Efficiency of cluster schemes implementation for discrete dynamic programming for systems of support of dispatching technological processes in transport systems

Within the framework of a computationally complex canonical scheduling problem, formulated by an optimization model for one-processor servicing of a finite deterministic flow of objects, a scheme of computational process of an algorithm of discrete dynamic programming in cluster implementation is considered. Variants of balancing of computational subtasks over network cluster array are investigated, purposed to reduce the volume and intensity of intranetwork interaction. It has been established that for practical improvement of efficiency of cluster algorithm, it is required not to increase the uniformity of distribution of subtasks among the cluster nodes, but to minimize the network traffic between the cluster nodes. Balancing options are proposed that allow to significantly increase localization of data in network computing. Experimental results are analytically confirmed, showing the scaling limits of implementation of discrete dynamic programming algorithms on a cluster architecture. The method for choosing the number of computational nodes and dimension of the problem being solved, which provide a threefold reduction in overhead costs for network exchange, is shown. The results obtained make it possible to objectively substantiate the choice of methodological and algorithmic approaches when choosing computer tools developing architectural and technological solutions for dispatching systems support in inland water transport.

Search for all solutions to the dynamic programming problem if their multicriteria estimates coincide

Among the mathematical methods used in economics, a prominent place is occupied by the dynamic programming method, with the help of which the optimal control of multi-stage processes is organized. The disadvantage of this method is the impossibility of calculating all solutions to the problem if their criteria-based estimates coincide. The fact of the existence of several optimal trajectories of a multi-step process may mean that the task is not set correctly, in the sense that the assigned criteria do not fully characterize the system under study. This means that the traditional method of dynamic programming needs to be refined in case of the existence of several optimal trajectories with the same value of the criterion. This article proposes the most general version of such refinement, namely, a multi-criteria numerical scheme is generalized. For a more visual representation of calculations and the result of the study, we will describe the discrete dynamic programming problem in terms of graph theory. In this case, it reduces to the problem of finding the optimal path on a directed graph. To solve it, a three-stage algorithm is proposed, the composition of which includes the following steps. The first stage is the construction of optimal criteria estimates for paths from the initial vertex to all the others. To perform this stage, the most universal method is the multicriteria version of the Ford – Bellman method. The second stage is the construction of a graph of optimal paths. In the original graph, arcs are selected that are part of the optimal paths. Of these, using the original algorithm, a subgraph is formed in which all paths are optimal. It is analytically proved that this algorithm gives the correct result (correct). The third stage is enumeration of all paths in the constructed subgraph. Numerical experiments showed that the proposed three-stage method works efficiently on oriented graphs of any type in a sufficiently large range of dimensions. The proposed algorithm with minimal changes can be used to solve an arbitrary discrete dynamic programming problem.

Convergence of dynamic programming principles for the p-Laplacian

We provide a unified strategy to show that solutions of dynamic programming principles associated to the p-Laplacian converge to the solution of the corresponding Dirichlet problem. Our approach includes all previously known cases for continuous and discrete dynamic programming principles, provides new results, and gives a convergence proof free of probability arguments.

Carbon sequestration and the optimal forest harvest decision under alternative baseline policies

The choice of a baseline against which to evaluate changes in carbon stocks is a critical component of any forest carbon offset market. In this paper, we use a discrete dynamic programming model and data from a lodgepole pine (Pinus contorta Douglas ex Loudon) stand in northeastern British Columbia, Canada, to demonstrate that different baselines have little or no effect on optimal harvest decision but can have a large impact on economic returns to a landowner. The results reveal that the magnitude of the financial return to the landowner is dependent on the starting conditions of both the predetermined baseline and the proposed carbon offset project. The study also shows that when given the choice between alternative baselines, a landowner will always choose a fixed baseline over a business-as-usual baseline.