831 research outputs found
Robust Stability Analysis of Sparsely Interconnected Uncertain Systems
In this paper, we consider robust stability analysis of large-scale sparsely
interconnected uncertain systems. By modeling the interconnections among the
subsystems with integral quadratic constraints, we show that robust stability
analysis of such systems can be performed by solving a set of sparse linear
matrix inequalities. We also show that a sparse formulation of the analysis
problem is equivalent to the classical formulation of the robustness analysis
problem and hence does not introduce any additional conservativeness. The
sparse formulation of the analysis problem allows us to apply methods that rely
on efficient sparse factorization techniques, and our numerical results
illustrate the effectiveness of this approach compared to methods that are
based on the standard formulation of the analysis problem.Comment: Provisionally accepted to appear in IEEE Transactions on Automatic
Contro
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
Optimal Rates of Statistical Seriation
Given a matrix the seriation problem consists in permuting its rows in such
way that all its columns have the same shape, for example, they are monotone
increasing. We propose a statistical approach to this problem where the matrix
of interest is observed with noise and study the corresponding minimax rate of
estimation of the matrices. Specifically, when the columns are either unimodal
or monotone, we show that the least squares estimator is optimal up to
logarithmic factors and adapts to matrices with a certain natural structure.
Finally, we propose a computationally efficient estimator in the monotonic case
and study its performance both theoretically and experimentally. Our work is at
the intersection of shape constrained estimation and recent work that involves
permutation learning, such as graph denoising and ranking.Comment: V2 corrects an error in Lemma A.1, v3 corrects appendix F on unimodal
regression where the bounds now hold with polynomial probability rather than
exponentia
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
Decomposition in conic optimization with partially separable structure
Decomposition techniques for linear programming are difficult to extend to
conic optimization problems with general non-polyhedral convex cones because
the conic inequalities introduce an additional nonlinear coupling between the
variables. However in many applications the convex cones have a partially
separable structure that allows them to be characterized in terms of simpler
lower-dimensional cones. The most important example is sparse semidefinite
programming with a chordal sparsity pattern. Here partial separability derives
from the clique decomposition theorems that characterize positive semidefinite
and positive-semidefinite-completable matrices with chordal sparsity patterns.
The paper describes a decomposition method that exploits partial separability
in conic linear optimization. The method is based on Spingarn's method for
equality constrained convex optimization, combined with a fast interior-point
method for evaluating proximal operators
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
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