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.
Bibliographical noteFunding Information:
The first author's work is supported partly by Hong Kong Research Grant Council grant PolyU5001/12p. The third author's work is supported partly by Hong Kong Research Grants Council grant PolyU253008/15p. This author's work is supported in part by an NSERC discovery grant.
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- Exact penalty
- Non-Lipschitz optimization
- Nonconvex optimization
- Proximal gradient method
- Sparse solution