1 research outputs found
On Convergence of Heuristics Based on Douglas-Rachford Splitting and ADMM to Minimize Convex Functions over Nonconvex Sets
Recently, heuristics based on the Douglas-Rachford splitting algorithm and
the alternating direction method of multipliers (ADMM) have found empirical
success in minimizing convex functions over nonconvex sets, but not much has
been done to improve the theoretical understanding of them. In this paper, we
investigate convergence of these heuristics. First, we characterize optimal
solutions of minimization problems involving convex cost functions over
nonconvex constraint sets. We show that these optimal solutions are related to
the fixed point set of the underlying nonconvex Douglas-Rachford operator.
Next, we establish sufficient conditions under which the Douglas-Rachford
splitting heuristic either converges to a point or its cluster points form a
nonempty compact connected set. In the case where the heuristic converges to a
point, we establish sufficient conditions for that point to be an optimal
solution. Then, we discuss how the ADMM heuristic can be constructed from the
Douglas-Rachford splitting algorithm. We show that, unlike in the convex case,
the algorithms in our nonconvex setup are not equivalent to each other and have
a rather involved relationship between them. Finally, we comment on convergence
of the ADMM heuristic and compare it with the Douglas-Rachford splitting
heuristic.Comment: 11 page