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A Non-Archimedean Interior Point Method and Its Application to the Lexicographic Multi-Objective Quadratic Programming

Author

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  • Lorenzo Fiaschi

    (Department of Information Engineering, University of Pisa, 56122 Pisa, Italy)

  • Marco Cococcioni

    (Department of Information Engineering, University of Pisa, 56122 Pisa, Italy)

Abstract

This work presents a generalized implementation of the infeasible primal-dual interior point method (IPM) achieved by the use of non-Archimedean values, i.e., infinite and infinitesimal numbers. The extended version, called here the non-Archimedean IPM (NA-IPM), is proved to converge in polynomial time to a global optimum and to be able to manage infeasibility and unboundedness transparently, i.e., without considering them as corner cases: by means of a mild embedding (addition of two variables and one constraint), the NA-IPM implicitly and transparently manages their possible presence. Moreover, the new algorithm is able to solve a wider variety of linear and quadratic optimization problems than its standard counterpart. Among them, the lexicographic multi-objective one deserves particular attention, since the NA-IPM overcomes the issues that standard techniques (such as scalarization or preemptive approach) have. To support the theoretical properties of the NA-IPM, the manuscript also shows four linear and quadratic non-Archimedean programming test cases where the effectiveness of the algorithm is verified. This also stresses that the NA-IPM is not just a mere symbolic or theoretical algorithm but actually a concrete numerical tool, paving the way for its use in real-world problems in the near future.

Suggested Citation

  • Lorenzo Fiaschi & Marco Cococcioni, 2022. "A Non-Archimedean Interior Point Method and Its Application to the Lexicographic Multi-Objective Quadratic Programming," Mathematics, MDPI, vol. 10(23), pages 1-34, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:23:p:4536-:d:989701
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    References listed on IDEAS

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    1. Cococcioni, Marco & Pappalardo, Massimo & Sergeyev, Yaroslav D., 2018. "Lexicographic multi-objective linear programming using grossone methodology: Theory and algorithm," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 298-311.
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    3. Pourkarimi, L. & Zarepisheh, M., 2007. "A dual-based algorithm for solving lexicographic multiple objective programs," European Journal of Operational Research, Elsevier, vol. 176(3), pages 1348-1356, February.
    4. Khorram, E. & Zarepisheh, M. & Ghaznavi-ghosoni, B.A., 2010. "Sensitivity analysis on the priority of the objective functions in lexicographic multiple objective linear programs," European Journal of Operational Research, Elsevier, vol. 207(3), pages 1162-1168, December.
    5. Kevin A. McShane & Clyde L. Monma & David Shanno, 1989. "An Implementation of a Primal-Dual Interior Point Method for Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 1(2), pages 70-83, May.
    6. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    7. Letsios, Dimitrios & Mistry, Miten & Misener, Ruth, 2021. "Exact lexicographic scheduling and approximate rescheduling," European Journal of Operational Research, Elsevier, vol. 290(2), pages 469-478.
    8. Sherali, Hanif D., 1982. "Equivalent weights for lexicographic multi-objective programs: Characterizations and computations," European Journal of Operational Research, Elsevier, vol. 11(4), pages 367-379, December.
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