In this paper, a scalable iterative projection-type algorithm for solving non-stationary systems of linear inequalities is considered. A non-stationary system is understood as a large-scale system of inequalities in which coefficients and constant terms can change during the calculation process. The proposed parallel algorithm uses the concept of pseudo-projection which generalizes the notion of orthogonal projection. The parallel pseudo-projection algorithm is implemented using the parallel BSF-skeleton. An analytical estimation of the algorithm scalability boundary is obtained on the base of the BSF cost metric. The large-scale computational experiments were performed on a cluster computing system. The obtained results confirm the efficiency of the proposed approach.

The article presents a new scalable iterative method for linear programming called the “apex method”. The key feature of this method is constructing a path close to optimal on the surface of the feasible region from a certain starting point to the exact solution of a linear programming problem. The optimal path refers to a path of the minimum length according to the Euclidean metric. The apex method is based on the predictor—corrector framework and proceeds in two stages: quest (predictor) and target (corrector). The quest stage calculates a rough initial approximation of the linear programming problem. The target stage refines the initial approximation with a given precision. The main operation used in the apex method is an operation that calculates the pseudoprojection, which is a generalization of the metric projection to a convex closed set. This operation is used both in the quest stage and in the target stage. A parallel algorithm using a Fejér mapping to compute the pseudoprojection is presented. An analytical estimation of the parallelism degree of this algorithm is obtained. AlsoAdditionally, an algorithm implementing the target stage is given. The convergence of this algorithm is proven. An experimental study of the scalability of the apex method on a cluster computing system is described. The results of applying the apex method to solve problems from the Netlib-LP repository are presented.