Abstract
Semidefinite optimization has an ever growing family of crucial applications, and large neighborhood interior point methods (IPMs) yield the method of choice to solve them. This chapter reviews the fundamental concepts and complexity results of Self-Regular (SR) IPMs for semidefinite optimizaion, that up to a log factor achieve the best polynomial complexity bound of small neighborhood IPMs. SR kernel functions are in the core of SR-IPMs. This chapter reviews several none SR kernel functions too. IPMs based on theses kernel functions enjoy similar iteration complexity bounds as SR-IPMs, though their complexity analysis requires additional tools.
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Notes
- 1.
As we see the right hand side of the last equation in (15.11) is the negative gradient of the classical logarithmic barrier function ψ(V ) (see Definition 3 on p. 9), where
$$\psi (t) = \frac{{t}^{2} - 1} {2} -\log t.$$
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Acknowledgements
Research of the second author was supported by a start-up grant of Lehigh University, and by the Hungarian National Development Agency and the European Union within the frame of the project TAMOP 4.2.2-08/1-2008-0021 at the Széchenyi István University entitled “Simulation and Optimization - basic research in numerical mathematics”.
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Salahi, M., Terlaky, T. (2012). Self-Regular Interior-Point Methods for Semidefinite Optimization. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_15
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