An Interior Points Method for Nonlinear Constrained Optimization

We describe a new general approach for interior points algorithms in nonlinear constrained optimization. It consists on the iterative solution, in the primal and dual variables, of Karush — Kuhn — Tucker first order optimality conditions. Based on this approach, different algorithms can be stated by taking advantage of the particular characteristics of the problem in consideration and of the order of the available information. This method is very strong and efficient, since at each iteration it only requires the solution of two linear systems with the same matrix. It is also particularly appropriated for Engineering Design Optimization, since feasible designs are obtained. We present a basic algorithm for inequality constrained problems and two of the possible particular versions. The first one is a first order algorithm and the second one uses a quasi — Newton approximation of the second derivative of the Lagrangian, in order to have superlinear asymptotic convergence. Equality constraints are introduced later.