Neural network for partitionable variational inequalities

ABSTRACT

Often in recent times industries have asked mathematicians to determine their "true" utility functions directly from the available data about used resources and about the corresponding profits in order to optimize the latter with respect to the former. The possibility of determining the utility function directly from the data is very important because in this way the exact situation of the company is described. Moreover, the biggest companies divide their investments into several activities. The optimization of their utility function can lead to problems that involve separable or partitionable functions. Two-layered feed-forward neural networks are able to approximate any separable function while fitting the data mantaining the separable structure with the desired approximation error. Thus the theory of partitionable variational inequalities can be used in order to find the optimum of the utility function, subject to some constraints. The presence of the partitionable structure is important because it simplifies the resolution algorithms and makes them more efficient. Moreover, with stronger assumptions about the function, the above results can be generalized to a larger class of utility functions: one problem of dimension n can be split into n problems of dimension one.

INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.1 COMPUTATION BY ABSTRACT DEVICES

General Terms:
Design, Languages, Theory