Back An Active Set Strategy for Solving Optimization Problems with up to 200,000,000 Nonlinear Constraints

K. Schittkowski: Applied Numerical Mathematics, Vol. 59, 2999-3007 (2009)

We propose a numerical algorithm 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 m_w for the maximum number of expected violated constraints, where m_w is a user-provided parameter less than the total number of constraints. A quadratic programming subproblem is generated with m_w linear constraints, the so-called working set, which are internally exchanged 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 takes the active constraints into account. Numerical results for some simple academic test problems show that nonlinear programs with up to 200,000,000 nonlinear constraints can be efficiently solved within a few minutes on a standard PC.

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