Minimization of Constrained Quadratic forms in Hilbert Spaces

TL;DR

This paper extends the minimization of symmetric positive definite quadratic forms with linear constraints from finite-dimensional matrices to infinite-dimensional Hilbert spaces, using generalized inverses of operators.

Contribution

It generalizes the solution for quadratic form minimization to infinite-dimensional Hilbert spaces, including cases with singular and non-invertible positive operators.

Findings

Extended Moore-Penrose inverse to operators in Hilbert spaces.

Provided solutions for positive diagonalizable and arbitrary positive operators.

Addressed minimization with singular constraints in infinite-dimensional spaces.

Abstract

A common optimization problem is the minimization of a symmetric positive definite quadratic form $< x,Tx >$ under linear constrains. The solution to this problem may be given using the Moore-Penrose inverse matrix. In this work we extend this result to infinite dimensional complex Hilbert spaces, making use of the generalized inverse of an operator. A generalization is given for positive diagonizable and arbitrary positive operators, not necessarily invertible, considering as constraint a singular operator. In particular, when $T$ is positive semidefinite, the minimization is considered for all vectors belonging to $\mathcal{N}(T)^\perp$.