Real life engineering systems involve a very large number of design variables and constraints. Evaluation of functions and of derivatives coming from engineering models is very expensive in terms of computer time. In practical applications, calculation and storage of second derivatives are impossible to be carried out. Then, numerical techniques for engineering optimization must be capable to solve very large problems with a reasonable number of function evaluations and without needing second derivatives. Robustness is also a crucial point for industrial applications.
Quasi-Newton techniques for nonlinear optimization construct a full matrix that is an approximation of the second derivative of the function, in the unconstrained case, or of the second derivative of the Lagrangian, when constraints are considered. Usually, numerical algorithms require positive definite quasi-Newton matrices. Classical techniques work with full quasi-Newton matrices requiring a very large storage area and a great number of computations. We present a new updating technique to obtain positive definite sparse quasi-Newton matrices. This technique can be included in the Feasible Arc Interior Point Algorithm (FAIPA),[
], in the Sequential Quadratic Programming Method (SQP) and in Primal-Dual optimization Algorithms. Several very large test constrained optimization problems, employing the present technique within FAIPA, were solved very efficiently.