Prox-Diagonal Method: Caracterization of the Limit

Transcription

1 International Journal of Mathematical Analysis Vol. 12, 2018, no. 9, HIKARI Ltd, Prox-Diagonal Method: Caracterization of the Limit M. Amin Bahraoui Faculty of Sciences and Techniques of Tangier B.P. 416 Tangier-Marocco Copyright c 2018 M. Amin Bahraoui. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The aim of this paper is to give a caracterization of the limit of the sequence generate by the diagonal version of the prox method. Keywords: Convex minimization, prox-diagonal method, variational principle, convergence 1 Introduction In [3], B. Lemaire annonced an asymptotical principle associated with the selection methods in continuous and discrete cases (proximal method of Martinet- Rockafellar), given so a caracterization of asymptotical limit of trajectory generated in the optimale set but under non uniqueness. The aim of this paper is to extend this result at diagonal exacte version of the proximal method. In Section 2 we recall some results of convergence given in [2]. We give in Section 3 a localization result of the asymptotic limit according to the initial point and the error of the approximation of f with {f n }. Section 4 is devoted to variational principle in the disret case. Caracterization of the limit is done in Section 5. 2 Main Results Let X be a real Hilbert space equipped with the inner product, and the associated norm. For n = 1, 2,, let f n Γ 0 (X) set of proper closed

2 404 M. Amin Bahraoui convex functions on X, λ n > 0 and x 0 X. The proximal trajectory x := {x n } in X is defined recursively by which is equivalent to x n (I + λ n f n ) 1 x n 1, n = 1, 2, (2.1) γ n := x n 1 x n λ n f n (x n ), n = 1, 2, (2.2) where f n denotes the subdifferential operator associted with f n. Remark 2.1 The iterative schema (2.2) appears like the implicit Euler discretization of the differential inclusion (see [4]) : du dt f(u), t > 0 u(0) = x 0. (2.3) The convergence result of the schema (2.1) (2.2) is given by the following theorem : [2] Theorem 2.2 Assume that (i) {f n } converge to f in the Mosco sense. (ii) Argminf (iii) x Argminf, θ n 0 such that f n (x) inf X f n + θ n (iv) (v) + n=0 λ n θ n < + 0 < λ λ n Then f n (x n ) inf X f, the sequence {x n} weakly converge to some point x Argminf and γ n = x n 1 x n λ n 0. The proof is given in [2] and based upon the following crucial lemma: Lemma 2.3 For all a, b, x in X, we have 2 a b, x b = x b 2 x a 2 + b a 2.

3 Prox-diagonal method: caracterization of the limit Localization of x We denotes S := Argminf. Proposition 3.1 If we suppose, in addition to the hypothesis of the theorem 2.2, that {θ n } is independent of the choice of x S then we have the following estimation : x x 0 2 4[d(x 0, S) θ k ]. From lemma 2.3 and scheme (2.2) we have easily (see [2]) : x S, x k x 2 x k 1 x θ k, k IN n IN, x n x 2 x 0 x θ k. Then, passing to the limit when n +, we have : + x x 2 x 0 x θ k. From which we get : Which acheved the proof. x x 0 2 2[ x x 2 + x 0 x 2 ] + 2[2 x 0 x θ k ] 4[d(x 0, S) θ k ]. Remark 3.2 If, for all n IN, f n = f then we recover the localization given in [3] (θ n = 0). For the following, let us assume satisfied the hypothesis of theorem Variational principle We define the set K(x 0 ) of discrete feasible trajectories from the initial point x 0 with : K(x 0 ) := { ỹ := (x 0, y 1, ); k 1, y k domf k, d k := y k 1 y k domf k, d k 0 and y k w y }.

