Tensors in Optimization

Abstract

Optimization problems can be formulated by using tensors and obtain in this way tensor field optimization problems introduced by (Rapcsak 1990. In differential geometry, theoretical physics and several applications of mathematics, the concept of tensor proved to be instrumental. In optimization theory, a new class of methods, called tensor methods, was introduced for solving systems of nonlinear equations (Schnabel and Frank, 1984) and for unconstrained optimization using second derivatives (Schnabel and Chowe, 1991). Tensor methods are of general purpose-methods intended especially for problems where the Jacobian matrix at the solution is singular or ill-conditioned. The description of a linear optimization problem in the tensor notation is proposed in order to study the integrability of vector and multivector fields associated with interior point methods by (Iri 1991). The most important feature of tensors is that their values do not change when they cause regular nonlinear coordinate transformations, and thus, this notion seems to be useful for the characterization of structural properties not depending on regular nonlinear coordinate transformations. This motivated the idea of using this notion within the framework of nonlinear optimization.