On sharp performance bounds for robust sparse signal recoveries

It is well known in compressive sensing that l_1 minimization can recover the sparsest solution for a large class of underdetermined systems of linear equations, provided the signal is sufficiently sparse. In this paper, we compute sharp performance bounds for several different notions of robustness in sparse signal recovery via l_1 minimization. In particular, we determine necessary and sufficient conditions for the measurement matrix A under which l_1 minimization guarantees the robustness of sparse signal recovery in the "weak", "sectional" and "strong" (e.g., robustness for "almost all" approximately sparse signals, or instead for "all" approximately sparse signals). Based on these characterizations, we are able to compute sharp performance bounds on the tradeoff between signal sparsity and signal recovery robustness in these various senses. Our results are based on a high-dimensional geometrical analysis of the null-space of the measurement matrix A. These results generalize the thresholds results for purely sparse signals and also present generalized insights on l_1 minimization for recovering purely sparse signals from a null-space perspective

Quasiconvex Constrained Multicriteria Continuous Location Problems: Structure of Nondominated Solution Sets

In this paper, we consider constrained multicriteria continuous location problems in two-dimensional spaces. In the literature, the continuous multicriteria location problem in two-dimensional spaces has received special attention in the last years, although only particular instances of convex functions have been considered. Our approach only requires the functions to be strictly quasiconvex and inf-compact. We obtain a geometrical description that provides a unified approach to handle multicriteria location models in two-dimensional spaces which has been implemented in MATHEMATIC