62 research outputs found
A Statistical Learning Theory Approach for Uncertain Linear and Bilinear Matrix Inequalities
In this paper, we consider the problem of minimizing a linear functional
subject to uncertain linear and bilinear matrix inequalities, which depend in a
possibly nonlinear way on a vector of uncertain parameters. Motivated by recent
results in statistical learning theory, we show that probabilistic guaranteed
solutions can be obtained by means of randomized algorithms. In particular, we
show that the Vapnik-Chervonenkis dimension (VC-dimension) of the two problems
is finite, and we compute upper bounds on it. In turn, these bounds allow us to
derive explicitly the sample complexity of these problems. Using these bounds,
in the second part of the paper, we derive a sequential scheme, based on a
sequence of optimization and validation steps. The algorithm is on the same
lines of recent schemes proposed for similar problems, but improves both in
terms of complexity and generality. The effectiveness of this approach is shown
using a linear model of a robot manipulator subject to uncertain parameters.Comment: 19 pages, 2 figures, Accepted for Publication in Automatic
Sequential Randomized Algorithms for Convex Optimization in the Presence of Uncertainty
In this paper, we propose new sequential randomized algorithms for convex
optimization problems in the presence of uncertainty. A rigorous analysis of
the theoretical properties of the solutions obtained by these algorithms, for
full constraint satisfaction and partial constraint satisfaction, respectively,
is given. The proposed methods allow to enlarge the applicability of the
existing randomized methods to real-world applications involving a large number
of design variables. Since the proposed approach does not provide a priori
bounds on the sample complexity, extensive numerical simulations, dealing with
an application to hard-disk drive servo design, are provided. These simulations
testify the goodness of the proposed solution.Comment: 18 pages, Submitted for publication to IEEE Transactions on Automatic
Contro
Randomized algorithms for control of uncertain systems with application to hand disk drives
Ph.DDOCTOR OF PHILOSOPH
Randomized Constraints Consensus for Distributed Robust Linear Programming
In this paper we consider a network of processors aiming at cooperatively
solving linear programming problems subject to uncertainty. Each node only
knows a common cost function and its local uncertain constraint set. We propose
a randomized, distributed algorithm working under time-varying, asynchronous
and directed communication topology. The algorithm is based on a local
computation and communication paradigm. At each communication round, nodes
perform two updates: (i) a verification in which they check-in a randomized
setup-the robust feasibility (and hence optimality) of the candidate optimal
point, and (ii) an optimization step in which they exchange their candidate
bases (minimal sets of active constraints) with neighbors and locally solve an
optimization problem whose constraint set includes: a sampled constraint
violating the candidate optimal point (if it exists), agent's current basis and
the collection of neighbor's basis. As main result, we show that if a processor
successfully performs the verification step for a sufficient number of
communication rounds, it can stop the algorithm since a consensus has been
reached. The common solution is-with high confidence-feasible (and hence
optimal) for the entire set of uncertainty except a subset having arbitrary
small probability measure. We show the effectiveness of the proposed
distributed algorithm on a multi-core platform in which the nodes communicate
asynchronously.Comment: Accepted for publication in the 20th World Congress of the
International Federation of Automatic Control (IFAC
On Repetitive Scenario Design
Repetitive Scenario Design (RSD) is a randomized approach to robust design
based on iterating two phases: a standard scenario design phase that uses
scenarios (design samples), followed by randomized feasibility phase that uses
test samples on the scenario solution. We give a full and exact
probabilistic characterization of the number of iterations required by the RSD
approach for returning a solution, as a function of , , and of the
desired levels of probabilistic robustness in the solution. This novel approach
broadens the applicability of the scenario technology, since the user is now
presented with a clear tradeoff between the number of design samples and
the ensuing expected number of repetitions required by the RSD algorithm. The
plain (one-shot) scenario design becomes just one of the possibilities, sitting
at one extreme of the tradeoff curve, in which one insists in finding a
solution in a single repetition: this comes at the cost of possibly high .
Other possibilities along the tradeoff curve use lower values, but possibly
require more than one repetition
Randomized constraints consensus for distributed robust mixed-integer programming
open4siThis work was supported in part
by the European Research Council (ERC) under the European Union’s
Horizon 2020 Research and Innovation Programme under Grant 638992
- OPT4SMART, and in part by a grant from the Singapore National
Research Foundation (NRF) under the ASPIRE Project under Grant
NCR-NCR001-040.In this article, we consider a network of processors aiming at cooperatively solving mixed-integer convex programs subject to uncertainty. Each node only knows a common cost function and its local uncertain constraint set. We propose a randomized, distributed algorithm working under asynchronous, unreliable, and directed communication. The algorithm is based on a local computation and communication paradigm. At each communication round, nodes perform two updates: 1) A verification in which they check - in a randomized fashion - the robust feasibility of a candidate optimal point, and 2) an optimization step in which they exchange their candidate basis (the minimal set of constraints defining a solution) with neighbors and locally solve an optimization problem. As a main result, we show that processors can stop the algorithm after a finite number of communication rounds (either because verification has been successful for a sufficient number of rounds or because a given threshold has been reached) so that candidate optimal solutions are consensual. The common solution has proven to be - with high confidence - feasible and, hence, optimal for the entire set of uncertainty except a subset having an arbitrarily small probability measure. We show the effectiveness of the proposed distributed algorithm using two examples: a random, uncertain mixed-integer linear program and a distributed localization in wireless sensor networks. The distributed algorithm is implemented on a multicore platform in which the nodes communicate asynchronously.embargoed_20210317Chamanbaz M.; Notarstefano G.; Sasso F.; Bouffanais R.Chamanbaz M.; Notarstefano G.; Sasso F.; Bouffanais R
An Overview of Integral Quadratic Constraints for Delayed Nonlinear and Parameter-Varying Systems
A general framework is presented for analyzing the stability and performance
of nonlinear and linear parameter varying (LPV) time delayed systems. First,
the input/output behavior of the time delay operator is bounded in the
frequency domain by integral quadratic constraints (IQCs). A constant delay is
a linear, time-invariant system and this leads to a simple, intuitive
interpretation for these frequency domain constraints. This simple
interpretation is used to derive new IQCs for both constant and varying delays.
Second, the performance of nonlinear and LPV delayed systems is bounded using
dissipation inequalities that incorporate IQCs. This step makes use of recent
results that show, under mild technical conditions, that an IQC has an
equivalent representation as a finite-horizon time-domain constraint. Numerical
examples are provided to demonstrate the effectiveness of the method for both
class of systems
Repetitive Scenario Design
Repetitive Scenario Design (RSD) is a randomized approach to robust design based on iterating two phases: a standard scenario design phase that uses N scenarios (design samples), followed by randomized feasibility phase that uses No test samples on the scenario solution. We give a full and exact probabilistic characterization of the number of iterations required by the RSD approach for returning a solution, as a function of N, No, and of the desired levels of probabilistic robustness in the solution. This novel approach broadens the applicability of the scenario technology, since the user is now presented with a clear tradeoff between the number N of design samples and the ensuing expected number of repetitions required by the RSD algorithm. The plain (one-shot) scenario design becomes just one of the possibilities, sitting at one extreme of the tradeoff curve, in which one insists in finding a solution in a single repetition: this comes at the cost of possibly high N. Other possibilities along the tradeoff curve use lower N values, but possibly require more than one repetition
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