83,211 research outputs found
Stability and Performance Verification of Optimization-based Controllers
This paper presents a method to verify closed-loop properties of
optimization-based controllers for deterministic and stochastic constrained
polynomial discrete-time dynamical systems. The closed-loop properties amenable
to the proposed technique include global and local stability, performance with
respect to a given cost function (both in a deterministic and stochastic
setting) and the gain. The method applies to a wide range of
practical control problems: For instance, a dynamical controller (e.g., a PID)
plus input saturation, model predictive control with state estimation, inexact
model and soft constraints, or a general optimization-based controller where
the underlying problem is solved with a fixed number of iterations of a
first-order method are all amenable to the proposed approach.
The approach is based on the observation that the control input generated by
an optimization-based controller satisfies the associated Karush-Kuhn-Tucker
(KKT) conditions which, provided all data is polynomial, are a system of
polynomial equalities and inequalities. The closed-loop properties can then be
analyzed using sum-of-squares (SOS) programming
Stochastic MPC Design for a Two-Component Granulation Process
We address the issue of control of a stochastic two-component granulation
process in pharmaceutical applications through using Stochastic Model
Predictive Control (SMPC) and model reduction to obtain the desired particle
distribution. We first use the method of moments to reduce the governing
integro-differential equation down to a nonlinear ordinary differential
equation (ODE). This reduced-order model is employed in the SMPC formulation.
The probabilistic constraints in this formulation keep the variance of
particles' drug concentration in an admissible range. To solve the resulting
stochastic optimization problem, we first employ polynomial chaos expansion to
obtain the Probability Distribution Function (PDF) of the future state
variables using the uncertain variables' distributions. As a result, the
original stochastic optimization problem for a particulate system is converted
to a deterministic dynamic optimization. This approximation lessens the
computation burden of the controller and makes its real time application
possible.Comment: American control Conference, May, 201
The stochastic matching problem
The matching problem plays a basic role in combinatorial optimization and in
statistical mechanics. In its stochastic variants, optimization decisions have
to be taken given only some probabilistic information about the instance. While
the deterministic case can be solved in polynomial time, stochastic variants
are worst-case intractable. We propose an efficient method to solve stochastic
matching problems which combines some features of the survey propagation
equations and of the cavity method. We test it on random bipartite graphs, for
which we analyze the phase diagram and compare the results with exact bounds.
Our approach is shown numerically to be effective on the full range of
parameters, and to outperform state-of-the-art methods. Finally we discuss how
the method can be generalized to other problems of optimization under
uncertainty.Comment: Published version has very minor change
Stochastic Budget Optimization in Internet Advertising
Internet advertising is a sophisticated game in which the many advertisers
"play" to optimize their return on investment. There are many "targets" for the
advertisements, and each "target" has a collection of games with a potentially
different set of players involved. In this paper, we study the problem of how
advertisers allocate their budget across these "targets". In particular, we
focus on formulating their best response strategy as an optimization problem.
Advertisers have a set of keywords ("targets") and some stochastic information
about the future, namely a probability distribution over scenarios of cost vs
click combinations. This summarizes the potential states of the world assuming
that the strategies of other players are fixed. Then, the best response can be
abstracted as stochastic budget optimization problems to figure out how to
spread a given budget across these keywords to maximize the expected number of
clicks.
We present the first known non-trivial poly-logarithmic approximation for
these problems as well as the first known hardness results of getting better
than logarithmic approximation ratios in the various parameters involved. We
also identify several special cases of these problems of practical interest,
such as with fixed number of scenarios or with polynomial-sized parameters
related to cost, which are solvable either in polynomial time or with improved
approximation ratios. Stochastic budget optimization with scenarios has
sophisticated technical structure. Our approximation and hardness results come
from relating these problems to a special type of (0/1, bipartite) quadratic
programs inherent in them. Our research answers some open problems raised by
the authors in (Stochastic Models for Budget Optimization in Search-Based
Advertising, Algorithmica, 58 (4), 1022-1044, 2010).Comment: FINAL versio
High-Dimensional Stochastic Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition
This paper presents a novel adaptive-sparse polynomial dimensional
decomposition (PDD) method for stochastic design optimization of complex
systems. The method entails an adaptive-sparse PDD approximation of a
high-dimensional stochastic response for statistical moment and reliability
analyses; a novel integration of the adaptive-sparse PDD approximation and
score functions for estimating the first-order design sensitivities of the
statistical moments and failure probability; and standard gradient-based
optimization algorithms. New analytical formulae are presented for the design
sensitivities that are simultaneously determined along with the moments or the
failure probability. Numerical results stemming from mathematical functions
indicate that the new method provides more computationally efficient design
solutions than the existing methods. Finally, stochastic shape optimization of
a jet engine bracket with 79 variables was performed, demonstrating the power
of the new method to tackle practical engineering problems.Comment: 18 pages, 2 figures, to appear in Sparse Grids and
Applications--Stuttgart 2014, Lecture Notes in Computational Science and
Engineering 109, edited by J. Garcke and D. Pfl\"{u}ger, Springer
International Publishing, 201
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