Exact Regularization of Polyhedral Norms

Abstract

We are concerned with linearly constrained convex programs with polyhedral norm as objective function. Friedlander and Tseng [SIAM J. Optim., 18 (2007), pp. 1326--1350] have shown that there exists an exact regularization parameter for the associated regularized problems. Possible values of the exact regularization parameter will in general depend on the given right-hand sides of the linear constraint. Here we prove that by taking the square of a polyhedral norm in the regularized objective function there exists an exact regularization parameter independent of the given right-hand sides. In the $\ell_1$-norm case, where one is interested in finding sparse solutions of underdetermined systems of equations, we give explicit expressions for exact regularization parameters, provided that the expected number of nonzeros of the solution is less than some upper bound. The bounds are those known to be sufficient for a minimum $\ell_1$-norm solution to be the sparsest solution as well. Furthermore, for the $\ell_1$-norm and the $\ell_\infty$-norm we compute the duality mappings of the dual regularized norms, which in turn can be used to solve the smooth unconstrained dual of the regularized problem. In the $\ell_1$-norm case the duality mapping involves a shrinkage operation where the value of the threshold depends on the point which is to be shrunk, in contrast to the well-known soft shrinkage operator, where the threshold is fixed. The $\ell_\infty$-norm case results in a projection operation onto an $\ell_\infty$-ball whose size depends on the point to be projected.