Self-Scaled Barrier Functions on Symmetric Cones and Their Classification
Foundations of Computational Mathematics
121 - 143
MetadataShow full item record
Please cite this item using this persistent URLhttp://hdl.handle.net/11693/24576
Self-scaled barrier functions on self-scaled cones were axiomatically introduced by Nesterov and Todd in 1994 as a tool for the construction of primal-dual long-step interior point algorithms. This paper provides firm foundations for these objects by exhibiting their symmetry properties, their close ties with the symmetry groups of their domains of definition, and subsequently their decomposition into irreducible parts and their algebraic classification theory. In the first part we recall the characterization of the family of self-scaled cones as the set of symmetric cones and develop a primal-dual symmetric viewpoint on self-scaled barriers, results that were first discovered by the second author. We then show in a short, simple proof that any pointed, convex cone decomposes into a direct sum of irreducible components in a unique way, a result which can also be of independent interest. We then proceed to showing that any self-scaled barrier function decomposes, in an essentially unique way, into a direct sum of self-scaled barriers defined on the irreducible components of the underlying symmetric cone. Finally, we present a complete algebraic classification of self-scaled barrier functions using the correspondence between symmetric cones and Euclidean-Jordan algebras.
Showing items related by title, author, creator and subject.
Salihoglu O.; Hostut, M.; Tansel, T.; Kutluer, K.; Kilic, A.; Alyoruk, M.; Sevik, C.; Turan, R.; Ergun, Y.; Aydinli, A. (Elsevier, 2013)We report on the development of a new structure for type II superlattice photodiodes that we call the "N" design. In this new design, we insert an electron barrier between InAs and GaSb in the growth direction. The barrier ...
Bennett, C. R.; Tanatar, B.; Constantinou, N. C.; Babiker, M. (Pergamon Press, 1994)The effect of cross-sectional geometry on both the intrasubband plasmon and intersubband plasmon of a quantum wire is investigated within a two-subband RPA scheme. Exact analytical electronic wavefunctions for circular, ...
Aksun, M. I.; Çalışkan, F.; Gürel, L. (IEEE, 2002)A numerically efficient technique, based on the spectral-domain method of moments (MoM) in conjunction with the generalized pencil-of-functions (GPOF) method, is developed for the characterization of two-dimensional ...