Existence and Approximation of Fixed Points of. Bregman Nonexpansive Operators. Banach Spaces

Transcription

1 Existence and Approximation of Fixed Points of in Reflexive Banach Spaces Department of Mathematics The Technion Israel Institute of Technology Haifa Joint work with Prof. Simeon Reich

2 General Assumptions X a reflexive Banach space f : X R a proper, convex and l.s.c function which is Gâteaux differentiable K a nonempty, closed and convex subset of X

3 Bregman Distance Definition [Bregman (1967), Censor and Lent (1981)] The Bregman distance D f : domf int domf [0, + ) is defined by D f (y, x) := f (y) f (x) f (x), y x.

4 Classes of Operators in Hilbert Spaces Let K be a nonempty, closed and convex subset of a Hilbert space H and let T : K K be an operator T is nonexpansive if for all x, y K Tx Ty x y T is quasi-nonexpansive if for all p F (T ) and x K p Tx p x T is firmly nonexpansive if for all x, y K Tx Ty 2 Tx Ty, x y

5 Classes of Operators in Banach Spaces Let f : X R be a function. Let K be a nonempty, closed and convex subset of X and let T : K K be an operator T is Bregman nonexpansive (BNE) if for all x, y K D f (Tx, Ty) D f (x, y) T is quasi-bregman nonexpansive (QBNE) if for all p F (T ) and x K D f (p, Tx) D f (p, x)

6 Classes of Operators in Banach Spaces Definition [Bauschke, Borwein and Combettes (2003)] T is Bregman firmly nonexpansive (BFNE) if for all x, y K f (Tx) f (Ty), Tx Ty f (x) f (y), Tx Ty or equivalently D f (Tx, Ty) + D f (Ty, Tx) + D f (Tx, x) + D f (Ty, y) D f (Tx, y) + D f (Ty, x) T is quasi-bregman firmly nonexpansive (QBFNE) if for all p F (T ) and x K D f (p, Tx) + D f (Tx, x) D f (p, x)

7 Bregman Strongly Nonexpansive Operaotrs Definition [Reich (1996)] T is Bregman strongly nonexpansive (BSNE) with respect to a nonempty set S if D f (p, Tx) D f (p, x) for all p S and x K, and if whenever {x n } n N K is bounded, p S, and lim (D f (p, x n ) D f (p, Tx n )) = 0, n + it follows that lim D f (Tx n, x n ) = 0 n +

8 Bregman Strongly Nonexpansive Operaotrs Definition [Reich (1996)] A point u K is said to be an asymptotic fixed point of T if there exists a sequence {x n } n N in K such that x n u and x n Tx n 0. We denote the set of asymptotic fixed points of T by F (T ). Let {T i : 1 i N} be N BSNE operators and denote the composition T N T N 1 T 1 by T and F = {F (T i ) : 1 i N} and ˆF } = {ˆF (Ti ) : 1 i N

9 Properties of BSNE Operators Lemma [Reich (1996)] Let f : X R be bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of K. If the sets ˆF (T ) and ˆF are nonempty, then T is BSNE with respect to ˆF (T ) and ˆF (T ) ˆF. If ˆF (T i ) = F (T i ) for each 1 i N, F and F (T ) are nonempty, then T is also BSNE with F (T ) = ˆF (T ). Indeed, F (T ) ˆF (T ) ˆF = F F (T )

10 Legendre Functions Definition [Bauschke, Borwein and Combettes (2001)] The convex function f is called a Legendre function if it satisfies the following two conditions: (L1) The interior of the domain of f, int dom f, is nonempty, f is Gâteaux differentiable on int dom f and dom f = int dom f ; (L2) The interior of the domain of f, int dom f, is nonempty, f is Gâteaux differentiable on int dom f and dom f = int dom f.

