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Computer Solution of Linear Programs: Non-simplex Algorithms

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Advances in Nonlinear Programming

Part of the book series: Applied Optimization ((APOP,volume 14))

Abstract

The far-reaching advances ushered in by Karmarkar’s pioneering work are rooted in well-known algorithmic paradigms of nonlinear programming/nonlinear equation-solving that have found fundamental new expression within the context of linear programming. We given an overview of interior-point and infeasible-interior-point LP algorithms from this perspective, concentrating on their underlying algebraic and geometric aspects. We formulate the direction-finding problems of primal-dual affine-scaling and potential-reduction algorithms in a unified, self-contained manner. We then explore the basic dichotomy of nonlinear-equation solving—minimizing a merit or potential function vis-a-vis following a homotopy path—which reasserts itself within the LP setting. In particular, this leads to a detailed categorization of homotopy and central paths and enables us to clarify the way in which different path-following algorithms operate. Finally, we observe that practical implementations must draw simultaneously on both potential-based and homotopy-based methods and reassess the popular formulations of Mehrotra’s breakthrough implementation in this light.

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Authors and Affiliations

  1. Department of Pure and Applied Mathematics, Washington State University, Pullman, WA, 99164-3113, USA

    J. L. Nazareth

  2. Department of Applied Mathematics, University of Washington, Seattle, WA, 98195, USA

    J. L. Nazareth

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  1. Institute of Computational Mathematics, and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, China

    Ya-xiang Yuan

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Nazareth, J.L. (1998). Computer Solution of Linear Programs: Non-simplex Algorithms. In: Yuan, Yx. (eds) Advances in Nonlinear Programming. Applied Optimization, vol 14. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3335-7_6

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  • DOI: https://doi.org/10.1007/978-1-4613-3335-7_6

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