An affine-scaling interior-point CBB method for box-constrained optimization
An affine-scaling interior-point algorithm for box-constrained nonlinear optimization problems is developed. The k-th direction in the algorithm is a quasi-Newton direction for the first order optimality (KKT) conditions for the problem, where in regard to large systems as they occur, for example, in the context of image reconstruction for positron emission tomography (PET), the current Hessian of the objective function is replaced by a suitable multiple of the unit matrix. The respective factor for the unit matrix is computed by a cyclic modification of a rule which is used in the Barzilai-Borwein (BB) quasi-Newton method. In order to relate to further original features of BB type methods which typically do not reduce the value of the cost function in a monotonic way, the method is connected with a nonmonotone line search procedure. The authors prove global convergence of the method as well as local r-linear convergence to a local minimizer which is nondegenerate and satisfies the second order sufficiency optimality conditions. Numerical experiments demonstrate that the speed of convergence of the method is relatively independent of the conditioning of the problem. They moreover indicate that the new method is more efficient than the earlier developed conjugate gradient based active set algorithm ASA if a not too small error tolerance is chosen, but that it is inferior to the latter algorithm if many iterations are needed and the error tolerance is tiny respectively.
scientific article; zbMATH DE number 5547227