On the Solution of Linear Programming Problems in the Age of Big Data

Abstract

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

Scalability Evaluation of Cimmino Algorithm for Solving Linear Inequality Systems on Multiprocessors with Distributed Memory

“…1) apply the Cimmino algorithm to implement the Qwest phase of the NSLP algorithm [2], designed to solve large-scale non-stationary linear programming problems;…”

“…As examples, we can mention linear programming [1,2], image reconstruction from projections [3], image processing in magnetic resonance imaging [4], intensity-modulated radiation therapy (IMRT) [5]. At the present time, a lot of methods for solving systems of linear inequalities are known, among which we can mark out a class of self-correcting iteration methods that allow efficient parallelization.…”

“…In many cases, systems of linear inequalities arising in the solution of practical problems can involve up to tens of millions of inequalities and up to hundreds of millions of variables [2]. In this case, the issue of developing scalable parallel algorithms for solving large-scale systems of linear inequalities on multiprocessor systems with distributed memory becomes very urgent.…”

Scalability Evaluation of NSLP Algorithm for Solving Non-Stationary Linear Programming Problems on Cluster Computing Systems

“…The Quest phase calculates a point z belonging to the polytope M t . This phase is described in detail in [12]. The Quest Phase is followed by the Targeting phase.…”

“…A distinctive feature of the Fejer process is its "self-guided" capability: the Fejer process automatically corrects its motion path according to the polytope position changes during the calculation of the pseudo-projection. The Quest phase was investigated in [12], where the convergence theorem was proved for the case when the polytope is translated with a fixed vector in the each unit of time. In the paper [17], the authors demonstrated that Intel Xeon Phi multi-core processors can be efficiently used for calculating the pseudo-projections.…”