# The many faces of degeneracy in conic optimization

## TL;DR

This paper explores the causes and implications of degeneracy in conic optimization, emphasizing the role of facial reduction techniques in addressing issues arising from the loss of strict feasibility.

## Contribution

It provides a comprehensive analysis of degeneracy causes in conic optimization and highlights the effectiveness of facial reduction methods for overcoming these challenges.

## Key findings

- Loss of strict feasibility affects optimality conditions and numerical methods.
- Facial reduction offers a mathematically elegant way to handle degeneracy.
- Rich problem structures can be exploited to improve optimization outcomes.

## Abstract

Slater's condition -- existence of a "strictly feasible solution" -- is a common assumption in conic optimization. Without strict feasibility, first-order optimality conditions may be meaningless, the dual problem may yield little information about the primal, and small changes in the data may render the problem infeasible. Hence, failure of strict feasibility can negatively impact off-the-shelf numerical methods, such as primal-dual interior point methods, in particular. New optimization modelling techniques and convex relaxations for hard nonconvex problems have shown that the loss of strict feasibility is a more pronounced phenomenon than has previously been realized. In this text, we describe various reasons for the loss of strict feasibility, whether due to poor modelling choices or (more interestingly) rich underlying structure, and discuss ways to cope with it and, in many pronounced cases, how to use it as an advantage. In large part, we emphasize the facial reduction preprocessing technique due to its mathematical elegance, geometric transparency, and computational potential.