In this paper, we conduct a thorough study on the first and second order properties of the Moreau-Yosida regularization of the vector k-norm function, the indicator function of its epigraph, and the indicator function of the vector k-norm ball. We start with settling the vector k-norm case via applying the existing breakpoint searching algorithms to the metric projector over its dual norm ball....

We consider computing optimal k-norm preemptive schedules of jobs that arrive over time. In particular, we show that computing the optimal k-norm of flow schedule, 1 | rj , pmtn | ∑ j(Cj − rj) k in standard 3-field scheduling notation, is strongly NP-hard for k ∈ (0, 1) and integers k ∈ (1,∞). Further we show that computing the optimal k-norm of stretch schedule, 1 | rj , pmtn | ∑ j((Cj − rj)/p...

We present an algorithm to compute the Euclidean minimum of an algebraic number field, which is a generalization of the algorithm restricted to the totally real case described by Cerri ([7]). With a practical implementation, we obtain unknown values of the Euclidean minima of algebraic number fields of degree up to 8 in any signature, especially for cyclotomic fields, and many new examples of n...

We study a regularizer which is defined as a parameterized infimum of quadratics, and which we call the box-norm. We show that the k-support norm, a regularizer proposed by Argyriou et al. (2012) for sparse vector prediction problems, belongs to this family, and the box-norm can be generated as a perturbation of the former. We derive an improved algorithm to compute the proximity operator of th...