An active set strategy for solving optimization problems with up to 200,000,000 nonlinear constraints

Numerical test results are presented for solving smooth nonlinear programming problems with a large number of constraints, but a moderate number of variables. The active set method proceeds from a given bound for the maximum number of expected active constraints at an optimal solution, which must be less than the total number of constraints. A quadratic programming subproblem is generated with a reduced number of linear constraints from the so-called working set, which is internally changed from one iterate to the next. Only for active constraints, i.e., a certain subset of the working set, new gradient values must be computed. The line search is adapted to avoid too many active constraints which do not fit into the working set. The active set strategy is an extension of an algorithm described earlier by the author together with a rigorous convergence proof. Numerical results for some simple academic test problems show that nonlinear programs with up to 200,000,000 nonlinear constraints are efficiently solved on a standard PC.