33 research outputs found
ITERATIVE COMPUTATION FOR SOLVING CONVEX OPTIMIZATION PROBLEMS OVER THE SET OF COMMON FIXED POINTS OF QUASI-NONEXPANSIVE AND DEMICONTRACTIVE MAPPINGS
In this paper, a new iterative method for solving convex minimization problems over the set of common fixed points of quasi-nonexpansive and demicontractive mappings is constructed. Convergence theorems are also proved in Hilbert spaces without any compactness assumption. As an application, we shall utilize our results to solve quadratic optimization problems involving bounded linear operator. Our theorems are significant improvements on several important recent results
Strong convergence of inertial extragradient algorithms for solving variational inequalities and fixed point problems
The paper investigates two inertial extragradient algorithms for seeking a
common solution to a variational inequality problem involving a monotone and
Lipschitz continuous mapping and a fixed point problem with a demicontractive
mapping in real Hilbert spaces. Our algorithms only need to calculate the
projection on the feasible set once in each iteration. Moreover, they can work
well without the prior information of the Lipschitz constant of the cost
operator and do not contain any line search process. The strong convergence of
the algorithms is established under suitable conditions. Some experiments are
presented to illustrate the numerical efficiency of the suggested algorithms
and compare them with some existing ones.Comment: 25 pages, 12 figure
A hybrid proximal point algorithm for finding minimizers and fixed points in CAT(0) spaces
We introduce a hybrid proximal point algorithm and establish its strong convergence to a common solution of a proximal point of a lower semi-continuous mapping and a fixed point of a demicontractive mapping in the framework of a CAT(0) space. As applications of our new result, we solve variational inequality problems for these mappings on a Hilbert space. Illustrative example is given to validate theoretical result obtained herein.The Deanship of Scientific Research (DSR) at King Fahd University of Petroleum and Minerals (KFUPM) for funding this work through project no. IN151014.http://link.springer.com/journal/117842019-06-01hj2018Mathematics and Applied Mathematic
Sharp estimation of local convergence radius for the Picard iteration
We investigate the local convergence radius of a general Picard iteration in the frame of a real Hilbert space. We propose a new algorithm to estimate the local convergence radius. Numerical experiments show that the proposed procedure gives sharp estimation (i.e., close to or even identical with the best one) for several well known or recent iterative methods and for various nonlinear mappings. Particularly, we applied the proposed algorithm for classical Newton method, for multi-step Newton method (in particular for third-order Potra-Ptak method) and for fifth-order "M5" method. We present also a new formula to estimate the local convergence radius for multi-step Newton method
Strict pseudocontractions and demicontractions, their properties and applications
We give properties of strict pseudocontractions and demicontractions defined
on a Hilbert space, which constitute wide classes of operators that arise in
iterative methods for solving fixed point problems. In particular, we give
necessary and sufficient conditions under which a convex combination and
composition of strict pseudocontractions as well as demicontractions that share
a common fixed point is again a strict pseudocontraction or a demicontraction,
respectively. Moreover, we introduce a generalized relaxation of composition of
demicontraction and give its properties. We apply these properties to prove the
weak convergence of a class of algorithms that is wider than the
Douglas-Rachford algorithm and projected Landweber algorithms. We have also
presented two numerical examples, where we compare the behavior of the
presented methods with the Douglas-Rachford method.Comment: 27 pages, 3 figure
Self-adaptive inertial algorithms for approximating solutions of split feasilbility, monotone inclusion, variational inequality and fixed point problems.
Masters Degree. University of KwaZulu-Natal, Durban.In this dissertation, we introduce a self-adaptive hybrid inertial algorithm for approximating
a solution of split feasibility problem which also solves a monotone inclusion problem
and a fixed point problem in p-uniformly convex and uniformly smooth Banach spaces.
We prove a strong convergence theorem for the sequence generated by our algorithm which
does not require a prior knowledge of the norm of the bounded linear operator. Numerical
examples are given to compare the computational performance of our algorithm with other
existing algorithms.
Moreover, we present a new iterative algorithm of inertial form for solving Monotone Inclusion
Problem (MIP) and common Fixed Point Problem (FPP) of a finite family of
demimetric mappings in a real Hilbert space. Motivated by the Armijo line search technique,
we incorporate the inertial technique to accelerate the convergence of the proposed
method. Under standard and mild assumptions of monotonicity and Lipschitz continuity
of the MIP associated mappings, we establish the strong convergence of the iterative
algorithm. Some numerical examples are presented to illustrate the performance of our
method as well as comparing it with the non-inertial version and some related methods in
the literature.
Furthermore, we propose a new modified self-adaptive inertial subgradient extragradient
algorithm in which the two projections are made onto some half spaces. Moreover, under
mild conditions, we obtain a strong convergence of the sequence generated by our proposed
algorithm for approximating a common solution of variational inequality problems
and common fixed points of a finite family of demicontractive mappings in a real Hilbert
space. The main advantages of our algorithm are: strong convergence result obtained
without prior knowledge of the Lipschitz constant of the the related monotone operator,
the two projections made onto some half-spaces and the inertial technique which speeds
up rate of convergence. Finally, we present an application and a numerical example to
illustrate the usefulness and applicability of our algorithm