Minimization over Nonconvex Sets

Abstract

Minimum norm problems consist of finding the distance of a closed subset of a normed space to the origin. Usually, the given closed subset is also asked to be convex, thus resulting in a convex minimum norm problem. There are plenty of techniques and algorithms to compute the distance of a closed convex set to the origin, which mostly exist in the Hilbert space setting. In this manuscript, we consider nonconvex minimum norm problems that arise from Bioengineering and reformulate them in such a way that the solution to their reformulation is already known. In particular, we tackle the problem of min∥x∥ subject to ∥Rk(x)∥ ≥ ak for k = 1,…,l, where x∈X and Rk:X→Y are continuous linear operators between real normed spaces X,Y, and ak > 0 for k = 1,…,l. Notice that the region of constraints of the previous problem is neither convex nor balanced. However, it is additively symmetric, which is also the case for the objective function, due to the properties satisfied by norms, which makes possible the analytic resolution of such a nonconvex minimization. The recent literature shows that the design of optimal coils for electronics applications can be achieved by solving problems like this. However, in this work, we apply our analytical solutions to design an optimal coil for an electromagnetic sensor.