Refining the partition for multifold conic optimization problems

ABSTRACT

In this paper, we give a unified treatment of two different definitions of complementarity partition of multifold conic programs introduced independently in Bonnans and Ramírez [Perturbation analysis of second-order cone programming problems, Math Program. 2005;104(2–30):205–227] for conic optimization problems, and in Peña and Roshchina [A complementarity partition theorem for multifold conic systems, Math Program. 2013;142(1–2):579–589] for homogeneous feasibility problems. We show that both can be treated within the same unified geometric framework and extend the latter notion to optimization problems. We also show that the two partitions do not coincide, and their intersection gives a seven-set index partition. Finally, we demonstrate that the partitions are preserved under the application of nonsingular linear transformations, and in particular, that a standard conversion of a second-order cone program into a semidefinite programming problem preserves the partitions.

Keywords:

• Linear conic programming
• optimal partition
• feasibility problems

2010 Mathematics Subject Classifications:

• 90C46
• 90C22
• 90C25

Acknowledgments

We thank two anonymous reviewers who have substantially contributed to the improved quality of the revision. This research was partially supported by ANID (Chile) under REDES project number 180032 and by the Australian Research Council grant DE150100240. The second author was supported by FONDECYT regular projects 1160204 and 1201982, and Basal Program CMM-AFB 170001, all from ANID (Chile).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was partially supported by ANID (Chile) under REDES project number 180032 and by the Australian Research Council grant DE150100240. The second author was supported by FONDECYT (Fondo de Fomento al Desarrollo Científico y Tecnológico) regular projects 1160204 and 1201982, and Basal Program CMM-AFB 170001 (Comisión Nacional de Investigación Científica y Tecnológica), all from ANID (Chile).