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Implicit Redundancy and Degeneracy in Conic Program

Haesol Im

TL;DR

The paper proves that for any conic program whose Slater condition fails, every feasible point is degenerate and the constraint system inherits implicit redundancies. Using facial reduction, it shows that the feasible set lies in a reduced minimal face, causing the linear map $\mathcal{A}$ to lose surjectivity on the restricted domain, which the authors quantify via the implicit problem singularity $\operatorname{ips}(\mathcal{F})$. These degeneracy and redundancy phenomena are connected to ill-conditioning in interior point methods and non-differentiability of the optimal value under perturbations, extending known results from linear and semidefinite programming to general conic programs. The findings have practical implications for bounds, solver design, and perturbation analysis in conic optimization, highlighting universal structural issues when strict feasibility fails.

Abstract

This paper examines the feasible region of a standard conic program represented as the intersection of a closed convex cone and a set of linear equalities. It is recently shown that when Slater constraint qualification (strict feasibility) fails for the classes of linear and semidefinite programs, two key properties emerge within the feasible region; (a) every point in the feasible region is degenerate; (b) the constraint system inherits implicit redundancies. In this paper we show that degeneracy and implicit redundancies are inherent and universal traits of all conic programs in the absence of strict feasibility.

Implicit Redundancy and Degeneracy in Conic Program

TL;DR

The paper proves that for any conic program whose Slater condition fails, every feasible point is degenerate and the constraint system inherits implicit redundancies. Using facial reduction, it shows that the feasible set lies in a reduced minimal face, causing the linear map to lose surjectivity on the restricted domain, which the authors quantify via the implicit problem singularity . These degeneracy and redundancy phenomena are connected to ill-conditioning in interior point methods and non-differentiability of the optimal value under perturbations, extending known results from linear and semidefinite programming to general conic programs. The findings have practical implications for bounds, solver design, and perturbation analysis in conic optimization, highlighting universal structural issues when strict feasibility fails.

Abstract

This paper examines the feasible region of a standard conic program represented as the intersection of a closed convex cone and a set of linear equalities. It is recently shown that when Slater constraint qualification (strict feasibility) fails for the classes of linear and semidefinite programs, two key properties emerge within the feasible region; (a) every point in the feasible region is degenerate; (b) the constraint system inherits implicit redundancies. In this paper we show that degeneracy and implicit redundancies are inherent and universal traits of all conic programs in the absence of strict feasibility.
Paper Structure (15 sections, 10 theorems, 55 equations, 1 algorithm)

This paper contains 15 sections, 10 theorems, 55 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Related Work
  4. Background
  5. Faces of Cones
  6. Facial Reduction for Conic Programs
  7. Degeneracy
  8. Main Result
  9. Discussions
  10. Other Singularity Notions
  11. Interior Point Methods
  12. Limiting Behaviour of the Self-Concordant Barrier based Interior Point Method
  13. Limiting Behaviour of the Extended Dual Embedding
  14. Differentiability of the Optimal Value Function
  15. Conclusions

Key Result

Lemma 2.1

For a feasible system ${\mathcal{F}}$, exactly one of the following holds.

Theorems & Definitions (20)

  • Lemma 2.1: Theorem of the Alternative
  • Example 2.2: FR on the exponential cone
  • Definition 2.3
  • Theorem 3.1: Degeneracy in the absence of strict feasibility
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • Lemma 3.4
  • proof
  • ...and 10 more