11 research outputs found
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
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
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
Stochastic AC Optimal Power Flow: A Data-Driven Approach
There is an emerging need for efficient solutions to stochastic AC Optimal
Power Flow ({AC-}OPF) to ensure optimal and reliable grid operations in the
presence of increasing demand and generation uncertainty. This paper presents a
highly scalable data-driven algorithm for stochastic AC-OPF that has extremely
low sample requirement. The novelty behind the algorithm's performance involves
an iterative scenario design approach that merges information regarding
constraint violations in the system with data-driven sparse regression.
Compared to conventional methods with random scenario sampling, our approach is
able to provide feasible operating points for realistic systems with much lower
sample requirements. Furthermore, multiple sub-tasks in our approach can be
easily paralleled and based on historical data to enhance its performance and
application. We demonstrate the computational improvements of our approach
through simulations on different test cases in the IEEE PES PGLib-OPF benchmark
library.Comment: a version of this paper will appear in the proceedings of the 21st
Power Systems Computation Conference (PSCC 2020
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
Analysis of Theoretical and Numerical Properties of Sequential Convex Programming for Continuous-Time Optimal Control
Sequential Convex Programming (SCP) has recently gained significant
popularity as an effective method for solving optimal control problems and has
been successfully applied in several different domains. However, the
theoretical analysis of SCP has received comparatively limited attention, and
it is often restricted to discrete-time formulations. In this paper, we present
a unifying theoretical analysis of a fairly general class of SCP procedures for
continuous-time optimal control problems. In addition to the derivation of
convergence guarantees in a continuous-time setting, our analysis reveals two
new numerical and practical insights. First, we show how one can more easily
account for manifold-type constraints, which are a defining feature of optimal
control of mechanical systems. Second, we show how our theoretical analysis can
be leveraged to accelerate SCP-based optimal control methods by infusing
techniques from indirect optimal control