Refining the partition for multifold conic optimization problems

TL;DR

This paper unifies two definitions of complementarity partitions in multifold conic optimization, extends the concept to optimization problems, and analyzes their properties and invariance under transformations.

Contribution

It provides a unified geometric framework for the partitions, shows they differ but intersect to form a seven-set partition, and proves their invariance under nonsingular linear transformations.

Findings

The two partitions do not coincide.

Their intersection forms a seven-set index partition.

Partitions are preserved under nonsingular linear transformations.

Abstract

In this paper we give a unified treatment of two different definitions of complementarity partition of multifold conic programs introduced independently in [J. F. Bonnans and H. Ram\'irez C., Math. Program. 104 (2005), no. 2-3, Ser. B, 205--227] for conic optimization problems, and in [J. Pe\~na and V. Roshchina, Math. Program. 142 (2013), no 1-2, Ser. A, 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.