Bound Constrained Semismooth Equations Trust Region Methods

In this paper, we develop and analyze a class of affine scaling trust-region methods in as-sociation with nonmonotone interior backtracking line technique for solving bound-constrainedsemismooth equations.An important practical aspect of trust-region method is the shape of the trust region itself,which is determined by the norm used in its subproblem. The choice of norm may have a seriousimpact on the core computations, the nature of the Cauchy point and model minimizer, as well.In our experience, we choose l∞and l2 norms in our trust-region methods. The l∞-norm trust-region is easy to use, as a point may be checked component by component to see if it lies inthe region. So, in the l∞-norm trust-region problem, we propose a way of computing trial stepsby a semismooth Newton-like method that is augmented by a projection onto the feasible set.Under a Dennis-More′-type condition, we prove that, close to a BD-regular solution, the trust-region algorithm turns into this projected Newton method, which is shown to local q-superlinearconverge. The l2 norm is not as easy to apply, but has a strong theoretical advantage in that thereare provably efficient methods for minimizing q within an l2 trust region. So, in the sense of l2norm, we define the trust region subproblem by minimizing a squared Eudidean norm of linearmodel with a new affine matrix called minimum-scaling. Under a reasonable assumption of thisnew affine-scaling matrix, we stress that the minimum-scaling has some additional properties thatallows us to prove stronger global convergence results than for the Coleman-Li-scaling.Both line search and trust region algorithm are well-accepted methods in the optimizationto assure global convergence. In this paper, we consider how to combine the nonmonotone linesearch with trust region strategy and use them to ensure the global convergence and speed up theconvergence progress in the contours of objective function with large curvature.