8,040 research outputs found
Accelerated Consensus via Min-Sum Splitting
We apply the Min-Sum message-passing protocol to solve the consensus problem
in distributed optimization. We show that while the ordinary Min-Sum algorithm
does not converge, a modified version of it known as Splitting yields
convergence to the problem solution. We prove that a proper choice of the
tuning parameters allows Min-Sum Splitting to yield subdiffusive accelerated
convergence rates, matching the rates obtained by shift-register methods. The
acceleration scheme embodied by Min-Sum Splitting for the consensus problem
bears similarities with lifted Markov chains techniques and with multi-step
first order methods in convex optimization
Joint Spectral Radius and Path-Complete Graph Lyapunov Functions
We introduce the framework of path-complete graph Lyapunov functions for
approximation of the joint spectral radius. The approach is based on the
analysis of the underlying switched system via inequalities imposed among
multiple Lyapunov functions associated to a labeled directed graph. Inspired by
concepts in automata theory and symbolic dynamics, we define a class of graphs
called path-complete graphs, and show that any such graph gives rise to a
method for proving stability of the switched system. This enables us to derive
several asymptotically tight hierarchies of semidefinite programming
relaxations that unify and generalize many existing techniques such as common
quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov
functions. We compare the quality of approximation obtained by certain classes
of path-complete graphs including a family of dual graphs and all path-complete
graphs with two nodes on an alphabet of two matrices. We provide approximation
guarantees for several families of path-complete graphs, such as the De Bruijn
graphs, establishing as a byproduct a constructive converse Lyapunov theorem
for maximum/minimum-of-quadratics Lyapunov functions.Comment: To appear in SIAM Journal on Control and Optimization. Version 2 has
gone through two major rounds of revision. In particular, a section on the
performance of our algorithm on application-motivated problems has been added
and a more comprehensive literature review is presente
Sensitivity Analysis for Mirror-Stratifiable Convex Functions
This paper provides a set of sensitivity analysis and activity identification
results for a class of convex functions with a strong geometric structure, that
we coined "mirror-stratifiable". These functions are such that there is a
bijection between a primal and a dual stratification of the space into
partitioning sets, called strata. This pairing is crucial to track the strata
that are identifiable by solutions of parametrized optimization problems or by
iterates of optimization algorithms. This class of functions encompasses all
regularizers routinely used in signal and image processing, machine learning,
and statistics. We show that this "mirror-stratifiable" structure enjoys a nice
sensitivity theory, allowing us to study stability of solutions of optimization
problems to small perturbations, as well as activity identification of
first-order proximal splitting-type algorithms. Existing results in the
literature typically assume that, under a non-degeneracy condition, the active
set associated to a minimizer is stable to small perturbations and is
identified in finite time by optimization schemes. In contrast, our results do
not require any non-degeneracy assumption: in consequence, the optimal active
set is not necessarily stable anymore, but we are able to track precisely the
set of identifiable strata.We show that these results have crucial implications
when solving challenging ill-posed inverse problems via regularization, a
typical scenario where the non-degeneracy condition is not fulfilled. Our
theoretical results, illustrated by numerical simulations, allow to
characterize the instability behaviour of the regularized solutions, by
locating the set of all low-dimensional strata that can be potentially
identified by these solutions
Tropical Kraus maps for optimal control of switched systems
Kraus maps (completely positive trace preserving maps) arise classically in
quantum information, as they describe the evolution of noncommutative
probability measures. We introduce tropical analogues of Kraus maps, obtained
by replacing the addition of positive semidefinite matrices by a multivalued
supremum with respect to the L\"owner order. We show that non-linear
eigenvectors of tropical Kraus maps determine piecewise quadratic
approximations of the value functions of switched optimal control problems.
This leads to a new approximation method, which we illustrate by two
applications: 1) approximating the joint spectral radius, 2) computing
approximate solutions of Hamilton-Jacobi PDE arising from a class of switched
linear quadratic problems studied previously by McEneaney. We report numerical
experiments, indicating a major improvement in terms of scalability by
comparison with earlier numerical schemes, owing to the "LMI-free" nature of
our method.Comment: 15 page
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
We show that for any positive integer , there are families of switched
linear systems---in fixed dimension and defined by two matrices only---that are
stable under arbitrary switching but do not admit (i) a polynomial Lyapunov
function of degree , or (ii) a polytopic Lyapunov function with facets, or (iii) a piecewise quadratic Lyapunov function with
pieces. This implies that there cannot be an upper bound on the size of the
linear and semidefinite programs that search for such stability certificates.
Several constructive and non-constructive arguments are presented which connect
our problem to known (and rather classical) results in the literature regarding
the finiteness conjecture, undecidability, and non-algebraicity of the joint
spectral radius. In particular, we show that existence of an extremal piecewise
algebraic Lyapunov function implies the finiteness property of the optimal
product, generalizing a result of Lagarias and Wang. As a corollary, we prove
that the finiteness property holds for sets of matrices with an extremal
Lyapunov function belonging to some of the most popular function classes in
controls
- …