Title: Solving convex optimization problems for which Slater's constraint qualification fails



For nonlinear convex optimization problems, it is known that strong duality need not hold in the absence of a constraint qualification. Moreover, constraint qualifications are closely tied to numerical stability and well-posedness. We discuss two known approaches to regularizing problems for which Slater's constraint qualification fails, and then explain our own improved method, which identifies and solves such problems, including those for which the duality gap is nonzero and finite.

Our procedure only requires the solution of well behaved convex programs of size not exceeding that of the original problem. We explain in some detail how our procedure is applied to semidefinite programs.

In practice, it might not be possible to distinguish between problems for which Slater's constraint qualification fails and those for which it ``almost fails''. We discuss how to obtain an approximate optimal solution in this case. Some numerical results will be given.