The Many Faces of Degeneracy in Conic Optimization describes various reasons for the loss of strict feasibility, whether due to poor modeling choices or (more interestingly) rich underlying structure, and discusses ways to cope with it and, in many pronounced cases, how to use it as an advantage.
The Many Faces of Degeneracy in Conic Optimization describes various reasons for the loss of strict feasibility, whether due to poor modeling choices ...