A tolerant algorithm for linearly constrained optimization calculations

Two extreme techniques when choosing a search direction in a linearly constrained optimization calculation are to take account of all the constraints or to use an active set method that satisfies selected constraints as equations, the remaining constraints being ignored. Preference is for an intermediate method that treats all inequality constraints with ‘small’ residuals as inequalities with zero right hand sides and that disregards the other inequality conditions. Thus the step along the search direction is not restricted by any constraints with small residuals, which can help efficiency greatly, particularly when some constraints are nearly degenerate. The implementation, convergence properties and performance of an algorithm that employs this idea are studied. The implementation considerations include the choice and automatic adjustment of the tolerance that defines the ‘small’ residuals, the calculation of the search directions, and the updating of second derivative approximations. The main convergence theorem imposes no conditions on the constraints except for boundedness of the feasible region. The numerical results indicate that a Fortran implementation of the present algorithm is much more reliable than the software that was tested by Hock and Schittkowski. Therefore the algorithm seems to be very suitable for general use, and it is particularly appropriate for semi-infinite programming calculations that have many linear constraints that come from discretizations of continua.