An algorithm and associated strategy for solving polynomial systems within the optimization framework is presented. The algorithm and strategy are named, respectively, the penetrating gradient algorithm and the deepest descent strategy. The most prominent feature of penetrating gradient algorithm, after which it was named, is its ability to “see and penetrate through” the obstacles in error spa...

To reduce the difficulty and complexity in computing the projection from a real Hilbert space onto a nonempty closed convex subset, researchers have provided a hybrid steepest-descent method for solving VI(F,K) and a subsequent three-step relaxed version of this method. In a previous study, the latter was used to develop a modified and relaxed hybrid steepest-descent (MRHSD) method. However, ch...

We introduce a family of numerical algorithms for the solution of linear system in higher dimensions with the matrix and right hand side given and the solution sought in the tensor train format. The proposed methods are rank–adaptive and follow the alternating directions framework, but in contrast to ALS methods, in each iteration a tensor subspace is enlarged by a set of vectors chosen similar...

This paper investigates the dynamics of batch learning of multilayer neural networks in the asymptotic case where the number of training data is much larger than the number of parameters. First, we present experimental results on the behavior in the steepest descent learning of multilayer perceptrons and three-layer linear neural networks. We see in these results that strong overtraining, which...