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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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TL;DR: It is shown that in general the usual constraint qualifications do not hold and the right constraint qualification is the calmness condition, and it is also shown that the linear bilevel programming problem and the minmax problem satisfy the Calmness condition automatically.
Abstract: The bilevel programming problem (BLPP) is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. To obtain optimality conditions, we reformulate BLPP as a single level mathematical programming problem (SLPP) which involves the value function of the lower level problem. For this mathematical programming problem, it is shown that in general the usual constraint qualifications do not hold and the right constraint qualification is the calmness condition. It is also shown that the linear bilevel programming problem and the minmax problem satisfy the calmness condition automatically. A sufficient condition for the calmness for the bilevel programming problem with quadratic lower level problem and nondegenerate linear complementar¬ity lower level problem are given. First order necessary optimality condition are given using nonsmooth analysis. Second order sufficient optimality conditions are also give...
298 citations
TL;DR: This paper establishes a link between reachability, viability and invariance problems and viscosity solutions of a special form of the Hamilton-Jacobi equation to address optimal control problems where the cost function is the minimum of a function of the state over a specified horizon.
293 citations
TL;DR: In this paper, a new characterization of excessive functions for arbitrary one-dimensional regular diffusion processes is provided, using the notion of concavity, and a new perspective and new facts about the principle of smooth-fit in the context of optimal stopping are presented.
289 citations
Proceedings Article•
21 Aug 2003TL;DR: In this article, the authors provide error bounds for approximate policy iterative using quadratic norms, and illustrate those results in the case of feature-based linear function approximation, where most function approximators (such as linear regression) select the best fit in a given class of parameterized functions by minimizing some (weighted) Quadratic norm.
Abstract: In Dynamic Programming, convergence of algorithms such as Value Iteration or Policy Iteration results -in discounted problems- from a contraction property of the back-up operator, guaranteeing convergence to its fixed-point. When approximation is considered, known results in Approximate Policy Iteration provide bounds on the closeness to optimality of the approximate value function obtained by successive policy improvement steps as a function of the maximum norm of value determination errors during policy evaluation steps. Unfortunately, such results have limited practical range since most function approximators (such as linear regression) select the best fit in a given class of parameterized functions by minimizing some (weighted) quadratic norm.
In this paper, we provide error bounds for Approximate Policy Iteration using quadratic norms, and illustrate those results in the case of feature-based linear function approximation.
288 citations
TL;DR: In this paper, a risk process modelled as a compound Poisson process is considered and the ruin probability of this risk process is minimized by the choice of a suitable investment strategy for a capital market index.
Abstract: We consider a risk process modelled as a compound Poisson process. The ruin probability of this risk process is minimized by the choice of a suitable investment strategy for a capital market index. The optimal strategy is computed using the Bellman equation. We prove the existence of a smooth solution and a verification theorem, and give explicit solutions in some cases with exponential claim size distribution, as well as numerical results in a case with Pareto claim size. For this last case, the optimal amount invested will not be bounded.
284 citations