Abstract
This paper is devoted to the study of optimal solutions of symmetric cone programs by means of the asymptotic behavior of central paths with respect to a broad class of barrier functions. This class is, for instance, larger than that typically found in the literature for semidefinite positive programming. In this general framework, we prove the existence and the convergence of primal, dual and primal–dual central paths. We are then able to establish concrete characterizations of the limit points of these central paths for specific subclasses. Indeed, for the class of barrier functions defined at the origin, we prove that the limit point of a primal central path minimizes the corresponding barrier function over the solution set of the studied symmetric cone program. In addition, we show that the limit points of the primal and dual central paths lie in the relative interior of the primal and dual solution sets for the case of the logarithm and modified logarithm barriers.
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Acknowledgments
This work is supported by CONICYT’s BASAL project PFB-03 “Centro de Modelamiento Matemático, Universidad de Chile.” Additionally, the first author is partially supported by Math-Amsud N\(^{\circ }\) 15 MATH-02 and by FONDECYT project 1160204, and the second author is partially supported by FONDECYT project 3150323. Both authors would like to thank the referees for their meticulous reading of the manuscript and for several suggestions that improved its presentation.
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Ramírez, H., Sossa, D. On the Central Paths in Symmetric Cone Programming. J Optim Theory Appl 172, 649–668 (2017). https://doi.org/10.1007/s10957-016-0989-8
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DOI: https://doi.org/10.1007/s10957-016-0989-8