11 Legendre Functions Proposition [Bauschke, Borwein and Combettes (2001)] f is Legendre if and only if f is Legendre. Proposition [Bauschke, Borwein and Combettes (2001)] If f is a Legendre function then: f is strictly convex on the interior of its domain. ran f = dom f = int dom f. Note: f = ( f ) 1.

12 Properties of the Gradient Proposition [Reich and Sabach (2009)] If f : X R is uniformly Fréchet differentiable and bounded on bounded subsets of X, then f is uniformly continuous on bounded subsets of X from the strong topology of X to the strong topology of X. Proposition [Reich and Sabach (2010)] If f : X (, + ] is a positively homogeneous function of degree α R, then f is a positively homogeneous function of degree α 1.

13 Totally Convex Functions Definition [Butnariu and Iusem (2000)] The function f is called totally convex at a point x int dom f if its modulus of total convexity at x, that is, the function υ f : int dom f [0, + ) [0, + ] defined by υ f (x, t) := inf {D f (y, x) : y dom f, y x = t} is positive whenever t > 0.

14 Totally Convex Functions Proposition [Resmerita (2004)] Let f : X (, + ] be a function and take x int dom f. Then f is totally convex at x if and only if lim n D f (y n, x) = 0 implies that lim n y n x = 0 for any sequence {y n } n N dom f.

15 Characterization of Bregman Projections Proposition [Butnariu and Resmerita (2006)] Suppose that f is totally convex on int dom f. Let x int domf and let K int domf be a nonempty, closed and convex set. If ˆx K, then the following statements are equivalent: The vector ˆx is the Bregman projection of x onto K with respect to f ; The vector ˆx is the unique solution of the variational inequality f (x) f (z), z y 0, y K ; The vector ˆx is the unique solution of the inequality D f (y, z) + D f (z, x) D f (y, x), y K.

16 Properties of Bregman Nonexpansive operators Lemma Let f : X (, + ] be a Legendre function. Let K be a nonempty, closed and convex subset of int dom f and let T : K K be a QBNE operator with respect to f. Then F (T ) is closed and convex.

17 Properties of Bregman Nonexpansive operators Lemma Let f : X (, + ] be a Legendre function which is uniformly Fréchet differentiable and bounded on bounded subsets of X. Let K be a nonempty, closed and convex subset of int dom f and let T : K K be a BFNE operator with respect to f. Then F (T ) = F (T )

18 Existence of Fixed Points Theorem Let f : X (, + ] be a Legendre function such that f is bounded on bounded subsets of X. Let K be a nonempty, closed and convex subset of int dom f and let T : K K be a QBNE operator with respect to f. If F (T ) is nonempty, then {T n y} n N is bounded for each y K.

19 Existence of Fixed Points Proof of Theorem For any x F (T ) and y K we have D f (x, T n y) D f (x, y) Then the nonnegative sequence {D f (x, T n y)} n N is bounded. Then f (x) f (T n y), x + f ( f (T n y)) = D f (x, T n y) M This implies that the sequence { f (T n y)} n N is contained in the sublevel set lev ψ (M f (x)) of the function ψ = f, x. Hence, the sequence { f (T n y)} n N is bounded. Thus the sequence T n y = f ( f (T n y)), n N, is bounded too.

20 Existence of Fixed Points Theorem Let f : X (, + ] be a Legendre function. Let K be a nonempty, closed and convex subset of int dom f and let T : K K be a BFNE operator with respect to f. If there exists y K such that S n (y) as n, then F (T ) is nonempty. Definition For an operator T : K K, let S n (z) := (1/n) n k=1 T k z for all z K.

21 Some Relevant Implications of our Theorems Corollary Let f : X (, + ] be a Legendre function. Every nonempty, bounded, closed and convex subset of int dom f has the fixed point property for BFNE self-mappings. Theorem Let f : X (, + ] be a Legendre function. Let K be a nonempty, bounded, closed and convex subset of int dom f. Let {T α } α A be a commutative family of BFNE operators with respect to f from K into itself. Then the family {T α } α A has a common fixed point.

