A fixed-point representation of the generalized complementarity problem

1985 ◽  
Vol 45 (3) ◽  
pp. 375-381
Author(s):  
S. C. Fang ◽  
E. L. Peterson
Keyword(s):  
Fixed Point ◽  
Complementarity Problem ◽  
Generalized Complementarity Problem ◽  
Point Representation

Convergence of ℓ2/3 Regularization for Sparse Signal Recovery

2015 ◽  
Vol 32 (04) ◽  
pp. 1550023 ◽  
Author(s):  
Lu Liu ◽  
Di-Rong Chen
Keyword(s):  
Fixed Point ◽  
Large Scale ◽  
Sparse Signal ◽  
Sparse Signal Recovery ◽  
Signal Recovery ◽  
Lipschitz Optimization ◽  
Iterative Thresholding ◽  
Point Representation ◽  
Iterative Thresholding Algorithm ◽  
Thresholding Algorithm

In this paper, we consider the problem of finding the sparsest solution to underdetermined linear systems. Unlike the literatures which use the ℓ1 regularization to approximate the original problem, we consider the ℓ2/3 regularization which leads to a better approximation but a nonconvex, nonsmooth, and non-Lipschitz optimization problem. Through developing a fixed point representation theory associated with the two thirds thresholding operator for ℓ2/3 regularization solutions, we propose a fixed point iterative thresholding algorithm based on two thirds norm for solving the k-sparsity problems. Relying on the restricted isometry property, we provide subsequentional convergence guarantee for this fixed point iterative thresholding algorithm on recovering a sparse signal. By discussing the preferred regularization parameters and studying the phase diagram, we get an adequate and efficient algorithm for the high-dimensional sparse signal recovery. Finally, comparing with the existing algorithms, such as the standard ℓ1 minimization, the iterative reweighted ℓ2 minimization, the iterative reweighted ℓ1 minimization, and iterative Half thresholding algorithm, we display the results of the experiment which indicate that the two thirds norm fixed point iterative thresholding algorithm applied to sparse signal recovery and large scale imageries from noisy measurements can be accepted as an effective solver for ℓ2/3 regularization.

scholarly journals Further Application ofH-Differentiability to Generalized Complementarity Problems Based on Generalized Fisher-Burmeister Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Wei-Zhe Gu ◽  
Mohamed A. Tawhid
Keyword(s):  
Stationary Point ◽  
Complementarity Problem ◽  
Global Minimum ◽  
Complementarity Problems ◽  
Merit Function ◽  
Merit Functions ◽  
Generalized Complementarity Problem ◽  
Nonlinear Complementarity ◽  
Generalized Complementarity Problems ◽  
The Given

We study nonsmooth generalized complementarity problems based on the generalized Fisher-Burmeister function and its generalizations, denoted by GCP(f,g) wherefandgareH-differentiable. We describeH-differentials of some GCP functions based on the generalized Fisher-Burmeister function and its generalizations, and their merit functions. Under appropriate conditions on theH-differentials offandg, we show that a local/global minimum of a merit function (or a “stationary point” of a merit function) is coincident with the solution of the given generalized complementarity problem. When specializing GCP(f,g)to the nonlinear complementarity problems, our results not only give new results but also extend/unify various similar results proved forC1, semismooth, and locally Lipschitzian.

scholarly journals Linear program fixed-point representation

MATEMATIKA ◽  
2017 ◽  
Vol 33 (1) ◽  
pp. 55
Author(s):  
Jalaluddin Morris Abdullah
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Symmetric Matrix ◽  
Fixed Point Problem ◽  
Linear Program ◽  
Point Problem ◽  
Central Elements ◽  
Point Representation ◽  
Primal And Dual Problems ◽  
Primal And Dual

From a linear program and its asymmetric dual, invariant primal and dual problems are constructed. Regular mappings are defined between the solution spaces of the original and invariant problems. The notion of centrality is introduced and subsets of regular mappings are shown to be inversely related surjections of central elements, thus representing the original problems as invariant problems. A fixed-point problem involving an idempotent symmetric matrix is constructed from the invariant problems and the notion of centrality carried over to it; the non-negative central fixed-points are shown to map one-to-one to the central solutions to the invariant problems, thus representing the invariant problems as a fixed-point problem and, by transitivity, the original problems as a fixed-point problem.

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