## Interative approaches to convex feasibility problems.

##### Abstract

Solutions to convex feasibility problems are generally found by iteratively constructing
sequences that converge strongly or weakly to it. In this study, four types
of iteration schemes are considered in an attempt to find a point in the intersection
of some closed and convex sets.
The iteration scheme Xn+l = (1 - λn+1)y + λn+1Tn+lxn is first considered for infinitely
many nonexpansive maps Tl , T2 , T3 , ... in a Hilbert space. A result of Shimizu
and Takahashi [33] is generalized, and it is shown that the sequence of iterates converge
to Py, where P is some projection. This is further generalized to a uniformly
smooth Banach space having a weakly continuous duality map. Here the iterates
converge to Qy, where Q is a sunny nonexpansive retraction. For this same iteration
scheme, with finitely many maps Tl , T2, ... , TN , a complementary result to a result of
Bauschke [2] is proved by introducing a new condition on the sequence of parameters
(λn). The iterates converge to Py, where P is the projection onto the intersection
of the fixed point sets of the Tis. Both this result and Bauschke's result [2] are then
generalized to a uniformly smooth Banach space, and to a reflexive Banach space
having a weakly continuous duality map and having Reich's property. Now the iterates
converge to Qy, where Q is the unique sunny nonexpansive retraction onto the
intersection of the fixed point sets of the Tis.
For a random map r : N {I, 2, ... ,N}, the iteration scheme xn+l = Tr(n+l)xn
is considered. In a finite dimensional Hilbert space with Tr(n) = Pr(n) , the iterates
converge to a point in the intersection of the fixed point sets of the PiS. In an arbitrary
Banach space, under certain conditions on the mappings, the iterates converge to a
point in the intersection of the fixed point sets of the Tis.
For the scheme xn+l = (1- λn+l)xn+λn+lTr(n+l)xn, in a finite dimensional Hilbert
space the iterates converge to a point in the intersection of the fixed point sets of the
Tis, and in an infinite dimensional Hilbert space with the added assumption that the
random map r is quasi-cyclic, then the iterates converge weakly to a point in the
intersection of the fixed point sets of the Tis.
Lastly, the minimization of a convex function θ is considered over some closed and
convex subset of a Hilbert space. For both the case where θ is a quadratic function
and for the general case, first the unique fixed points of some maps Tλ are shown
to converge to the unique minimizer of θ and then an algorithm is proposed that
converges to this unique minimizer.