We consider Continuous Linear Programs over a continuous finite time horizon T , with linear cost coefficient functions and linear right hand side functions and a constant coefficient matrix, where we search for optimal solutions in the space of measures or of functions of bounded variation. These models generalize the Separated Continuous Linear Programming models and their various duals, as formulated in the past by Anderson, by Pullan, and by Weiss. We present simple necessary and sufficient conditions for feasibility. We formulate a symmetric dual and investigate strong duality by considering discrete time approximations. We prove that under a Slater type condition there is no duality gap and there exist optimal solutions which have impulse controls at 0 and T and have piecewise constant densities in (0,T ). Moreover, we show that under non-degeneracy assumptions all optimal solutions are of this form, and are uniquely determined over (0,T ). Key words. Continuous linear programming, symmetric dual, strong duality. AMS subject classifications. 34H99,49N15,65K99,90C48 P(t) ≥ 0, P(t) non-decreasing and right continuous on [0,T].with K unknown dual functions P with the same convention P(0−) = 0. It is convenient to think of dual time as running backwards, so that P(T − t) corresponds to U(t).The main feature to note here is that the objective as well as the left hand side of the constraints are formulated as Lebesgue-Stieltjes integrals with respect to a vector of monotone non-decreasing control function U(t), in other words our controls are in the space of measures. This is in contrast to most formulations in which the objective and left hand side of the constraints are Lebesgue integrals with respect to a measurable bounded control u(t), in other words controls which are in the space of densities. In particular, while in the usual formulation the left hand side of the constraints is an absolutely continuous function, our formulation allows the left hand side of the constraint to have jumps, as a result of jumps in U(t), which correspond to impulse controls.Our main results in this paper include the following: *
We consider Continuous Linear Programs over a continuous finite time horizon T , with linear cost coefficient functions, linear right hand side functions, and a constant coefficient matrix, as well as their symmetric dual. We search for optimal solutions in the space of measures or of functions of bounded variation. These models generalize the Separated Continuous Linear Programming models and their various duals, as formulated in the past by Anderson, by Pullan, and by Weiss. In a recent paper we have shown that under a Slater type condition, these problems possess optimal strongly dual solutions. In this paper we give a detailed description of optimal solutions and define a combinatorial analog to basic solutions of standard LP. We also show that feasibility implies existence of strongly dual optimal solutions without requiring the Slater condition. We present several examples to illustrate the richness and complexity of these solutions.Key words. Continuous linear programming, symmetric dual, strong duality, structure of solutions, optimization in the space of measures, optimal sequence of bases AMS subject classifications. 34H99,49N15,65K99,90C48
This paper is devoted to the testing of of automatic well logs interpretation testing, developed on the basis of machine learning methods. The basis of the method presented in the paper is recurrent artificial neural networks. For their training and adjustment, log curves and their corresponding interpretation of different years are used. The set of well data is divided into training, validation and test samples. The resulting tool is set up on training and validation samples, and then used on a test sample of wells for which the interpretatoin was hidden, in order to automatically predict net pays and compare the results with the interpretation performed by an expert petrophysicist. For the test sample, traditional machine learning metrics metrics and special geological were calculated to assess the quality of the algorithm. During the work a number of experiments were carried out, in which the dependence of the forecast quality was estimated not only on the different architecture and settings of the artificial neural network, but also on the amount of input data. The iterative approach in the research allowed to determine the best parameters for the solution of the task. For each well of the test set, a forecast of reservoir intervals distribution is made. The resulting interpretation shows high accuracy, both in terms of different mathematical metrics, and the results of analysis and evaluation of the expert petrophysics. Also, during the experiments, an important conclusion was made about the generalizing ability of the proposed methodology. The use of several variants of interpretation of well log data, performed by different specialists at different times and on the basis of different petrophysical models, allows to generalize and use all the accumulated experience of well logs interpretation, thereby improving the quality of the forecast. The main conclusion of the study can be considered a statement about the efficient applicability of machine learning algorithms for automatic well logs interpretation.
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