### Abstract

The objective of this research is to realize structural learning within a Boltzmann machine (BM), which enables the effective solution of problems defined in terms of mixed integer quadratic programming. Simulation results show that computation time is up to one fifth faster than conventional BMs. The computational efficiency of the resulting double-layer BM is approximately expressed as the ratio n divided by N, where n denotes the number of selected units (neurons/nodes), and N the total number of units. The double-layer BM is applied to efficiently solve the mean-variance problem using mathematical programming with two objectives: the minimization of risk and the maximization of expected return. Finally, the effectiveness of our method is illustrated by way of a light emitting diodes (LED) signal retrofit example. The double-layer BM enables us to not only obtain a more effective selection of results, but also enhance effective decision making. The results also enable us to reduce the computational overhead, as well as to more easily understand the structure. In other words, decision makers are able to select the best solution given their respective points of view, by means of the alternative solution provided by the proposed method.

Original language | English |
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Title of host publication | Studies in Computational Intelligence |

Pages | 55-74 |

Number of pages | 20 |

Volume | 372 |

DOIs | |

Publication status | Published - 2011 |

Externally published | Yes |

### Publication series

Name | Studies in Computational Intelligence |
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Volume | 372 |

ISSN (Print) | 1860949X |

### Keywords

- Boltzmann machine (BM)
- double-layer BM
- Hopfield network
- mean-variance analysis
- quadratic programming
- Structural learning

### ASJC Scopus subject areas

- Artificial Intelligence

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## Cite this

*Studies in Computational Intelligence*(Vol. 372, pp. 55-74). (Studies in Computational Intelligence; Vol. 372). https://doi.org/10.1007/978-3-642-11739-8_3