Gauge optimization is the class of problems for finding the element of

a convex set that is minimal with respect to a gauge (e.g., the

least-norm solution of a linear system). These conceptually simple

problems appear in a remarkable array of applications of sparse

optimization. Their structure allows for a special kind of duality

framework that can lead to new algorithmic approaches to challenging

problems. Low-rank spectral optimization problems that arise in two

signal-recovery application, phase retrieval and blind deconvolution,

illustrate the benefits of the approach.