12,891 research outputs found
On common fixed points approximation of countable families of certain multi-valued maps in hilbert spaces.
Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban 2017.Fixed point theory and its applications have been widely studied by many researchers.
Di erent iterative algorithms have been used extensively to approximate solutions of xed
point problems and other related problems such as equilibrium problems, variational in-
equality problems, optimization problems and so on. In this dissertation, we rst introduce
an iterative algorithm for nding a common solution of multiple-set split equality mixed
equilibrium problem and xed point problem for in nite families of generalized ki-strictly
pseudo-contractive multi-valued mappings in real Hilbert spaces. Using our iterative algo-
rithm, we obtain weak and strong convergence results for approximating a common solution
of multiple-set split equality mixed equilibrium problem and xed point problem. As ap-
plication, we utilize our result to study the split equality mixed variational inequality and
split equality convex minimization problems .
Also, we present another iterative algorithm that does not require the knowledge of the oper-
ator norm for approximating a common solution of split equilibrium problem and xed point
problem for in nite family of multi-valued quasi-nonexpansive mappings in real Hilbert
spaces. Using our iterative algorithm, we state and prove a strong convergence result for
approximating a common solution of split equilibrium problem and xed point problem
for in nite family of multi-valued quasi-nonexpansive mappings in real Hilbert spaces. We
apply our result to convex minimization problem and also present a numerical example
Iterative algorithms for approximating solutions of variational inequality problems and monotone inclusion problems.
Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban, 2017.In this work, we introduce and study an iterative algorithm independent of the operator
norm for approximating a common solution of split equality variational inequality prob-
lem and split equality xed point problem. Using our algorithm, we state and prove a
strong convergence theorem for approximating an element in the intersection of the set
of solutions of a split equality variational inequality problem and the set of solutions of
a split equality xed point problem for demicontractive mappings in real Hilbert spaces.
We then considered nite families of split equality variational inequality problems and
proposed an iterative algorithm for approximating a common solution of this problem and
the multiple-sets split equality xed point problem for countable families of multivalued
type-one demicontractive-type mappings in real Hilbert spaces. A strong convergence re-
sult of the sequence generated by our proposed algorithm to a solution of this problem was
also established. We further extend our study from the frame work of real Hilbert spaces
to more general p-uniformly convex Banach spaces which are also uniformly smooth. In
this space, we introduce an iterative algorithm and prove a strong convergence theorem for
approximating a common solution of split equality monotone inclusion problem and split
equality xed point problem for right Bregman strongly nonexpansive mappings. Finally,
we presented numerical examples of our theorems and applied our results to study the
convex minimization problems and equilibrium problems
Successive Concave Sparsity Approximation for Compressed Sensing
In this paper, based on a successively accuracy-increasing approximation of
the norm, we propose a new algorithm for recovery of sparse vectors
from underdetermined measurements. The approximations are realized with a
certain class of concave functions that aggressively induce sparsity and their
closeness to the norm can be controlled. We prove that the series of
the approximations asymptotically coincides with the and
norms when the approximation accuracy changes from the worst fitting to the
best fitting. When measurements are noise-free, an optimization scheme is
proposed which leads to a number of weighted minimization programs,
whereas, in the presence of noise, we propose two iterative thresholding
methods that are computationally appealing. A convergence guarantee for the
iterative thresholding method is provided, and, for a particular function in
the class of the approximating functions, we derive the closed-form
thresholding operator. We further present some theoretical analyses via the
restricted isometry, null space, and spherical section properties. Our
extensive numerical simulations indicate that the proposed algorithm closely
follows the performance of the oracle estimator for a range of sparsity levels
wider than those of the state-of-the-art algorithms.Comment: Submitted to IEEE Trans. on Signal Processin
Recovery of Low-Rank Matrices under Affine Constraints via a Smoothed Rank Function
In this paper, the problem of matrix rank minimization under affine
constraints is addressed. The state-of-the-art algorithms can recover matrices
with a rank much less than what is sufficient for the uniqueness of the
solution of this optimization problem. We propose an algorithm based on a
smooth approximation of the rank function, which practically improves recovery
limits on the rank of the solution. This approximation leads to a non-convex
program; thus, to avoid getting trapped in local solutions, we use the
following scheme. Initially, a rough approximation of the rank function subject
to the affine constraints is optimized. As the algorithm proceeds, finer
approximations of the rank are optimized and the solver is initialized with the
solution of the previous approximation until reaching the desired accuracy.
On the theoretical side, benefiting from the spherical section property, we
will show that the sequence of the solutions of the approximating function
converges to the minimum rank solution. On the experimental side, it will be
shown that the proposed algorithm, termed SRF standing for Smoothed Rank
Function, can recover matrices which are unique solutions of the rank
minimization problem and yet not recoverable by nuclear norm minimization.
Furthermore, it will be demonstrated that, in completing partially observed
matrices, the accuracy of SRF is considerably and consistently better than some
famous algorithms when the number of revealed entries is close to the minimum
number of parameters that uniquely represent a low-rank matrix.Comment: Accepted in IEEE TSP on December 4th, 201
Spectrum optimization in multi-user multi-carrier systems with iterative convex and nonconvex approximation methods
Several practical multi-user multi-carrier communication systems are
characterized by a multi-carrier interference channel system model where the
interference is treated as noise. For these systems, spectrum optimization is a
promising means to mitigate interference. This however corresponds to a
challenging nonconvex optimization problem. Existing iterative convex
approximation (ICA) methods consist in solving a series of improving convex
approximations and are typically implemented in a per-user iterative approach.
However they do not take this typical iterative implementation into account in
their design. This paper proposes a novel class of iterative approximation
methods that focuses explicitly on the per-user iterative implementation, which
allows to relax the problem significantly, dropping joint convexity and even
convexity requirements for the approximations. A systematic design framework is
proposed to construct instances of this novel class, where several new
iterative approximation methods are developed with improved per-user convex and
nonconvex approximations that are both tighter and simpler to solve (in
closed-form). As a result, these novel methods display a much faster
convergence speed and require a significantly lower computational cost.
Furthermore, a majority of the proposed methods can tackle the issue of getting
stuck in bad locally optimal solutions, and hence improve solution quality
compared to existing ICA methods.Comment: 33 pages, 7 figures. This work has been submitted for possible
publicatio
A new ADMM algorithm for the Euclidean median and its application to robust patch regression
The Euclidean Median (EM) of a set of points in an Euclidean space
is the point x minimizing the (weighted) sum of the Euclidean distances of x to
the points in . While there exits no closed-form expression for the EM,
it can nevertheless be computed using iterative methods such as the Wieszfeld
algorithm. The EM has classically been used as a robust estimator of centrality
for multivariate data. It was recently demonstrated that the EM can be used to
perform robust patch-based denoising of images by generalizing the popular
Non-Local Means algorithm. In this paper, we propose a novel algorithm for
computing the EM (and its box-constrained counterpart) using variable splitting
and the method of augmented Lagrangian. The attractive feature of this approach
is that the subproblems involved in the ADMM-based optimization of the
augmented Lagrangian can be resolved using simple closed-form projections. The
proposed ADMM solver is used for robust patch-based image denoising and is
shown to exhibit faster convergence compared to an existing solver.Comment: 5 pages, 3 figures, 1 table. To appear in Proc. IEEE International
Conference on Acoustics, Speech, and Signal Processing, April 19-24, 201
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