8,108 research outputs found
A scenario approach for non-convex control design
Randomized optimization is an established tool for control design with
modulated robustness. While for uncertain convex programs there exist
randomized approaches with efficient sampling, this is not the case for
non-convex problems. Approaches based on statistical learning theory are
applicable to non-convex problems, but they usually are conservative in terms
of performance and require high sample complexity to achieve the desired
probabilistic guarantees. In this paper, we derive a novel scenario approach
for a wide class of random non-convex programs, with a sample complexity
similar to that of uncertain convex programs and with probabilistic guarantees
that hold not only for the optimal solution of the scenario program, but for
all feasible solutions inside a set of a-priori chosen complexity. We also
address measure-theoretic issues for uncertain convex and non-convex programs.
Among the family of non-convex control- design problems that can be addressed
via randomization, we apply our scenario approach to randomized Model
Predictive Control for chance-constrained nonlinear control-affine systems.Comment: Submitted to IEEE Transactions on Automatic Contro
Distributed Random Convex Programming via Constraints Consensus
This paper discusses distributed approaches for the solution of random convex
programs (RCP). RCPs are convex optimization problems with a (usually large)
number N of randomly extracted constraints; they arise in several applicative
areas, especially in the context of decision under uncertainty, see [2],[3]. We
here consider a setup in which instances of the random constraints (the
scenario) are not held by a single centralized processing unit, but are
distributed among different nodes of a network. Each node "sees" only a small
subset of the constraints, and may communicate with neighbors. The objective is
to make all nodes converge to the same solution as the centralized RCP problem.
To this end, we develop two distributed algorithms that are variants of the
constraints consensus algorithm [4],[5]: the active constraints consensus (ACC)
algorithm, and the vertex constraints consensus (VCC) algorithm. We show that
the ACC algorithm computes the overall optimal solution in finite time, and
with almost surely bounded communication at each iteration. The VCC algorithm
is instead tailored for the special case in which the constraint functions are
convex also w.r.t. the uncertain parameters, and it computes the solution in a
number of iterations bounded by the diameter of the communication graph. We
further devise a variant of the VCC algorithm, namely quantized vertex
constraints consensus (qVCC), to cope with the case in which communication
bandwidth among processors is bounded. We discuss several applications of the
proposed distributed techniques, including estimation, classification, and
random model predictive control, and we present a numerical analysis of the
performance of the proposed methods. As a complementary numerical result, we
show that the parallel computation of the scenario solution using ACC algorithm
significantly outperforms its centralized equivalent
On the Sample Size of Random Convex Programs with Structured Dependence on the Uncertainty (Extended Version)
The "scenario approach" provides an intuitive method to address chance
constrained problems arising in control design for uncertain systems. It
addresses these problems by replacing the chance constraint with a finite
number of sampled constraints (scenarios). The sample size critically depends
on Helly's dimension, a quantity always upper bounded by the number of decision
variables. However, this standard bound can lead to computationally expensive
programs whose solutions are conservative in terms of cost and violation
probability. We derive improved bounds of Helly's dimension for problems where
the chance constraint has certain structural properties. The improved bounds
lower the number of scenarios required for these problems, leading both to
improved objective value and reduced computational complexity. Our results are
generally applicable to Randomized Model Predictive Control of chance
constrained linear systems with additive uncertainty and affine disturbance
feedback. The efficacy of the proposed bound is demonstrated on an inventory
management example.Comment: Accepted for publication at Automatic
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