4,128 research outputs found
Randomized Solutions to Convex Programs with Multiple Chance Constraints
The scenario-based optimization approach (`scenario approach') provides an
intuitive way of approximating the solution to chance-constrained optimization
programs, based on finding the optimal solution under a finite number of
sampled outcomes of the uncertainty (`scenarios'). A key merit of this approach
is that it neither assumes knowledge of the uncertainty set, as it is common in
robust optimization, nor of its probability distribution, as it is usually
required in stochastic optimization. Moreover, the scenario approach is
computationally efficient as its solution is based on a deterministic
optimization program that is canonically convex, even when the original
chance-constrained problem is not. Recently, researchers have obtained
theoretical foundations for the scenario approach, providing a direct link
between the number of scenarios and bounds on the constraint violation
probability. These bounds are tight in the general case of an uncertain
optimization problem with a single chance constraint. However, this paper shows
that these bounds can be improved in situations where the constraints have a
limited `support rank', a new concept that is introduced for the first time.
This property is typically found in a large number of practical
applications---most importantly, if the problem originally contains multiple
chance constraints (e.g. multi-stage uncertain decision problems), or if a
chance constraint belongs to a special class of constraints (e.g. linear or
quadratic constraints). In these cases the quality of the scenario solution is
improved while the same bound on the constraint violation probability is
maintained, and also the computational complexity is reduced.Comment: This manuscript is the preprint of a paper submitted to the SIAM
Journal on Optimization and it is subject to SIAM copyright. SIAM maintains
the sole rights of distribution or publication of the work in all forms and
media. If accepted, the copy of record will be available at
http://www.siam.or
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
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
A Posteriori Probabilistic Bounds of Convex Scenario Programs with Validation Tests
Scenario programs have established themselves as efficient tools towards
decision-making under uncertainty. To assess the quality of scenario-based
solutions a posteriori, validation tests based on Bernoulli trials have been
widely adopted in practice. However, to reach a theoretically reliable
judgement of risk, one typically needs to collect massive validation samples.
In this work, we propose new a posteriori bounds for convex scenario programs
with validation tests, which are dependent on both realizations of support
constraints and performance on out-of-sample validation data. The proposed
bounds enjoy wide generality in that many existing theoretical results can be
incorporated as particular cases. To facilitate practical use, a systematic
approach for parameterizing a posteriori probability bounds is also developed,
which is shown to possess a variety of desirable properties allowing for easy
implementations and clear interpretations. By synthesizing comprehensive
information about support constraints and validation tests, improved risk
evaluation can be achieved for randomized solutions in comparison with existing
a posteriori bounds. Case studies on controller design of aircraft lateral
motion are presented to validate the effectiveness of the proposed a posteriori
bounds
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
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