Application of the modified barrier method in large-scale quadratic programming problems

The application of the penalty/modified barrier function method (PE/MBF) is presented for the solution of large-scale positive-semidefinite quadratic programming problems (QP). A review of the recent literature on QP methods is presented and the choice of the PE/MBF method for QP problems is justified by previous experience in very large-scale bound-constrained problems. The proposed algorithm performs two types of iterations: an outer iteration in which the Lagrange multipliers of the bounds are adjusted, and an inner iteration for the solution of an equality constrained subproblem. The inner iteration solves a modified problem, containing penalty/modified barrier terms for the bounds in the objective, and is subject to equality constraints only. The equality constraints are handled directly via the use of additional Lagrange multipliers during the inner iteration and thus, instead of an unconstrained problem, the inner iteration solves a modified equality constrained problem. Any inequality constraints, other than bounds, are formulated as equalities via the use of slack variables. Computational results show this method to be promising, and motivate further investigation for the general case of nonlinear programming problems.