An additive eigenvalue problem of physics related to linear programming.

*(English)*Zbl 0639.65033Starting from a functional equation for the ground state energy per atom by R. B. Griffiths a discrete approximation of this equation by \(\min_{j=1,...,n}(K_{ij}+x_ j)=\lambda +x_ i,\) \(i=1,...,n\) is investigated. Here \(K_{ij}\) is taken to be an arbitrary real square matrix. \(\lambda\) is termed an additive eigenvalue and x is termed an additive eigenvector. This additive eigenvalue equation had previously arisen in an entirely different area-management science. A motivation problem was cost efficient scheduling of industrial processes.

Brouwers fixed point theorem is used to show that a solution exists, that the eigenvalue is unique, but possibly there is more than one associated eigenvector. It is then shown that this equation can be solved by two linear programs. The first program has maximum value \(\lambda\). Then the second linear program furnishes a corresponding eigenvector.

Brouwers fixed point theorem is used to show that a solution exists, that the eigenvalue is unique, but possibly there is more than one associated eigenvector. It is then shown that this equation can be solved by two linear programs. The first program has maximum value \(\lambda\). Then the second linear program furnishes a corresponding eigenvector.

Reviewer: J.Born

##### MSC:

65K05 | Numerical mathematical programming methods |

90B35 | Deterministic scheduling theory in operations research |

90C08 | Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.) |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

##### Keywords:

infinite elastic chain of atoms; periodic potential field; crystal lattice; ground state energy per atom; eigenvalue; eigenvector; cost efficient scheduling; Brouwers fixed point theorem; linear programs
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\textit{W. Chou} and \textit{R. J. Duffin}, Adv. Appl. Math. 8, 486--498 (1987; Zbl 0639.65033)

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