Optimization codes and modeling environment

For the numerical solution of all problems considered in the following section a combination of the AMPL [12] algebraic modeling language and the interior point solver LOQO [23] proved to be both convenient and powerful. In order to make the formulation of mathematical optimization problems generic and independent of both the actual solver used and the programming language it is written in, modeling languages were developed. AMPL provides interfaces to a large number of solvers, both commercial and free-for-research codes. One of the latter ones is LOQO which grew out of an interior point LP optimizer to a convex QP and very recently to a general NLP solver implementing an interior point approach. Although the code is currently still being perfected it proved to be very efficient for the solution of large-scale nonlinear problems in the benchmarks of [21]. It was thus chosen for the following computations; see also the comparison in Example 5.1 below. It should be remarked that LOQO implements an infeasible primal-dual path-following method. The KKT necessary conditions are essentially solved as a system of nonlinear equations with a Newton-like method. Therefore, it causes no problem if iterates are not feasible because it only means that residuals or right-hand sides corresponding to the equality constraints are not zero. At least asymptotically feasibility will be attained. Another feature that makes AMPL attractive and that was exploited is its automatic differentiation capability. Only functions for objective and constraints need to be provided.