Sequential constant rank constraint qualifications for nonlinear semidefinite programming with applications

Abstract

We present new constraint qualification conditions for nonlinear semidefinite programming that extend some of the constant rank-type conditions from nonlinear programming. As an application of these conditions, we provide a unified global convergence proof of a class of algorithms to stationary points without assuming neither uniqueness of the Lagrange multiplier nor boundedness of the Lagrange multipliers set. This class of algorithm includes, for instance, general forms of augmented Lagrangian, sequential quadratic programming, and interior point methods. We also compare these new conditions with some of the existing ones, including the nondegeneracy condition, Robinson's constraint qualification, and the metric subregularity constraint qualification.

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

“…As it can be easily verified, the same conclusion holds for the constraint qualification called "weak-nondegeneracy" for NSDP that was introduced in [10]. Thus, a specialized analysis is required to obtain results similar to [9,10], for NSOCP. In fact, the analysis we present in this section regarding those weak conditions is, in a sense, more refined than the one presented in [10] since there are some important questions that were left open in [10], which we are able to answer here.…”

“…For such, we will first introduce weak variants of the nondegeneracy condition and Robinson's CQ -here called weak-nondegeneracy and weak-Robinson's CQ -which are weaker than their original versions but that still reduce to LICQ and MFCQ, respectively, when an NLP problem is modelled as an NSOCP problem with several one-dimensional constraints. Moreover, we show that weak-nondegeneracy is strictly weaker than nondegeneracy, and we also clarify some relations between weak-nondegeneracy (weak-Robinson's CQ) and standard nondegeneracy (Robinson's CQ), which were only partially addressed in [9]. In particular, we show a new characterization of nondegeneracy in terms of the validity of weak-nondegeneracy plus the linear independence of a partial Jacobian of the constraints.…”

“…Besides, these conditions shall pave the way for a more radical relaxation in terms of local constant rank, which will be discussed in the next section. A similar approach has been conducted in [9,10] for NSDP problems, but although NSOCP can be seen as a particular case of NSDP via an arrowhead matrix transformation…”

“…However, no application towards algorithms was provided or suggested in [12]. Since the sequential approach from [11] seems more suitable for algorithms, we developed it even further for NSDP problems [9,10] by directly exploiting the eigenvector structure of the problem, overcoming the limitations of the naive approach.…”

“…This paper introduces new constraint qualifications for NSOCP problems following similar ideas to those used in [9] and [10], but taking into account the structure of the second-order cone. For such, we will first introduce weak variants of the nondegeneracy condition and Robinson's CQ -here called weak-nondegeneracy and weak-Robinson's CQ -which are weaker than their original versions but that still reduce to LICQ and MFCQ, respectively, when an NLP problem is modelled as an NSOCP problem with several one-dimensional constraints.…”