Two popular methods for unconstrained minimization are: Trust Region methods and Quasi-Newton methods. 

Trust Region Subproblems consist in minimizing a quadratic function subject to a quadratic constraint. This directory contains programs for solving the trust region subproblem. This routine is in MATLAB. It uses the MATLAB eigs function, i.e., it finds the smallest eigenvalue using the Lanczos algorithm. This program can solve large sparse problems very quickly. The research report is available.

Also available are two related research reports:
Indefinite trust region subproblems provide characterizations of optimality and relations to perturbed eigenvalue problems for the case where both the objective and constraint can be indefinite;

Nonsymmetric eigenvalue perturbations provides relationships between trust region subproblems and eigenvalue perturbations.
 
Quasi-Newton Methods or Least Change Secant Methods are based on approximating the Hessian using gradient information. The following three papers deal with measures for finding best Hessian approximations:
Sizing and Least Change Secant Methods
Measures for Symmetric Rank-One Updates
An All Inclusive Efficient Region of Updates for Least Change Secant Methods

Also available
Measures for Least Change Secant Methods,
Master's thesis of Qing Zhao.

Bibliography for Optimization with Sensitivity Analysis,
long list of links related to sensitivity analysis.