4 406 M. Amin Bahraoui Proposition 4.1 The set K(x 0 ) is a convex set containing proximal trajectory x. Let ỹ, z K(x 0 ) and λ ]0, 1[. We have : ũ := λỹ + (1 λ) z = (x 0, λy 1 + (1 λ)z 1, ). As f k is convex then we get : k IN, and u k = λy k + (1 λ)z k domf k d k := u k 1 u k The remainder is trivial. = λ y k 1 y k + (1 λ) z k 1 z k domf k. From the proximal trajectory x, we have : x k domf k and γ k domf k, because x k f k ( γ k ). Elsewhere, from theorem 2.2, we have : Therefor x K(x 0 ). γ k 0 and x k w x S. We define the sequence of cost function for finite horizon {J n } (see [3]) with : n IN, J n : K(x 0 ) IR + J n (ỹ) := [f k (y k ) + f k ( d k ) + y k, d k ]. Proposition 4.2 i) For all n IN, J n is convex. ii) The sequence {J n } is increasing. i) From lemma 2.3, we have: Therefor J n (ỹ) = y k, d k = 1 2 y k y k λ2 k 2 d k 2, [f k (y k ) + f k ( d k ) + 2 d k 2 ] ( y n 2 x 0 2 ). In this form, we verify easily that J n is convex, for all n IN.

5 Prox-diagonal method: caracterization of the limit 407 ii) Comes immediatly from: f k (y k ) + f k ( d k ) + y k, d k 0, k IN. Now, we define the cost function for infinite horizon (convex) J with : J : K(x 0 ) IR + {+ } J(ỹ) := lim J n(ỹ) = sup J n (ỹ). n + n 1 Theorem 4.3 (Variational principle : discret case). x is the unique minimizer of J in K(x 0 ) with J( x) = 0. From extremality condition: k IN, f k (x k ) + f k ( γ k ) + x k, γ k = 0, then n IN, J n ( x) = 0. Therefor J( x) = 0 = inf K(x 0 ) J, because J(ỹ) 0, x K(x 0). Let z := {z n } K(x 0 ) such that J( z) = 0. i.e. n IN, J n ( z) = 0. Therefor z verify extremality condition. Then x = z. Proposition 4.4 We have, ỹ K(x 0 ) J(ỹ) 1 2 y x 2. i) First, let proof that: n IN, ỹ K(x 0 ) J n (ỹ) 1 2 y n x n 2. We take: n IN J n (ỹ) = A k, where A k := f k (y k ) + f k ( d k ) + y k, d k. From extremality condition and as γ k := x k 1 x k f k (x k ) x k f k ( γ k )

6 408 M. Amin Bahraoui we have A k = f k (y k ) + f k ( d k ) + y k, d k f k (x k ) f k ( γ k ) x k, γ k γ k, y k x k + x k, d k + γ k + y k, d k x k, γ k γ k, y k x k + y k, d k + x k, d k γ k d k, x k y k. Therefor, n IN J n (ỹ) γ k d k, x k y k. Taking a k := x k y k, we have : γ k d k, x k y k = a k a k 1, a k = a k 2 a k 1, a k. Using anew the lemma 2.3, we get: a k 2 a k 1, a k = 1 2 a k a k a k a k 2 Summing from k = 1 to n, 1 2 a k a k 1 2. J n (ỹ) 1 2 [ a k 2 a k 1 2 ] 1 2 a n a 0 2 with a 0 = x 0 y 0 = 0 and a n = x n y n. Then n IN, J n (ỹ) 1 2 y n x n 2. ii) Passing to the limit as n +, we get: J(ỹ) 1 2 y x 2. Proposition 4.5 i) For every ỹ K(x 0 ) such that y / S then J(ỹ) = +.

7 Prox-diagonal method: caracterization of the limit 409 ii) Under condition (C) { k > 0 such that f k is coercive, k k, f k f k and x k domf k we have: x S, ỹ K(x 0 ) such that y = x and J(ỹ) < +. i) Taking Let ỹ K(x 0 ) such that y / S. Then we have A k := f k (y k ) + f k ( d k ) + y k, d k. lim inf k + A k λ [ f(y ) + f (0) ] > 0, and J(ỹ) = + A k = +. ii) Let x S. Let n be an integer such that n > k. We define the trajectory ỹ := {y k } with : x k if 0 k k y k = x k + (t k t k) x x k t n t k if k < k n x if k > n k where t k = λ l. We have: l=1 y k domf k. Indeed, in the case that k { k + 1,, n}, we get f k (y k ) t k t k f k (x) + (1 t k t k )f k (x k). t n t k t n t k From the condition (C), f k (x k) < +. The condition (iii) of theorem 2.2, f k (x) < +. Therefor f k (y k ) < +.

