Solving Karush--Kuhn--Tucker Systems via the trust region and the conjugate gradient methods

Abstract

A popular approach to solving the Karush-Kuhn-Tucker (KKT) system, mainly arising from the variational inequality problem, is to reformulate it as a constrained minimization problem with simple bounds. In this paper, we propose a trust region method for solving the reformulation problem with the trust region subproblems being solved by the truncated conjugate gradient (CG) method, which is cost effective. Other advantages of the proposed method over existing ones include the fact that a good approximated solution to the trust region subproblem can be found by the truncated CG method and is judged in a simple way; also, the working matrix in each iteration is H, instead of the condensed H[sup T]H, where H is a matrix element of the generalized Jacobian of the function used in the reformulation. As a matter of fact, the matrix used is of reduced dimension. We pay extra attention to ensure the success of the truncated CG method as well as the feasibility of the iterates with respect to the simple constraints. Another feature of the proposed method is that we allow the merit function value to be increased at some iterations to speed up the convergence. Global and superlinear/quadratic convergence is shown under standard assumptions. Numerical results are reported on a subset of problems from the MCPLIB collection [S. P. Dirkse and M. C. Ferris, Optim. Methods Softw., 5 (1995), pp. 319-345].