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## Interative approaches to convex feasibility problems.

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. / Thesis (Ph.D.)-University of Durban-Westville, 2001.

Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:ukzn/oai:http://researchspace.ukzn.ac.za:10413/3967 |

Date | January 2001 |

Creators | Pillay, Paranjothi. |

Contributors | Xu, Hong-Kun., O'Hara, John G. |

Source Sets | South African National ETD Portal |

Language | English |

Detected Language | English |

Type | Thesis |

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