GRAS uses three algorisms for solving constrained nonlinear programming problems. The first algorism is quasi Newton method, which solves extended lagrange equation. The second is the algorism for implementing nonlinear constrained inequality into the extended lagrangean and for adjusting penalty and lagrangean. The third is the linear optimizing algorism.

This method is slightly mentioned in the book, Numerical Optimization, second edt., by J.Nocedal and Stephen J. Wright, Springer 2006. The method is mentioned in the pages 519-524. In the book, the algorism is for equality constraint. However, I employ inequality constraint. Whole algorism that I refer to is in a Japanese book. So, in the following pdf file, I add a little explanations for the mathematical base of the algorism that GRAS uses.