120,448 research outputs found
Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics
We analyze a stochastic optimal control problem, where the state process
follows a McKean-Vlasov dynamics and the diffusion coefficient can be
degenerate. We prove that its value function V admits a nonlinear Feynman-Kac
representation in terms of a class of forward-backward stochastic differential
equations, with an autonomous forward process. We exploit this probabilistic
representation to rigorously prove the dynamic programming principle (DPP) for
V. The Feynman-Kac representation we obtain has an important role beyond its
intermediary role in obtaining our main result: in fact it would be useful in
developing probabilistic numerical schemes for V. The DPP is important in
obtaining a characterization of the value function as a solution of a
non-linear partial differential equation (the so-called Hamilton-Jacobi-Belman
equation), in this case on the Wasserstein space of measures. We should note
that the usual way of solving these equations is through the Pontryagin maximum
principle, which requires some convexity assumptions. There were attempts in
using the dynamic programming approach before, but these works assumed a priori
that the controls were of Markovian feedback type, which helps write the
problem only in terms of the distribution of the state process (and the control
problem becomes a deterministic problem). In this paper, we will consider
open-loop controls and derive the dynamic programming principle in this most
general case. In order to obtain the Feynman-Kac representation and the
randomized dynamic programming principle, we implement the so-called
randomization method, which consists in formulating a new McKean-Vlasov control
problem, expressed in weak form taking the supremum over a family of equivalent
probability measures. One of the main results of the paper is the proof that
this latter control problem has the same value function V of the original
control problem.Comment: 41 pages, to appear in Transactions of the American Mathematical
Societ
Minimizing bed occupancy variance by scheduling patients under uncertainty
International audienceIn this paper we consider the problem of scheduling patients in allocated surgery blocks in a Master Surgical Schedule. We pay attention to both the available surgery blocks and the bed occupancy in the hospital wards. More specifically, large probabilities of overtime in each surgery block are undesirable and costly, while large fluctuations in the number of used beds requires extra buffer capacity and makes the staff planning more challenging. The stochastic nature of surgery durations and length of stay on a ward hinders the use of classical techniques. Transforming the stochastic problem into a deterministic problem does not result into practically feasible solutions. In this paper we develop a technique to solve the stochastic scheduling problem, whose primary objective it to minimize variation in the necessary bed capacity, while maximizing the number of patients operated, and minimizing the maximum waiting time, and guaranteeing a small probability of overtime in surgery blocks. The method starts with solving an Integer Linear Programming (ILP) formulation of the problem, and then simulation and local search techniques are applied to guarantee small probabilities of overtime and to improve upon the ILP solution. Numerical experiments applied to a Dutch hospital show promising results
Stochastic Constraint Programming
To model combinatorial decision problems involving uncertainty and
probability, we introduce stochastic constraint programming. Stochastic
constraint programs contain both decision variables (which we can set) and
stochastic variables (which follow a probability distribution). They combine
together the best features of traditional constraint satisfaction, stochastic
integer programming, and stochastic satisfiability. We give a semantics for
stochastic constraint programs, and propose a number of complete algorithms and
approximation procedures. Finally, we discuss a number of extensions of
stochastic constraint programming to relax various assumptions like the
independence between stochastic variables, and compare with other approaches
for decision making under uncertainty.Comment: Proceedings of the 15th Eureopean Conference on Artificial
Intelligenc
Probabilistic Constraint Logic Programming
This paper addresses two central problems for probabilistic processing
models: parameter estimation from incomplete data and efficient retrieval of
most probable analyses. These questions have been answered satisfactorily only
for probabilistic regular and context-free models. We address these problems
for a more expressive probabilistic constraint logic programming model. We
present a log-linear probability model for probabilistic constraint logic
programming. On top of this model we define an algorithm to estimate the
parameters and to select the properties of log-linear models from incomplete
data. This algorithm is an extension of the improved iterative scaling
algorithm of Della-Pietra, Della-Pietra, and Lafferty (1995). Our algorithm
applies to log-linear models in general and is accompanied with suitable
approximation methods when applied to large data spaces. Furthermore, we
present an approach for searching for most probable analyses of the
probabilistic constraint logic programming model. This method can be applied to
the ambiguity resolution problem in natural language processing applications.Comment: 35 pages, uses sfbart.cl
Minimizing value-at-risk in the single-machine total weighted tardiness problem
The vast majority of the machine scheduling literature focuses on deterministic
problems, in which all data is known with certainty a priori. This may be a reasonable assumption when the variability in the problem parameters is low. However, as variability in the parameters increases incorporating this uncertainty explicitly into a scheduling model is essential to mitigate the resulting adverse effects. In this paper, we consider the celebrated single-machine total weighted tardiness (TWT) problem in the presence of uncertain problem parameters. We impose a probabilistic constraint on the random TWT and introduce a risk-averse stochastic programming model. In particular, the objective of the proposed model is to find a non-preemptive static job processing sequence that minimizes the value-at-risk (VaR) measure on the random
TWT at a specified confidence level. Furthermore, we develop a lower bound on the optimal VaR that may also benefit alternate solution approaches in the future. In this study, we implement a tabu-search heuristic to obtain reasonably good feasible solutions and present results to demonstrate the effect of the risk parameter and the value of the proposed model with respect to a corresponding risk-neutral approach
Controlled diffusion processes
This article gives an overview of the developments in controlled diffusion
processes, emphasizing key results regarding existence of optimal controls and
their characterization via dynamic programming for a variety of cost criteria
and structural assumptions. Stochastic maximum principle and control under
partial observations (equivalently, control of nonlinear filters) are also
discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
- …