Alternative second-order conditions in constrained optimization

New formulations are given for the second-order necessary conditions in parameter optimization with equality constraints. The new conditions are shown to be equivalent to previously known conditions by using the properties of projection matrices. The various conditions require different computational tools. As one of the obtained conditions requires standard computational tools (i.e., matrix inversion and eigen-value computation), it might be useful in applications (e.g., real-time optimization schemes), when these standard tools are already in use.