, + g : H R {+ } convex, lower semicontinuous, + P α k 0, ν k 0, + k=0 α P k+1 α k, + k=0 ν k+1 ν k Algorithm: >< >: x k+1 = argmin{f(ξ) + β 2 ξ y k 2 + α k 2 ξ x k 2 : ξ H} then y k+1 = argmin{g(η) + µ 2 η x k ν k 2 η y k 2 : η H} Alternating minimization with anchoring (inertia) of L β,µ (ξ, η) = µf(ξ) + βg(η) + βµ 2 ξ η 2 Equilibrium: solutions of the minimization problem min{µf(ξ) + βg(η) + βµ 2 ξ η 2 : ξ H, η H} Theorem (Attouch-Redont-Soubeyran, 2007) Assume S = argmin L β,µ. Then, a) (x k, y k ) k N is a minimizing sequence for L β,µ. b) x k w H x, y k w H y with (x, y ) S = argminl β,µ c) x k+1 x k s H 0, y k+1 y k s H 0 x k y k s H x y as k +

15 Alternating projection with anchoring versus classical version x 0 x 1 C x 3 x 2 u 1 u 2 v1 u 3 v2 v3 y 3 y 2 y 1 y 0 D C x 1 x x 3 x 2 y y 3 y 2 D y 1 y 0

16 4.2 Weak coupling Setting: X, Y, Z Hilbert spaces. The coupling between X and Y is realized via Z (trace, interface,...) f : X R {+ } convex, lower semicontinuous, + g : Y R {+ } convex, lower semicontinuous, + A: X Z linear, continuous operator. B: Y Z linear, continuous operator. Algorithm: Alternating minimization with anchoring (inertia) of L λ (ξ, η) = f(ξ) + g(η) + 1 2λ A(ξ) B(η) 2 >< >: x k+1 = argmin{f(ξ) + 1 2λ A(ξ) B(y k) 2 + α 2 ξ x k 2 : ξ X} then y k+1 = argmin{g(η) + 1 2λ B(η) A(x k+1) 2 + ν 2 η y k 2 : η Y } Equilibrium: solutions of the minimization problem min{f(ξ) + g(η) + 1 2λ A(ξ) B(η) 2 : ξ X, η Y } Theorem (Attouch-Redont-Soubeyran, 2007) Assume S = argmin L λ. Then, a) (x k, y k ) k N is a minimizing sequence for L λ. b) x k w X x, y k w Y y with (x, y ) S = argminl λ c) x k+1 x k s X 0, y k+1 y k s Y 0 A(x k ) B(y k ) s Z A(x ) B(y ) as k +.

17 4.3 Decomposition and splitting for P.D.E Variational problem min{f(ξ) + g(η) + 1 2λ A(ξ) B(η) 2 : ξ X, η Y } Transmission through a thin weakly conducting layer, contact problems. Ω 1 Σ Ω 2 X = H 1 (Ω 1 ), A : H 1 (Ω 1 ) Z = L 2 (Σ) Y = H 1 (Ω 2 ), B : H 1 (Ω 2 ) Z = L 2 (Σ) trace operator trace operator min{ 1 2 Z Ω 1 u Z Ω 2 u λ Z Σ Z [u] 2 Ω fu : u 1 X, u 2 Y } < : u = u 1 on Ω 1 u = u 2 on Ω 2 [u] = A(u 1 ) B(u 2 ) : jump of u through Σ. Alternating minimization decomposition of domain

18 5 Alternating proximal minimization: the analytic and subanalytic cases. f : R n R {+ }, g : R m R {+ } lower semicontinuous, A: R n R p, B: R m R p linear (continuous) operators. Equilibrium: critical points of the bivariate function L λ : R n R m R {+ } L λ (ξ, η) = f(ξ) + g(η) + 1 A(ξ) B(η) 2 2λ Algorithm: Alternating minimization with anchoring (inertia) of L λ >< >: x k+1 argmin{f(ξ) + 1 2λ Aξ By k θ k ξ x k 2 : ξ R n } then y k+1 argmin{g(η) + 1 2λ Bη Ax k µ k η y k 2 : η R m } where 0 θ θ k θ + +, 0 µ µ k µ + +. Let us assume that inf R nf and inf R mg and that f and g are respectively continuous on their domains dom f and dom g. Theorem (Attouch-Bolte, 2007) Assume that the bivariate function L λ : R n R m R {+ } satisfy the Lojasiewicz property. Let (x k, y k ) k N be a bounded sequence generated by the alternating proximal algorithm with anchoring, then + X k=0 ( x k+1 x k + y k+1 y k ) < +, and the whole sequence (x k, y k ) k N converges to some critical point of L λ.