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.
Highlights
Positive Semidefinite Programming (SDP) problems have attracted a lot of attention in the literature for more than two decades, and have been used to model a plethora of Communicated by Florian Potra.Journal of Optimization Theory and Applications different problems arising from control theory [3, Chapter 14], power systems [13], stochastic optimization [6], truss optimization [28], and many other application areas
A viable and successful alternative to Interior Point Method (IPM) for SDP problems, which circumvents the issue of ill-conditioning without significantly compromising convergence speed, is based on the so-called Augmented Lagrangian method (ALM), which can be seen as the dual application of the proximal point method
We develop and analyze an IP-Proximal Method of Multipliers (PMM) for linear SDP problems, which allows for inexactness in the solution of the linear systems that have to be solved at every iteration
Summary
Positive Semidefinite Programming (SDP) problems have attracted a lot of attention in the literature for more than two decades, and have been used to model a plethora of Communicated by Florian Potra. While IPMs enjoy fast convergence, in theory and in practice, each IPM iteration requires the solution of a very large-scale linear system, even for small-scale SDP problems What is worse, such linear systems are inherently ill-conditioned. Viable alternatives based on proximal splitting algorithms have been studied in [12,25] Such schemes are very efficient and require significantly less computations and memory per iteration, as compared to IPM or ALM. The algorithm in [19] was developed for convex quadratic programming problems and assumed that the resulting linear systems are solved exactly Under this framework, it was proved that IP-PMM converges in a polynomial number of iterations, under mild assumptions, and an infeasibility detection mechanism was established.
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