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An Interior Point-Proximal Method of Multipliers for Linear Positive Semi-Definite Programming

Author

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  • Spyridon Pougkakiotis

    (University of Edinburgh)

  • Jacek Gondzio

    (University of Edinburgh)

Abstract

In this paper we generalize the Interior Point-Proximal Method of Multipliers (IP-PMM) presented in Pougkakiotis and Gondzio (Comput Optim Appl 78:307–351, 2021. https://doi.org/10.1007/s10589-020-00240-9 ) for the solution of linear positive Semi-Definite Programming (SDP) problems, allowing inexactness in the solution of the associated Newton systems. In particular, we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM) and interpret the algorithm (IP-PMM) as a primal-dual regularized IPM, suitable for solving SDP problems. We apply some iterations of an IPM to each sub-problem of the PMM until a satisfactory solution is found. We then update the PMM parameters, form a new IPM neighbourhood, and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under mild assumptions, and without requiring exact computations for the Newton directions. We furthermore provide a necessary condition for lack of strong duality, which can be used as a basis for constructing detection mechanisms for identifying pathological cases within IP-PMM.

Suggested Citation

  • Spyridon Pougkakiotis & Jacek Gondzio, 2022. "An Interior Point-Proximal Method of Multipliers for Linear Positive Semi-Definite Programming," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 97-129, January.
    • Handle: RePEc:spr:joptap:v:192:y:2022:i:1:d:10.1007_s10957-021-01954-4
    DOI: 10.1007/s10957-021-01954-4
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    References listed on IDEAS

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    1. Spyridon Pougkakiotis & Jacek Gondzio, 2019. "Dynamic Non-diagonal Regularization in Interior Point Methods for Linear and Convex Quadratic Programming," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 905-945, June.
    2. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
    3. A. Ben-Tal & A. Nemirovski, 1998. "Robust Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 769-805, November.
    4. Spyridon Pougkakiotis & Jacek Gondzio, 2021. "An interior point-proximal method of multipliers for convex quadratic programming," Computational Optimization and Applications, Springer, vol. 78(2), pages 307-351, March.
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    Cited by:

    1. Jacek Gondzio & Spyridon Pougkakiotis & John W. Pearson, 2022. "General-purpose preconditioning for regularized interior point methods," Computational Optimization and Applications, Springer, vol. 83(3), pages 727-757, December.
    2. Sabir, Zulqurnain & Said, Salem Ben & Baleanu, Dumitru, 2022. "Swarming optimization to analyze the fractional derivatives and perturbation factors for the novel singular model," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).

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