8 410 M. Amin Bahraoui d k := y k 1 y k = x x k t n t k Condition (C) implies that f k is continuous at 0 and domf k domf k. Then for n large enough, d k domf k. We have: k > n, y k = x. Therefor k > n, d k = 0. n > n, we have: J n (ỹ) = k= k+1 k= k+1 A k + A k + k= n+1 k= n+1 [f k (x) + f k (0)] θ k J n (ỹ) < +. Remark 4.6 Suppose that f is coercive. i) Condition (C) is satisfied when f k = f, k, with k = 1. Then we recover proposition 4.3 (discret case) from [3]. ii) Condition (C) is also satisfied in case of the approximation with external penalization in convex programming (finite dimensional), see [2]. This comes directly from theorem 3.6 of [1] or from theorem 9 of [5] and owing to the fact that domf k = X, k. 5 Caracterization of x From [3], we define the asymptotic cost function ϕ x0 : X IR {+ } with: ϕ x0 (x) := Proposition 5.1 i) ϕ x0 is convex. ii) ϕ x0 (x ) = J( x) = 0. inf J(ỹ). ỹ K(x 0 ) y =x iii) x X, ϕ x0 (x) 1 2 x x 2.

9 Prox-diagonal method: caracterization of the limit 411 i) Lets x 1 and x 2 in X, λ ]0, 1[ and ɛ > 0. From ϕ x0 (x 1 )ϕ x0 (x 2 ) 0 we have : ỹ 1 K(x 0 ) such that y 1, = x 1 and J(ỹ 1 ) ϕ x0 (x 1 ) + ɛ 2 and ỹ 2 K(x 0 ) such that y 2, = x 2 and J(ỹ 2 ) ϕ x0 (x 2 ) + ɛ 2. Then ϕ xo (λx 1 + (1 λ)x 2 ) = inf J(ỹ) ỹ K(x 0 ) y =λx 1 +(1 λ)x 2 J(λỹ 1 + (1 λ)ỹ 2 ) λj(ỹ 1 ) + (1 λ)j(ỹ 2 ) λϕ x0 (x 1 ) + (1 λ)ϕx 0 (x 2 ) + ɛ. This is true for all ɛ > 0. We get the result. ii) We have: then ϕ x0 (x ) = J( x) = 0. 0 ϕ x0 (x ) = inf ỹ K(x 0 ) y =x J(ỹ) J( x) = 0, iii) From proposition 4.4, we have J(ỹ) 1 2 y x 2 inf J(ỹ) 1 ỹ K(x 0 ) 2 x x 2. y =x Then ϕ x0 (x) 1 2 x x 2. As a direct cosequence of proposition 5.1, we can give the result of caracterization of x : Theorem 5.2 The limit x is the unique minimizer of ϕ x0 on S (on X) with ϕ x0 (x ) = 0.

10 412 M. Amin Bahraoui x is the minimizer of ϕ x0 on S, because ϕ x0 (x ) = 0. If a S such that ϕ x0 (a) = 0 then 0 = ϕ x0 (a) 1 2 a x 2 0 a = x, then uniqueness. Remark 5.3 The theorem 5.2 show that x is the unique optimal solution of convex optimization probleme on S. This caracterization is true when the asymptotic cost function ϕ x0 is finite on S and this is guaranted with the following proposition : Proposition 5.4 If the condition (C) is satisfied then domϕ x0 = S. From Proposition 4.4, we have x S inf J(ỹ) < +. ỹ K(x 0 ) y =x References [1] A. Auslender, R. Cominetti and J.P. Crouzeix, Convex functions with unbounded level sets and applications to duality theory, SIAM J. Optimization, 3 (1993), no. 4, [2] M.A. Bahraoui, Suites Diagonalement Stationnaires en Optimisation Convexe, Diss, Université de Monpellier II, [3] B. Lemaire, An asymptotical variational principle associated with the steepest descent method for a convex function: Continuous and discrete cases, ICAA-93, Hanoi. [4] B. Lemaire, The Proximal Algorithm, International Series of Numerical Mathematics, Vol. 87, Birkhauser Verlag, [5] R. Wets, A Formula for the Level Sets of Epi-Limits and Some Applications, IIASA Working paper 82-81, A-2361 Laxenburg, Austria, Received: June 27, 2018; Published: August 20, 2018