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

Spyridon Pougkakiotis () and Jacek Gondzio ()
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Spyridon Pougkakiotis: University of Edinburgh
Jacek Gondzio: University of Edinburgh

Journal of Optimization Theory and Applications, 2022, vol. 192, issue 1, No 5, 97-129

Abstract: 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.

Keywords: Regularized Interior Point Methods; Semidefinite programming; Proximal method of multipliers; Interior Point Method; Proximal point method (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10957-021-01954-4

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