Exact penalization for cardinality and rank-constrained optimization problems via partial regularization

In this paper, we consider a class of constrained optimization problems whose constraints involve a cardinality or rank constraint. The penalty formulation based on a partial regularization has recently been promoted in the literature to approximate these problems, which usually outperforms the penalty formulation based on a full regularization in terms of solution quality. Nevertheless, the relation between the penalty formulation with a partial regularizer and the original problem was not much studied yet. Under some suitable assumptions, we show that the penalty formulation based on a partial regularization is an exact reformulation of the original problem in the sense that they both share the same global minimizers. We also show that a local minimizer of the original problem is that of the penalty reformulation. These results provide some theoretical justification for the often-observed superior performance of the penalty model based on a partial regularizer over a corresponding full regularizer.