The Big Data phenomenon has spawned large-scale linear programming problems. In many cases, these problems are non-stationary. In this paper, we describe a new scalable algorithm called NSLP for solving high-dimensional, non-stationary linear programming problems on modern cluster computing systems. The algorithm consists of two phases: Quest and Targeting. The Quest phase calculates a solution of the system of inequalities defining the constraint system of the linear programming problem under the condition of dynamic changes in input data. To this end, the apparatus of Fejer mappings is used. The Targeting phase forms a special system of points having the shape of an n-dimensional axisymmetric cross. The cross moves in the n-dimensional space in such a way that the solution of the linear programming problem is located all the time in an "-vicinity of the central point of the cross.Comment: Parallel Computational Technologies - 11th International Conference, PCT 2017, Kazan, Russia, April 3-7, 2017, Proceedings (to be published in Communications in Computer and Information Science, vol. 753

In the development of industrial digital twins, the optimization problem of technological and business processes often arises. In many cases, this problem can be reduced to a large-scale linear programming (LP) problem. The article is devoted to the new method for solving large-scale LP problems. This method is called the "apex-method". The apex-method uses the predictor-corrector framework. The predictor step calculates a point belonging to the feasible region of LP problem. The corrector step calculates a sequence of points converging to the exact solution of the LP problem. The article gives a formal description of the apex-method and provides information about its parallel implementation in C++ language by using the MPI library. The results of large-scale computational experiments on a cluster computing system to study the scalability of the apex method are presented.

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The paper is devoted to a scalability study of the NSLP algorithm for solving non-stationary high-dimension linear programming problem on the cluster computing systems. The analysis is based on the BSF model of parallel computations. The BSF model is a new parallel computation model designed on the basis of BSP and SPMD models. The brief descriptions of the NSLP algorithm and the BSF model are given. The NSLP algorithm implementation in the form of a BSF program is considered. On the basis of the BSF cost metric, the upper bound of the NSLP algorithm scalability is derived and its parallel efficiency is estimated. NSLP algorithm implementation using BSF skeleton is described. A comparison of scalability estimations obtained analytically and experimentally is provided.Keywords: non-stationary linear programming problem · large-scale linear programming · NSLP algorithm · BSF parallel computation model · cost metric · scalability bound · parallel efficiency estimation.

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