In http://apmath.spbu.ru/cnsa/pdf/monograf/Numerical_Optimization2006.pdf on page 151, we are told that approximations of the Hessian matrix with the BFGS formula are always symmetric and positive definite if the initial matrix is symmetric and positive definite - if the Wolfe condition is met in the choosing of the step length. My question: What if the initial matrix is symmetric and negative definite? (The reason: I want to use BFGS for finding the maximum instead of the minimum.) Are all subsequent matrices calculated with the BFGS formula symmetric and negative definite? How to proof this?
EDIT: For future readers, the following questions might be usefull:
- How to solve the matrix minimization for BFGS update in Quasi-Newton optimization
- Why does Frobenius norm make BFGS scale-invariant?
- Why does the standard BFGS update rule preserve positive definiteness?
(I am still trying to get around this ;-).)