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Bellman equation
About: Bellman equation is a research topic. Over the lifetime, 5884 publications have been published within this topic receiving 135589 citations.
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23 Mar 2000
TL;DR: It is shown that the discrete value function converges to the viscosity solution of the Hamilton-Jacobi-Bellman equation as a discretization parameter tends to zero.
Abstract: We consider the synthesis of optimal controls for continuous feedback systems by recasting the problem to a hybrid optimal control problem: to synthesize optimal enabling conditions for switching between locations in which the control is constant. An algorithmic solution is obtained by translating the hybrid automaton to a finite automaton using a bisimulation and formulating a dynamic programming problem with extra conditions to ensure non-Zenoness of trajectories. We show that the discrete value function converges to the viscosity solution of the Hamilton-Jacobi-Bellman equation as a discretization parameter tends to zero.
33 citations
TL;DR: This paper shows that solutions obtained by this approach are always efficient solutions and is based on the principle of optimality in dynamic programming and fuzzy decision approach.
33 citations
TL;DR: The multi-period technician scheduling problem with experience-based service times and stochastic customers is introduced and an approximate dynamic programming-based solution approach is introduced that is adapted to handle cases of worker attrition and new task types.
33 citations
TL;DR: For the special case when the so-called lower-level problem is convex, it is shown how the general optimality conditions can be strengthened, thereby giving a generalization of Theorem 4.2 in RA¼ckmann and Stein (2001).
Abstract: We present a general framework for the derivation of first-order optimality conditions in generalized semi-infinite programming. Since in our approach no constraint qualifications are assumed for the index set, we can generalize necessary conditions given by RA¼ckmann and Shapiro (1999) as well as the characterizations of local minimizers of order one, which were derived by Stein and Still (2000). Moreover, we obtain a short proof for Theorem 1.1 in Jongen et al. (1998).For the special case when the so-called lower-level problem is convex, we show how the general optimality conditions can be strengthened, thereby giving a generalization of Theorem 4.2 in RA¼ckmann and Stein (2001). Finally, if the directional derivative of a certain optimal value function exists and is subadditive with respect to the direction, we propose a Mangasarian-Fromovitz-type constraint qualification and show that it implies an Abadie-type constraint qualification.
33 citations
TL;DR: In this article, a general method for deriving closed reduced models of Hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation, where a cost function is constructed to quantify the lack-of-fit to the microscopic dynamics of any feasible path of densities from the statistical model; it is an ensemble-averaged, weighted, squared-norm of the residual that results from submitting the path of density to the Liouville equation.
Abstract: A general method for deriving closed reduced models of Hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation. Given a vector of resolved variables, selected to describe the macroscopic state of the system, a family of quasi-equilibrium probability densities on phase space corresponding to the resolved variables is employed as a statistical model, and the evolution of the mean resolved vector is estimated by optimizing over paths of these densities. Specifically, a cost function is constructed to quantify the lack-of-fit to the microscopic dynamics of any feasible path of densities from the statistical model; it is an ensemble-averaged, weighted, squared-norm of the residual that results from submitting the path of densities to the Liouville equation. The path that minimizes the time integral of the cost function determines the best-fit evolution of the mean resolved vector. The closed reduced equations satisfied by the optimal path are derived by Hamilton-Jacobi theory. When expressed in terms of the macroscopic variables, these equations have the generic structure of governing equations for nonequilibrium thermodynamics. In particular, the value function for the optimization principle coincides with the dissipation potential that defines the relation between thermodynamic forces and fluxes. The adjustable closure parameters in the best-fit reduced equations depend explicitly on the arbitrary weights that enter into the lack-of-fit cost function. Two particular model reductions are outlined to illustrate the general method. In each example the set of weights in the optimization principle contracts into a single effective closure parameter.
33 citations