Ellipsoidal bounds on state trajectories for discrete-time systems with linear fractional uncertainties

Computation of exact ellipsoidal bounds on the state trajectories of discrete-time linear systems that have time-varying or time-invariant linear fractional parameter uncertainties and ellipsoidal uncertainty in the initial state is known to be NP-hard. This paper proposes three algorithms to compute ellipsoidal bounds on such a state trajectory set and discusses the tradeoffs between computational complexity and conservatism of the algorithms. The approach employs linear matrix inequalities to determine an initial estimate of the ellipsoid that is refined by the subsequent application of the skewed structured singular value ν. Numerical examples are used to illustrate the application of the proposed algorithms and to compare the differences between them, where small conservatism for the tightest bounds is observed.Institute for Advanced Computing Applications and Technologie

Adaptive Cut Generation Algorithm for Improved Linear Programming Decoding of Binary Linear Codes

Linear programming (LP) decoding approximates maximum-likelihood (ML) decoding of a linear block code by relaxing the equivalent ML integer programming (IP) problem into a more easily solved LP problem. The LP problem is defined by a set of box constraints together with a set of linear inequalities called "parity inequalities" that are derived from the constraints represented by the rows of a parity-check matrix of the code and can be added iteratively and adaptively. In this paper, we first derive a new necessary condition and a new sufficient condition for a violated parity inequality constraint, or "cut," at a point in the unit hypercube. Then, we propose a new and effective algorithm to generate parity inequalities derived from certain additional redundant parity check (RPC) constraints that can eliminate pseudocodewords produced by the LP decoder, often significantly improving the decoder error-rate performance. The cut-generating algorithm is based upon a specific transformation of an initial parity-check matrix of the linear block code. We also design two variations of the proposed decoder to make it more efficient when it is combined with the new cut-generating algorithm. Simulation results for several low-density parity-check (LDPC) codes demonstrate that the proposed decoding algorithms significantly narrow the performance gap between LP decoding and ML decoding

Semi-proximal Mirror-Prox for Nonsmooth Composite Minimization

We propose a new first-order optimisation algorithm to solve high-dimensional non-smooth composite minimisation problems. Typical examples of such problems have an objective that decomposes into a non-smooth empirical risk part and a non-smooth regularisation penalty. The proposed algorithm, called Semi-Proximal Mirror-Prox, leverages the Fenchel-type representation of one part of the objective while handling the other part of the objective via linear minimization over the domain. The algorithm stands in contrast with more classical proximal gradient algorithms with smoothing, which require the computation of proximal operators at each iteration and can therefore be impractical for high-dimensional problems. We establish the theoretical convergence rate of Semi-Proximal Mirror-Prox, which exhibits the optimal complexity bounds, i.e. $O(1/\epsilon^2)$, for the number of calls to linear minimization oracle. We present promising experimental results showing the interest of the approach in comparison to competing methods

Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.Comment: 28 pages, survey pape

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

Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference

We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF inference problems. The core of our method is a very efficient bounding procedure, which combines scalable semidefinite programming (SDP) and a cutting-plane method for seeking violated constraints. In order to further speed up the computation, several strategies have been exploited, including model reduction, warm start and removal of inactive constraints. We analyze the performance of the proposed method under different settings, and demonstrate that our method either outperforms or performs on par with state-of-the-art approaches. Especially when the connectivities are dense or when the relative magnitudes of the unary costs are low, we achieve the best reported results. Experiments show that the proposed algorithm achieves better approximation than the state-of-the-art methods within a variety of time budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page

Algorithms for Highly Symmetric Linear and Integer Programs

This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of lower dimension. Combining this approach with knowledge of the geometry of feasible integer solutions yields an algorithm for solving highly symmetric integer linear programs which only takes time which is linear in the number of constraints and quadratic in the dimension.Comment: 21 pages, 1 figure; some references and further comments added, title slightly change