4,930 research outputs found
NLP Solutions as Asymptotic Values of ODE Trajectories
In this paper, it is shown that the solutions of general differentiable
constrained optimization problems can be viewed as asymptotic solutions to sets
of Ordinary Differential Equations (ODEs). The construction of the ODE
associated to the optimization problem is based on an exact penalty formulation
in which the weighting parameter dynamics is coordinated with that of the
decision variable so that there is no need to solve a sequence of optimization
problems, instead, a single ODE has to be solved using available efficient
methods. Examples are given in order to illustrate the results. This includes a
novel systematic approach to solve combinatoric optimization problems as well
as fast computation of a class of optimization problems using analogic circuits
leading to fast, parallel and highly scalable solutions
A Convex Feasibility Approach to Anytime Model Predictive Control
This paper proposes to decouple performance optimization and enforcement of
asymptotic convergence in Model Predictive Control (MPC) so that convergence to
a given terminal set is achieved independently of how much performance is
optimized at each sampling step. By embedding an explicit decreasing condition
in the MPC constraints and thanks to a novel and very easy-to-implement convex
feasibility solver proposed in the paper, it is possible to run an outer
performance optimization algorithm on top of the feasibility solver and
optimize for an amount of time that depends on the available CPU resources
within the current sampling step (possibly going open-loop at a given sampling
step in the extreme case no resources are available) and still guarantee
convergence to the terminal set. While the MPC setup and the solver proposed in
the paper can deal with quite general classes of functions, we highlight the
synthesis method and show numerical results in case of linear MPC and
ellipsoidal and polyhedral terminal sets.Comment: 8 page
Enlarging the domain of attraction of MPC controllers
This paper presents a method for enlarging the domain of attraction of nonlinear model predictive control (MPC). The usual way of guaranteeing stability of nonlinear MPC is to add a terminal constraint and a terminal cost to the optimization problem such that the terminal region is a positively invariant set for the system and the terminal cost is an associated Lyapunov function. The domain of attraction of the controller depends on the size of the terminal region and the control horizon. By increasing the control horizon, the domain of attraction is enlarged but at the expense of a greater computational burden, while increasing the terminal region produces an enlargement without an extra cost.
In this paper, the MPC formulation with terminal cost and constraint is modified, replacing the terminal constraint by a contractive terminal constraint. This constraint is given by a sequence of sets computed off-line that is based on the positively invariant set. Each set of this sequence does not need to be an invariant set and can be computed by a procedure which provides an inner approximation to the one-step set. This property allows us to use one-step approximations with a trade off between accuracy and computational burden for the computation of the sequence. This strategy guarantees closed loop-stability ensuring the enlargement of the domain of attraction and the local optimality of the controller. Moreover, this idea can be directly translated to robust MPC.Ministerio de Ciencia y TecnologÃa DPI2002-04375-c03-0
Asymptotic Stability of POD based Model Predictive Control for a semilinear parabolic PDE
In this article a stabilizing feedback control is computed for a semilinear
parabolic partial differential equation utilizing a nonlinear model predictive
(NMPC) method. In each level of the NMPC algorithm the finite time horizon open
loop problem is solved by a reduced-order strategy based on proper orthogonal
decomposition (POD). A stability analysis is derived for the combined POD-NMPC
algorithm so that the lengths of the finite time horizons are chosen in order
to ensure the asymptotic stability of the computed feedback controls. The
proposed method is successfully tested by numerical examples
Efficient Optimization of Loops and Limits with Randomized Telescoping Sums
We consider optimization problems in which the objective requires an inner
loop with many steps or is the limit of a sequence of increasingly costly
approximations. Meta-learning, training recurrent neural networks, and
optimization of the solutions to differential equations are all examples of
optimization problems with this character. In such problems, it can be
expensive to compute the objective function value and its gradient, but
truncating the loop or using less accurate approximations can induce biases
that damage the overall solution. We propose randomized telescope (RT) gradient
estimators, which represent the objective as the sum of a telescoping series
and sample linear combinations of terms to provide cheap unbiased gradient
estimates. We identify conditions under which RT estimators achieve
optimization convergence rates independent of the length of the loop or the
required accuracy of the approximation. We also derive a method for tuning RT
estimators online to maximize a lower bound on the expected decrease in loss
per unit of computation. We evaluate our adaptive RT estimators on a range of
applications including meta-optimization of learning rates, variational
inference of ODE parameters, and training an LSTM to model long sequences
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