An affine scaling interior trust-region method for LC1 minimization subject to bounds on variables

摘要

In this literature, we extend the classical affine scaling interior trust-region algorithm for smooth bounded-constrained nonlinear programming to the nonsmooth case where the objective function is only locally Lipschitzian. We propose and analyze a new affine scaling trust-region method in association with nonmonotonic interior backtracking line search technique for solving the linear equality constrained LC1 optimization subject to bounds on variables. At each iteration, the trust-region subproblem is defined by minimizing a quadratic function which is required both first- and second-order information of the objective function and a quadratic affine scaling matrix model subject only to an ellipsoidal constraint in a null space of the affine scaling linear equality constraints. The second-order derivative of the objective function is explicitly required to be Lipschitzian. Each iterate switches to a very general setting of computing trial backtracking steps generated by the general trust-region subproblem and then the line search technique projects the trial backtracking steps onto the strictly feasible set of the bounded constraints on variables. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases. Applications of the algorithm to various nonsmooth optimization problems are discussed.