Exact solutions to max∥x∥=1∑i=1∞∥Ti(x)∥2 with applications to Physics, Bioengineering and Statistics

Abstract The supporting vectors of a matrix A are the solutions of max ∥ x ∥ 2 = 1 ∥ A x ∥ 2 2 . The generalized supporting vectors of matrices A 1 , ⋯ , A k are the solutions of max ∥ x ∥ 2 = 1 ∥ A 1 x ∥ 2 2 + ⋯ + ∥ A k x ∥ 2 2 . Notice that the previous optimization problem is also a boundary element problem since the maximum is attained on the unit sphere. Many problems in Physics, Statistics and Engineering can be modeled by using generalized supporting vectors. In this manuscript we first raise the generalized supporting vectors to the infinite dimensional case by solving the optimization problem max ∥ x ∥ = 1 ∑ i = 1 ∞ ∥ T i ( x ) ∥ 2 where ( T i ) i ∈ N is a sequence of bounded linear operators between Hilbert spaces H and K of any dimension. Observe that the previous optimization problem generalizes the first two. Then a unified MATLAB code is presented for computing generalized supporting vectors of a finite number of matrices. Some particular cases are considered and three novel examples are provided to which our technique applies: optimized observable magnitudes by a pure state in a quantum mechanical system, a TMS optimized coil and an optimal location problem using statistics multivariate analysis. These three examples show the wide applicability of our theoretical and computational model.