Convergence of meta-controlled Boltzmann machine and its application for bilevel programming problem

Abstract

In this paper, meta controlled Boltzmann machine; the double-layered Boltzmann machine consisting of upper (Hopfield network) and lower (Boltzmann network) layers, is efficiently applied to solve mean-variance problem using mathematical programming with two objectives: the minimization of risk and the maximization of expected return. It is demonstrated that the proposed structural learning method has various advantages in a way such as an investment for a power system. As a result, it was shown that the structural learning can provide an alternative solution for decision makers to select the best solution from their respective point of view, as a numerical example shows. The simulation also showed that computational cost is significantly decreased compared with a conventional BM. The obtained results showed that the selection, investment expense rate to substations, and reduced computation time can be prolonged to increase cost savings.