In this paper we prove some inequalities for finite sums and infinite series with positive terms. As an application of these results we obtain some inequalities for entropies of discrete probability distributions.

In this paper we propose an easy-to-implement algorithm for solving general nonlinear optimization problems with nonlinear equality constraints. A nonmonotonic trust region strategy is suggested which does not require the merit function to reduce its value in every iteration. In order to deal with large problems, a reduced Hessian is used to replace a full Hessian matrix. To avoid solving quadratic trust region subproblems exactly which usually takes substantial computation, we only require an approximate solution which requires less computation. The calculation of correction steps, necessary from a theoretical view point to overcome the Maratos effect but which often brings in negative results in practice, is avoided in most cases by setting a criterion to judge its necessity. Global convergence and a local superlinear rate are then proved. This algorithm has a good performance.

In this paper, we discuss a general model for multiple criteria linear cost network flow problems. This model includes several classes of existing models in the operations research literature as special cases. Based on this model, a search algorithm for finding a feasible solution of the concurrent flow problem is suggested and illustrative numerical examples are given. This search algorithm is also extended to obtain a new algorithm for finding the efficient frontier of a multiple criteria linear program.