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Apex Method: A New Scalable Iterative Method for Linear Programming

Author

Listed:
  • Leonid B. Sokolinsky

    (School of Electronic Engineering and Computer Science, South Ural State University (National Research University), 76, Lenin Prospekt, 454080 Chelyabinsk, Russia
    These authors contributed equally to this work.)

  • Irina M. Sokolinskaya

    (School of Electronic Engineering and Computer Science, South Ural State University (National Research University), 76, Lenin Prospekt, 454080 Chelyabinsk, Russia
    These authors contributed equally to this work.)

Abstract

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.

Suggested Citation

  • Leonid B. Sokolinsky & Irina M. Sokolinskaya, 2023. "Apex Method: A New Scalable Iterative Method for Linear Programming," Mathematics, MDPI, vol. 11(7), pages 1-28, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:7:p:1654-:d:1111012
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    References listed on IDEAS

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    1. Rujira Visuthirattanamanee & Krung Sinapiromsaran & Aua-aree Boonperm, 2020. "Self-Regulating Artificial-Free Linear Programming Solver Using a Jump and Simplex Method," Mathematics, MDPI, vol. 8(3), pages 1-15, March.
    2. Liu, Xiaolan & Zhou, Mi, 2016. "A one-layer recurrent neural network for non-smooth convex optimization subject to linear inequality constraints," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 39-46.
    3. Jonathan Brogaard & Terrence Hendershott & Ryan Riordan, 2014. "High-Frequency Trading and Price Discovery," The Review of Financial Studies, Society for Financial Studies, vol. 27(8), pages 2267-2306.
    4. ManMohan S. Sodhi, 2005. "LP Modeling for Asset-Liability Management: A Survey of Choices and Simplifications," Operations Research, INFORMS, vol. 53(2), pages 181-196, April.
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