[Paper Review] Sequential constant rank constraint qualifications for nonlinear semidefinite programming with applications

Roberto Andreani, Gabriel Haeser+2 more|arXiv (Cornell University)|Jun 1, 2021

Advanced Optimization Algorithms Research22 references4 citations

TL;DR

This paper introduces sequential constant rank constraint qualifications—seq-CRCQ and seq-CPLD—for nonlinear semidefinite programming (NSDP), extending NLP-type conditions to conic optimization. These conditions ensure global convergence of augmented Lagrangian, SQP, and interior point methods to stationary points without requiring boundedness or uniqueness of Lagrange multipliers, and they are strictly weaker than nondegeneracy and Robinson’s CQ while implying metric subregularity.

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.

Motivation & Objective

• To extend constant rank-type constraint qualifications (CRCQ and CPLD) from nonlinear programming (NLP) to nonlinear semidefinite programming (NSDP), preserving their desirable convergence properties.
• To develop stronger, algorithmically robust variants—sequential-CRCQ and sequential-CPLD—by incorporating perturbations into the definitions, enabling application to a broader class of algorithms.
• To establish global convergence of augmented Lagrangian, sequential quadratic programming (SQP), and interior point methods to stationary points under weaker assumptions than nondegeneracy or Robinson’s CQ.
• To compare the new conditions with existing ones (nondegeneracy, Robinson’s CQ, metric subregularity) and clarify their relative strength and independence.
• To provide both sequential and non-sequential characterizations of the conditions, supporting theoretical and algorithmic applications.

Proposed method

• Propose weak-CRCQ and weak-CPLD for NSDP by leveraging eigenvector structures of matrix constraints, inspired by weak-nondegeneracy and weak-Robinson’s CQ from prior work.
• Introduce sequential variants—seq-CRCQ and seq-CPLD—by incorporating perturbations in the constraint dependence structure, enhancing robustness to approximate multipliers.
• Use Carathéodory’s lemma and the infinite pigeonhole principle to analyze linear dependence of gradient directions in the limit, establishing contradiction under failure of the conditions.
• Prove that seq-CPLD implies the metric subregularity constraint qualification, linking it to error bounds and convergence stability.
• Demonstrate that weak-CRCQ and weak-CPLD reduce to standard NLP counterparts when NLP is embedded via structurally diagonal semidefinite constraints.
• Establish a hierarchy of conditions: nondegeneracy ⇒ weak-nondegeneracy ⇒ weak-Robinson’s CQ ⇒ weak-CRCQ ⇒ weak-CPLD ⇒ Robinson’s CQ ⇒ seq-CRCQ ⇒ seq-CPLD ⇒ metric subregularity CQ.

Experimental results

Research questions

• RQ1 Can constant rank-type constraint qualifications from NLP be meaningfully extended to nonlinear semidefinite programming while preserving their convergence properties?
• RQ2 How do the proposed sequential variants of CRCQ and CPLD compare in strength to existing CQs like nondegeneracy, Robinson’s CQ, and metric subregularity?
• RQ3 Can these new CQs support global convergence of major NSDP algorithms (e.g., augmented Lagrangian, SQP, interior point) without assuming boundedness or uniqueness of Lagrange multipliers?
• RQ4 What is the relationship between the new sequential CQs and classical error bound or second-order optimality conditions in NSDP?
• RQ5 Do the weak and sequential CQs admit non-sequential characterizations that facilitate theoretical analysis and algorithm design?

Key findings

• The proposed sequential-CRCQ (seq-CRCQ) is strictly weaker than the nondegeneracy condition and independent of Robinson’s CQ, yet ensures global convergence of a broad class of algorithms.
• Sequential-CPLD (seq-CPLD) is strictly weaker than Robinson’s CQ and implies the metric subregularity constraint qualification, enabling error bound analysis.
• The weak-CRCQ and weak-CPLD conditions are shown to be consistent with the standard NLP counterparts when an NLP problem is embedded into NSDP via a structurally diagonal matrix constraint.
• The conditions are robust under perturbations and support a unified global convergence proof for augmented Lagrangian, SQP, and interior point methods to stationary points.
• A hierarchy of constraint qualifications is established: nondegeneracy ⇒ weak-nondegeneracy ⇒ weak-Robinson’s CQ ⇒ weak-CRCQ ⇒ weak-CPLD ⇒ Robinson’s CQ ⇒ seq-CRCQ ⇒ seq-CPLD ⇒ metric subregularity CQ.
• The paper provides both sequential and non-sequential characterizations of the new CQs, enhancing their theoretical and algorithmic applicability.