28 research outputs found
A weakly convergent fully inexact Douglas-Rachford method with relative error tolerance
Douglas-Rachford method is a splitting algorithm for finding a zero of the
sum of two maximal monotone operators. Each of its iterations requires the
sequential solution of two proximal subproblems. The aim of this work is to
present a fully inexact version of Douglas-Rachford method wherein both
proximal subproblems are solved approximately within a relative error
tolerance. We also present a semi-inexact variant in which the first subproblem
is solved exactly and the second one inexactly. We prove that both methods
generate sequences weakly convergent to the solution of the underlying
inclusion problem, if any
New Douglas-Rachford algorithmic structures and their convergence analyses
In this paper we study new algorithmic structures with Douglas- Rachford (DR)
operators to solve convex feasibility problems. We propose to embed the basic
two-set-DR algorithmic operator into the String-Averaging Projections (SAP) and
into the Block-Iterative Pro- jection (BIP) algorithmic structures, thereby
creating new DR algo- rithmic schemes that include the recently proposed cyclic
Douglas- Rachford algorithm and the averaged DR algorithm as special cases. We
further propose and investigate a new multiple-set-DR algorithmic operator.
Convergence of all these algorithmic schemes is studied by using properties of
strongly quasi-nonexpansive operators and firmly nonexpansive operators.Comment: SIAM Journal on Optimization, accepted for publicatio
A feasibility approach for constructing combinatorial designs of circulant type
In this work, we propose an optimization approach for constructing various
classes of circulant combinatorial designs that can be defined in terms of
autocorrelations. The problem is formulated as a so-called feasibility problem
having three sets, to which the Douglas-Rachford projection algorithm is
applied. The approach is illustrated on three different classes of circulant
combinatorial designs: circulant weighing matrices, D-optimal matrices, and
Hadamard matrices with two circulant cores. Furthermore, we explicitly
construct two new circulant weighing matrices, a and a
, whose existence was previously marked as unresolved in the most
recent version of Strassler's table