Penalty methods for a class of non-Lipschitz optimization problems

Understand

We consider a class of constrained optimization problems with a possibly nonconvex non-Lipschitz objective and a convex feasible set being the intersection of a polyhedron and a possibly degenerate ellipsoid.

• Such problems have a wide range of applications in data science, where the objective is used for inducing sparsity in the solutions while the constraint set models the noise tolerance and incorporates other prior information for data fitting.
• To solve this class of constrained optimization problems, a common approach is the penalty method.
• However, there is little theory on exact penalization for problems with nonconvex and non-Lipschitz objective functions.