Because the moment of inertia of a damaged member can take on any positive real value, the structure contains continuous design variables and, in turn, can be formulated to have continuous constraint and objective functions. This engineering problem lends itself well to gradient-based optimization. Several gradient based optimization procedures have been developed over the years, but most follow a similar form. The typical constrained optimization problem has the form:
Minimize ^ Ф = f (xj, x2,x3,…,xn) s. t. ^ gi (x^x2,x3,…,xn ) = 0
Note, Eq(5) and Eq(6) provide sufficient equations to solve for all lagrange multipliers.