192 research outputs found
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
Block Factor-width-two Matrices and Their Applications to Semidefinite and Sum-of-squares Optimization
Semidefinite and sum-of-squares (SOS) optimization are fundamental
computational tools in many areas, including linear and nonlinear systems
theory. However, the scale of problems that can be addressed reliably and
efficiently is still limited. In this paper, we introduce a new notion of
\emph{block factor-width-two matrices} and build a new hierarchy of inner and
outer approximations of the cone of positive semidefinite (PSD) matrices. This
notion is a block extension of the standard factor-width-two matrices, and
allows for an improved inner-approximation of the PSD cone. In the context of
SOS optimization, this leads to a block extension of the \emph{scaled
diagonally dominant sum-of-squares (SDSOS)} polynomials. By varying a matrix
partition, the notion of block factor-width-two matrices can balance a
trade-off between the computation scalability and solution quality for solving
semidefinite and SOS optimization. Numerical experiments on large-scale
instances confirm our theoretical findings.Comment: 26 pages, 5 figures. Added a new section on the approximation quality
analysis using block factor-width-two matrices. Code is available through
https://github.com/zhengy09/SDPf
Linear optimization over homogeneous matrix cones
A convex cone is homogeneous if its automorphism group acts transitively on
the interior of the cone, i.e., for every pair of points in the interior of the
cone, there exists a cone automorphism that maps one point to the other. Cones
that are homogeneous and self-dual are called symmetric. The symmetric cones
include the positive semidefinite matrix cone and the second order cone as
important practical examples. In this paper, we consider the less well-studied
conic optimization problems over cones that are homogeneous but not necessarily
self-dual. We start with cones of positive semidefinite symmetric matrices with
a given sparsity pattern. Homogeneous cones in this class are characterized by
nested block-arrow sparsity patterns, a subset of the chordal sparsity
patterns. We describe transitive subsets of the automorphism groups of the
cones and their duals, and important properties of the composition of log-det
barrier functions with the automorphisms in this set. Next, we consider
extensions to linear slices of the positive semidefinite cone, i.e.,
intersection of the positive semidefinite cone with a linear subspace, and
review conditions that make the cone homogeneous. In the third part of the
paper we give a high-level overview of the classical algebraic theory of
homogeneous cones due to Vinberg and Rothaus. A fundamental consequence of this
theory is that every homogeneous cone admits a spectrahedral (linear matrix
inequality) representation. We conclude by discussing the role of homogeneous
cone structure in primal-dual symmetric interior-point methods.Comment: 59 pages, 10 figures, to appear in Acta Numeric
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