Global Convergence of Algorithms Under Constant Rank Conditions for Nonlinear Second-Order Cone Programming

TL;DR

This paper extends the concept of nondegeneracy and constraint qualifications from nonlinear semidefinite programming to nonlinear second-order cone programming, enabling weaker conditions that still guarantee algorithmic convergence.

Contribution

It introduces weaker constraint qualifications based on problem structure, ensuring global convergence without boundedness of Lagrange multipliers.

Findings

Weaker constraint qualifications are sufficient for algorithm convergence.

Structural properties of second-order cones are exploited to simplify conditions.

The approach guarantees convergence without requiring bounded Lagrange multipliers.

Abstract

In [R. Andreani, G. Haeser, L. M. Mito, H. Ram\'irez C., Weak notions of nondegeneracy in nonlinear semidefinite programming, arXiv:2012.14810, 2020] the classical notion of nondegeneracy (or transversality) and Robinson's constraint qualification have been revisited in the context of nonlinear semidefinite programming exploiting the structure of the problem, namely, its eigendecomposition. This allows formulating the conditions equivalently in terms of (positive) linear independence of significantly smaller sets of vectors. In this paper we extend these ideas to the context of nonlinear second-order cone programming. For instance, for an $m$-dimensional second-order cone, instead of stating nondegeneracy at the vertex as the linear independence of $m$ derivative vectors, we do it in terms of several statements of linear independence of $2$ derivative vectors. This allows embedding the structure of the second-order cone into the formulation of nondegeneracy and, by extension, Robinson's constraint qualification as well. This point of view is shown to be crucial in defining significantly weaker constraint qualifications such as the constant rank constraint qualification and the constant positive linear dependence condition. Also, these conditions are shown to be sufficient for guaranteeing global convergence of several algorithms, while still implying metric subregularity and without requiring boundedness of the set of Lagrange multipliers.