82,540 research outputs found
Strong Stationarity Conditions for Optimal Control of Hybrid Systems
We present necessary and sufficient optimality conditions for finite time
optimal control problems for a class of hybrid systems described by linear
complementarity models. Although these optimal control problems are difficult
in general due to the presence of complementarity constraints, we provide a set
of structural assumptions ensuring that the tangent cone of the constraints
possesses geometric regularity properties. These imply that the classical
Karush-Kuhn-Tucker conditions of nonlinear programming theory are both
necessary and sufficient for local optimality, which is not the case for
general mathematical programs with complementarity constraints. We also present
sufficient conditions for global optimality.
We proceed to show that the dynamics of every continuous piecewise affine
system can be written as the optimizer of a mathematical program which results
in a linear complementarity model satisfying our structural assumptions. Hence,
our stationarity results apply to a large class of hybrid systems with
piecewise affine dynamics. We present simulation results showing the
substantial benefits possible from using a nonlinear programming approach to
the optimal control problem with complementarity constraints instead of a more
traditional mixed-integer formulation.Comment: 30 pages, 4 figure
Optimal designs for active controlled dose finding trials with efficacy-toxicity outcomes
Nonlinear regression models addressing both efficacy and toxicity outcomes
are increasingly used in dose-finding trials, such as in pharmaceutical drug
development. However, research on related experimental design problems for
corresponding active controlled trials is still scarce. In this paper we derive
optimal designs to estimate efficacy and toxicity in an active controlled
clinical dose finding trial when the bivariate continuous outcomes are modeled
either by polynomials up to degree 2, the Michaelis- Menten model, the Emax
model, or a combination thereof. We determine upper bounds on the number of
different doses levels required for the optimal design and provide conditions
under which the boundary points of the design space are included in the optimal
design. We also provide an analytical description of the minimally supported
-optimal designs and show that they do not depend on the correlation between
the bivariate outcomes. We illustrate the proposed methods with numerical
examples and demonstrate the advantages of the -optimal design for a trial,
which has recently been considered in the literature.Comment: Keywords and Phrases: Active controlled trials, dose finding, optimal
design, admissible design, Emax model, Equivalence theorem, Particle swarm
optimization, Tchebycheff syste
Primal-dual extragradient methods for nonlinear nonsmooth PDE-constrained optimization
We study the extension of the Chambolle--Pock primal-dual algorithm to
nonsmooth optimization problems involving nonlinear operators between function
spaces. Local convergence is shown under technical conditions including metric
regularity of the corresponding primal-dual optimality conditions. We also show
convergence for a Nesterov-type accelerated variant provided one part of the
functional is strongly convex.
We show the applicability of the accelerated algorithm to examples of inverse
problems with - and -fitting terms as well as of
state-constrained optimal control problems, where convergence can be guaranteed
after introducing an (arbitrary small, still nonsmooth) Moreau--Yosida
regularization. This is verified in numerical examples
On the policy function in continuos time economic models
In this paper, I consider a general class of continuous-time economic models with unbounded horizon. I study the sets of conditions under which the policy function is continuous, Lipschitz continuous, and Cl differentiable. 1 also single out certain postulates which may prevent higher-order differentiability. The analysis provides, therefore, a fmn foundation to the use of dynamic programming methods in continuous time models with unbounded horizo
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