22 Approximation of Fixed Points Theorem Let f : X (, + ] be a Legendre, totally convex function which is positively homogeneous of degree α > 1 and uniformly Fréchet differentiable on bounded subsets of X. Let K be a nonempty, bounded, closed and convex subset of int dom f with 0 K, and let T be a BFNE self-mapping with respect to f. Then the following two assertions hold: (i) For each t (0, 1), there exists a unique u t K satisfying u t = ttu t ; (ii) The net {u t } t (0,1) converges strongly to proj f F (T ) ( f (0)) as t 1.

23 Approximation of Fixed Points Proof of part (i) Denote S t = tt then S t is a BFNE operator too. Indeed, f (S t x) f (S t y), S t x S t y = t α f (Tx) f (Ty), Tx Ty t α f (x) f (y), Tx Ty = t α 1 f (x) f (y), S t x S t y f (x) f (y), S t x S t y Since K is bounded, it follows from Corollary 1 that S t has a fixed point. By the monotonicity of f it follows that the fixed point is unique.

24 Approximation of Fixed Points Proof of part (ii) From Lemma 1 and Theorem 2, F (T ) is nonempty, closed and convex. Thus the Bregman projection proj f F (T ) is well defined. Put x n = u tn. Since K is bounded, there is a subsequence {x nk } k N such that x nk v. In addition, we have that x n Tx n = (t n 1) Tx n and therefore x n Tx n 0 and hence v F (T ). Lemma 2 now implies that v F (T ). Since f is totally convex function, T is a BFNE operator and Tx nk v we get that Tx nk v. Now the characterization of the Bregman projection implies that v = proj f F (T ) ( f (0)) and this completes the proof.

25 Examples of BFNE operators Protoresolvent of A relative to f : Prt f A := ( f + A) 1 : X 2 X Resolvent of A relative to f : Res f A := Prtf A f : X 2X If A = ϕ then we denote: Prox f ϕ := Prt f ϕ and proxf ϕ := Resf ϕ If K is a nonempty, closed and convex subset of X, then we denote: proj f K := proxf ι K

26 Examples of BFNE operators Properties [Bauschke, Borwein and Combettes (2003)] If f : X R is a cofinite Legendre function and A : X 2 X is maximal monotone mapping, then: (i) dom (Res f A ) = X; (ii) Res f A is single-valued; (iii) Res f A is a BFNE operator; (iv) F(Res f A ) = A 1 (0 ).

27 Examples of BFNE operators Conditions Let K be a nonempty, closed and convex subset of X. Let g : K K R be a bifunction that satisfies the following conditions: (C1) g (x, x) = 0 for all x K ; (C2) g is monotone, i.e., g (x, y) + g (y, x) 0 for all x, y K ; (C3) for all x, y, z K, lim sup g (tz + (1 t) x, y) g (x, y) ; t 0 (C4) for each x K, g (x, ) is convex and lower semicontinuous.

28 Examples of BFNE operators The resolvent of bifunction g Res f g (x) = {z K : g (z, y) + f (z) f (x), y z 0 y K } Properties [Reich and Sabach (2010)] If f is a coercive Legendre function and g satisfies conditions (C1) (C4), then: (i) dom (Res f g) = X; (ii) Res f g is single-valued; (iii) Res f g is a BFNE operator; (iv) F(Res f g) = EP(g).

29 References Bauschke, H. H., Borwein, J. M. and Combettes, P. L.: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Comm. Contemp. Math. 3 (2001), Bauschke, H. H., Borwein, J. M. and Combettes, P. L.: Bregman monotone optimization algorithms, SIAM J. Control Optim. 42 (2003), Bregman, L. M.: A relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. and Math. Phys. 7 (1967), Butnariu, D. and Iusem, A. N.: Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer Academic Publishers, Dordrecht, 2000.