306 research outputs found
New Relaxation Modulus Based Iterative Method for Large and Sparse Implicit Complementarity Problem
This article presents a class of new relaxation modulus-based iterative
methods to process the large and sparse implicit complementarity problem (ICP).
Using two positive diagonal matrices, we formulate a fixed-point equation and
prove that it is equivalent to ICP. Also, we provide sufficient convergence
conditions for the proposed methods when the system matrix is a -matrix or
an -matrix.
Keyword: Implicit complementarity problem, -matrix, -matrix, matrix
splitting, convergenceComment: arXiv admin note: substantial text overlap with arXiv:2303.1251
Applications of a splitting algorithm to decomposition in convex programming and variational inequalities
Cover title.Includes bibliographical references.Partially supported by the U.S. Army Research Office (Center for Intelligent Control Systems) DAAL03-86-K-0171 Partially supported by the National Science Foundation. NSF-ECS-8519058by Paul Tseng
Some recent advances in projection-type methods for variational inequalities
AbstractProjection-type methods are a class of simple methods for solving variational inequalities, especially for complementarity problems. In this paper we review and summarize recent developments in this class of methods, and focus mainly on some new trends in projection-type methods
GMRES-Accelerated ADMM for Quadratic Objectives
We consider the sequence acceleration problem for the alternating direction
method-of-multipliers (ADMM) applied to a class of equality-constrained
problems with strongly convex quadratic objectives, which frequently arise as
the Newton subproblem of interior-point methods. Within this context, the ADMM
update equations are linear, the iterates are confined within a Krylov
subspace, and the General Minimum RESidual (GMRES) algorithm is optimal in its
ability to accelerate convergence. The basic ADMM method solves a
-conditioned problem in iterations. We give
theoretical justification and numerical evidence that the GMRES-accelerated
variant consistently solves the same problem in iterations
for an order-of-magnitude reduction in iterations, despite a worst-case bound
of iterations. The method is shown to be competitive against
standard preconditioned Krylov subspace methods for saddle-point problems. The
method is embedded within SeDuMi, a popular open-source solver for conic
optimization written in MATLAB, and used to solve many large-scale semidefinite
programs with error that decreases like , instead of ,
where is the iteration index.Comment: 31 pages, 7 figures. Accepted for publication in SIAM Journal on
Optimization (SIOPT
On general fixed point method based on matrix splitting for solving linear complementarity problem
In this article, we introduce a modified fixed point method to process the large and sparse linear complementarity problem (LCP) and formulate an equivalent fixed point equation for the LCP and show the equivalence. Also, we provide convergence conditions when the system matrix is a -matrix and two sufficient convergence conditions when the system matrix is an -matrix. To show the efficiency of our proposed method, we illustrate two numerical examples for different parameters
On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence
We introduce a framework for quasi-Newton forward--backward splitting
algorithms (proximal quasi-Newton methods) with a metric induced by diagonal
rank- symmetric positive definite matrices. This special type of
metric allows for a highly efficient evaluation of the proximal mapping. The
key to this efficiency is a general proximal calculus in the new metric. By
using duality, formulas are derived that relate the proximal mapping in a
rank- modified metric to the original metric. We also describe efficient
implementations of the proximity calculation for a large class of functions;
the implementations exploit the piece-wise linear nature of the dual problem.
Then, we apply these results to acceleration of composite convex minimization
problems, which leads to elegant quasi-Newton methods for which we prove
convergence. The algorithm is tested on several numerical examples and compared
to a comprehensive list of alternatives in the literature. Our quasi-Newton
splitting algorithm with the prescribed metric compares favorably against
state-of-the-art. The algorithm has extensive applications including signal
processing, sparse recovery, machine learning and classification to name a few.Comment: arXiv admin note: text overlap with arXiv:1206.115
On the convergence of a matrix splitting algorithm for the symmetric linear complementarity problem
Cover title.Includes bibliographical references (p. 45-48).Partially supported by the U.S. Army Research Office (Center for Intelligent Control Systems) DAAL03-86-K-0171 Partially supported by the Office of Naval Research. N00014-84-K-0519 (NR64-003) Partially supported by the National Science Foundation. NSF-ECS-8519058by Zhi-Quan Luo, Paul Tseng
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