39 research outputs found
A new projection method for finding the closest point in the intersection of convex sets
In this paper we present a new iterative projection method for finding the
closest point in the intersection of convex sets to any arbitrary point in a
Hilbert space. This method, termed AAMR for averaged alternating modified
reflections, can be viewed as an adequate modification of the Douglas--Rachford
method that yields a solution to the best approximation problem. Under a
constraint qualification at the point of interest, we show strong convergence
of the method. In fact, the so-called strong CHIP fully characterizes the
convergence of the AAMR method for every point in the space. We report some
promising numerical experiments where we compare the performance of AAMR
against other projection methods for finding the closest point in the
intersection of pairs of finite dimensional subspaces
Global Behavior of the Douglas-Rachford Method for a Nonconvex Feasibility Problem
In recent times the Douglas-Rachford algorithm has been observed empirically
to solve a variety of nonconvex feasibility problems including those of a
combinatorial nature. For many of these problems current theory is not
sufficient to explain this observed success and is mainly concerned with
questions of local convergence. In this paper we analyze global behavior of the
method for finding a point in the intersection of a half-space and a
potentially non-convex set which is assumed to satisfy a well-quasi-ordering
property or a property weaker than compactness. In particular, the special case
in which the second set is finite is covered by our framework and provides a
prototypical setting for combinatorial optimization problems
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
The Cyclic Douglas-Rachford Algorithm with r-sets-Douglas-Rachford Operators
The Douglas-Rachford (DR) algorithm is an iterative procedure that uses
sequential reflections onto convex sets and which has become popular for convex
feasibility problems. In this paper we propose a structural generalization that
allows to use -sets-DR operators in a cyclic fashion. We prove convergence
and present numerical illustrations of the potential advantage of such
operators with over the classical -sets-DR operators in a cyclic
algorithm.Comment: Accepted for publication in Optimization Methods and Software (OMS)
July 17, 201
The Boosted Double-Proximal Subgradient Algorithm for Nonconvex Optimization
In this paper we introduce the Boosted Double-proximal Subgradient Algorithm
(BDSA), a novel splitting algorithm designed to address general structured
nonsmooth and nonconvex mathematical programs expressed as sums and differences
of composite functions. BDSA exploits the combined nature of subgradients from
the data and proximal steps, and integrates a line-search procedure to enhance
its performance. While BDSA encompasses existing schemes proposed in the
literature, it extends its applicability to more diverse problem domains. We
establish the convergence of BDSA under the Kurdyka--Lojasiewicz property and
provide an analysis of its convergence rate. To evaluate the effectiveness of
BDSA, we introduce a novel family of challenging test functions with an
abundance of critical points. We conduct comparative evaluations demonstrating
its ability to effectively escape non-optimal critical points. Additionally, we
present two practical applications of BDSA for testing its efficacy, namely, a
constrained minimum-sum-of-squares clustering problem and a nonconvex
generalization of Heron's problem
The superiorization method with restarted perturbations for split minimization problems with an application to radiotherapy treatment planning
In this paper we study the split minimization problem that consists of two
constrained minimization problems in two separate spaces that are connected via
a linear operator that maps one space into the other. To handle the data of
such a problem we develop a superiorization approach that can reach a feasible
point with reduced (not necessarily minimal) objective function values. The
superiorization methodology is based on interlacing the iterative steps of two
separate and independent iterative processes by perturbing the iterates of one
process according to the steps dictated by the other process. We include in our
developed method two novel elements. The first one is the permission to restart
the perturbations in the superiorized algorithm which results in a significant
acceleration and increases the computational efficiency. The second element is
the ability to independently superiorize subvectors. This caters to the needs
of real-world applications, as demonstrated here for a problem in
intensity-modulated radiation therapy treatment planning.Comment: Revised version, October 10, 2022; accepted for publication in:
Applied Mathematics and Computatio
Distributed Forward-Backward Methods for Ring Networks
In this work, we propose and analyse forward-backward-type algorithms for
finding a zero of the sum of finitely many monotone operators, which are not
based on reduction to a two operator inclusion in the product space. Each
iteration of the studied algorithms requires one resolvent evaluation per
set-valued operator, one forward evaluation per cocoercive operator, and two
forward evaluations per monotone operator. Unlike existing methods, the
structure of the proposed algorithms are suitable for distributed,
decentralised implementation in ring networks without needing global summation
to enforce consensus between nodes.Comment: 19 page