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Constraint Qualification for Generic Parameter Families of Constraints in Optimization

Naoki Hamada, Kenta Hayano, Hiroshi Teramoto

TL;DR

The paper delivers a complete generic map-germ–level portrait of constraint qualifications for four-parameter constraint families by proving invariance of LICQ, MFCQ, ACQ, and GCQ under $\,\mathcal{K}[G]$-equivalence and reduction. Leveraging the prior classification of fully reduced constraint map-germs, the authors verify CQ validity across all generic classes, producing a comprehensive generic CQ landscape (Theorem). A key finding is that GCQ holds in a broad array of generic germs even as the stronger CQs fail, revealing that the GCQ–stronger CQ gap is a generic phenomenon. The work also outlines how this framework can aid in generating representative CQ examples, benchmarking algorithms, and guiding automated CQ recognition in future software, while noting limitations to smooth, four-parameter, differentiable constraints and the specific invariances considered.

Abstract

Constraint qualifications (CQs) are central to the local analysis of constrained optimization. In this paper, we completely determine the validity of the four classical CQs -- LICQ, MFCQ, ACQ, and GCQ -- for constraint map-germs that arise in generic four-parameter families. Our approach begins by proving that all four CQs are invariant under the action of the group $\mathcal{K}[G]$ and under the operation of reduction. As a consequence, the verification of CQ-validity for a generic constraint reduces to checking CQ-validity on the $\mathcal{K}[G]$-normal forms of fully reduced map-germs. Such normal forms have been classified in our recent work. In the present paper, we verify which CQs hold in each germ appearing in the classification tables from that work. This analysis provides a complete picture of the generic landscape of the four classical CQs. Most notably, we find that there exist numerous generic map-germs for which GCQ holds while all stronger CQs fail, showing that the gap between GCQ and the other qualifications is not an exceptional phenomenon but arises generically.

Constraint Qualification for Generic Parameter Families of Constraints in Optimization

TL;DR

The paper delivers a complete generic map-germ–level portrait of constraint qualifications for four-parameter constraint families by proving invariance of LICQ, MFCQ, ACQ, and GCQ under -equivalence and reduction. Leveraging the prior classification of fully reduced constraint map-germs, the authors verify CQ validity across all generic classes, producing a comprehensive generic CQ landscape (Theorem). A key finding is that GCQ holds in a broad array of generic germs even as the stronger CQs fail, revealing that the GCQ–stronger CQ gap is a generic phenomenon. The work also outlines how this framework can aid in generating representative CQ examples, benchmarking algorithms, and guiding automated CQ recognition in future software, while noting limitations to smooth, four-parameter, differentiable constraints and the specific invariances considered.

Abstract

Constraint qualifications (CQs) are central to the local analysis of constrained optimization. In this paper, we completely determine the validity of the four classical CQs -- LICQ, MFCQ, ACQ, and GCQ -- for constraint map-germs that arise in generic four-parameter families. Our approach begins by proving that all four CQs are invariant under the action of the group and under the operation of reduction. As a consequence, the verification of CQ-validity for a generic constraint reduces to checking CQ-validity on the -normal forms of fully reduced map-germs. Such normal forms have been classified in our recent work. In the present paper, we verify which CQs hold in each germ appearing in the classification tables from that work. This analysis provides a complete picture of the generic landscape of the four classical CQs. Most notably, we find that there exist numerous generic map-germs for which GCQ holds while all stronger CQs fail, showing that the gap between GCQ and the other qualifications is not an exceptional phenomenon but arises generically.

Paper Structure

This paper contains 12 sections, 69 theorems, 145 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries: constraints, $\mathcal{K}[G]$-equivalence, reduction, and classification of constraint map-germs
  3. Relation between reduction and $\mathcal{K} \left[ G \right]$-equivalence
  4. Classification of generic constraint map-germs
  5. Definitions and properties of constraint qualifications
  6. Verification of constraint qualifications for $\mathcal{K}[G]$-classes of generic constraints
  7. LICQ and MFCQ
  8. ACQ and GCQ
  9. ACQ and GCQ in Table \ref{['table:generic constraint q=0']}
  10. ACQ and GCQ in Table \ref{['table:generic constraint r=0']}
  11. ACQ and GCQ in Table \ref{['table:generic constraint q>0 r=1']}
  12. Genericity of counter examples to show the strictness of the hierarchy of the four classical constraint qualifications

Key Result

Lemma 2.1

Let $\left( g, h \right)$ and $\left( g', h' \right)$ be two constraint map-germs with the equal number of inequality and equality constraints. Then, $\left( g, h \right)$ and $\left( g', h' \right)$ are $\mathcal{K} \left[ G \right]$-equivalent if their reductions $\left( g, h \right)_{\iota_{\left

Theorems & Definitions (149)

  • Definition 2.1: Constraint map-germ and feasible set
  • Definition 2.2: Reduction
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 139 more