A new semidefinite programming bound for indefinite quadratic forms over a simplex
The author describes a method for computing a lower bound of the minimum of an indefinite quadratic form \(f(x):= x^T Fx\) over the \(n\)-dimensional standard simplex \(\Delta_n\). This method requires two steps. Firstly, by solving a certain semidefinite program there is found a symmetric matrix \(W\) which satisfies \(W\leq F\) (componentwise) and for which the quadratic form \(w(x):= x^T Wx\) is convex on \(\Delta_n\). Secondly, the minimum of \(w\) over \(\Delta_n\) is determined. It is proved that this minimum is a lower bound of the minimum of \(f\) over \(\Delta_n\).
scientific article; zbMATH DE number 1339671