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Efficient Use of Quantum Linear System Algorithms in Inexact Infeasible IPMs for Linear Optimization

Author

Listed:
  • Mohammadhossein Mohammadisiahroudi

    (Lehigh University)

  • Ramin Fakhimi

    (Lehigh University)

  • Tamás Terlaky

    (Lehigh University)

Abstract

Quantum computing has attracted significant interest in the optimization community because it potentially can solve classes of optimization problems faster than conventional supercomputers. Several researchers proposed quantum computing methods, especially quantum interior point methods (QIPMs), to solve convex conic optimization problems. Most of them have applied a quantum linear system algorithm at each iteration to compute a Newton step. However, using quantum linear solvers in QIPMs comes with many challenges, such as having ill-conditioned systems and the considerable error of quantum solvers. This paper investigates in detail the use of quantum linear solvers in QIPMs. Accordingly, an Inexact Infeasible Quantum Interior Point (II-QIPM) is developed to solve linear optimization problems. We also discuss how we can get an exact solution by iterative refinement (IR) without excessive time of quantum solvers. The proposed IR-II-QIPM shows exponential speed-up with respect to precision compared to previous II-QIPMs. Additionally, we present a quantum-inspired classical variant of the proposed IR-II-QIPM where QLSAs are replaced by conjugate gradient methods. This classic IR-II-IPM has some advantages compared to its quantum counterpart, as well as previous classic inexact infeasible IPMs. Finally, computational results with a QISKIT implementation of our QIPM using quantum simulators are presented and analyzed.

Suggested Citation

  • Mohammadhossein Mohammadisiahroudi & Ramin Fakhimi & Tamás Terlaky, 2024. "Efficient Use of Quantum Linear System Algorithms in Inexact Infeasible IPMs for Linear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 146-183, July.
    • Handle: RePEc:spr:joptap:v:202:y:2024:i:1:d:10.1007_s10957-024-02452-z
    DOI: 10.1007/s10957-024-02452-z
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    References listed on IDEAS

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    1. Roland W. Freund & Florian Jarre & Shinji Mizuno, 1999. "Convergence of a Class of Inexact Interior-Point Algorithms for Linear Programs," Mathematics of Operations Research, INFORMS, vol. 24(1), pages 50-71, February.
    2. Ambros M. Gleixner & Daniel E. Steffy & Kati Wolter, 2016. "Iterative Refinement for Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 28(3), pages 449-464, August.
    3. S. Bellavia, 1998. "Inexact Interior-Point Method," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 109-121, January.
    4. G. Al-Jeiroudi & J. Gondzio, 2009. "Convergence Analysis of the Inexact Infeasible Interior-Point Method for Linear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 231-247, May.
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