267 research outputs found
Peaceman-Rachford splitting for a class of nonconvex optimization problems
We study the applicability of the Peaceman-Rachford (PR) splitting method for
solving nonconvex optimization problems. When applied to minimizing the sum of
a strongly convex Lipschitz differentiable function and a proper closed
function, we show that if the strongly convex function has a large enough
strong convexity modulus and the step-size parameter is chosen below a
threshold that is computable, then any cluster point of the sequence generated,
if exists, will give a stationary point of the optimization problem. We also
give sufficient conditions guaranteeing boundedness of the sequence generated.
We then discuss one way to split the objective so that the proposed method can
be suitably applied to solving optimization problems with a coercive objective
that is the sum of a (not necessarily strongly) convex Lipschitz differentiable
function and a proper closed function; this setting covers a large class of
nonconvex feasibility problems and constrained least squares problems. Finally,
we illustrate the proposed algorithm numerically
Projection Methods: Swiss Army Knives for Solving Feasibility and Best Approximation Problems with Halfspaces
We model a problem motivated by road design as a feasibility problem.
Projections onto the constraint sets are obtained, and projection methods for
solving the feasibility problem are studied. We present results of numerical
experiments which demonstrate the efficacy of projection methods even for
challenging nonconvex problems
A convergent relaxation of the Douglas-Rachford algorithm
This paper proposes an algorithm for solving structured optimization
problems, which covers both the backward-backward and the Douglas-Rachford
algorithms as special cases, and analyzes its convergence. The set of fixed
points of the algorithm is characterized in several cases. Convergence criteria
of the algorithm in terms of general fixed point operators are established.
When applying to nonconvex feasibility including the inconsistent case, we
prove local linear convergence results under mild assumptions on regularity of
individual sets and of the collection of sets which need not intersect. In this
special case, we refine known linear convergence criteria for the
Douglas-Rachford algorithm (DR). As a consequence, for feasibility with one of
the sets being affine, we establish criteria for linear and sublinear
convergence of convex combinations of the alternating projection and the DR
methods. These results seem to be new. We also demonstrate the seemingly
improved numerical performance of this algorithm compared to the RAAR algorithm
for both consistent and inconsistent sparse feasibility problems
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