The algorithm is called FAIPA – Feasible Arc Interior Point Algorithm 17. FAIPA requires an initial point at the interior of the inequality constraints, and produces a sequence of interior points. When the problem is characterised by inequality constraints only, the objective function become smaller at each iteration. In cases in which also equality constraints are present, an auxiliary potential function is used. However, the orbit determination problem this thesis tries to address does not require equality constraints. FAIPA can be considered an efficient optimization tool because it gives a series of interior points even when the constraints are nonlinear. Since it returns the objective function value at each iteration, it can be stopped according to many commonly adopted criteria. For example, when the objective function reduction per iteration becomes smaller of a certain threshold.
The most important remark, which has also been one of the main design driver for our orbit determination software, is that FAIPA algorithm never performs function evaluations outside the feasible region. “FAIPA, that is an extension of the Feasible Directions Interior Point Algorithm, integrates ideas coming from the modern Interior Point Algorithms for Linear Programming with Feasible Directions Methods. At each point, FAIPA defines a “Feasible Descent Arc”. Then, it finds on the arc a new interior point with a lower objective. Newton, quasi – Newton and first order versions of FAIPA can be obtained. FAIPA is supported by strong theoretical results. In particular, the search along an arc ensures superlinear convergence for the quasi – Newton version, even when there are highly non-linear constraints, avoiding the so called “Maratos’ effect”.
The present method, that is simple to code, does not require the solution of quadratic programs and it is not a penalty neither a barrier method. It merely requires the solution of three linear systems with the same matrix per iteration. Several practical applications of the present and previous versions of FAIPA, as well as several numerical results, show that the present is a very strong and efficient technique for engineering design optimization” 17.
The following lines are extensively based on “Mathematical Programming Models and Algorithms for Engineering Design Optimization”, J. Herskovits.