Penalty methods for a class of non-Lipschitz optimization problems
This work addresses a theoretical gap in exact penalization for non-Lipschitz optimization, which is incremental but important for data science applications like sparse modeling.
The paper tackles constrained optimization problems with nonconvex, non-Lipschitz objectives, common in data science for sparsity induction, by developing a penalty method with theoretical guarantees for exact penalization and convergence to KKT points, showing efficiency in finding sparse solutions for underdetermined linear systems.
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. In this paper, we study the existence of exact penalty parameters regarding local minimizers, stationary points and $ε$-minimizers under suitable assumptions. Moreover, we discuss a penalty method whose subproblems are solved via a nonmonotone proximal gradient method with a suitable update scheme for the penalty parameters, and prove the convergence of the algorithm to a KKT point of the constrained problem. Preliminary numerical results demonstrate the efficiency of the penalty method for finding sparse solutions of underdetermined linear systems.