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An additive eigenvalue problem of physics related to linear programming. (English) Zbl 0639.65033
Starting 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.
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
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##### References:
  Griffiths, R.B; Chou, W, Effective potentials, a new approach and new results for one-dimensional systems with competing length scales, Phys. rev. lett., 56, 1929-1931, (1986)  Chou, W; Griffiths, R.B, Ground states of one-dimensional systems using effective potentials, Phys. rev. B, 34, 6219-6234, (1986)  Duffin, R.J, Finding Griffiths additive eigenvalues by linear programming, () · Zbl 0259.90032  Aubry, S, J. phys. (Paris), 44, 147-157, (1983)  Moser, J, Recent developments in the theory of Hamiltonian systems, SIAM rev., 28, 459-485, (1986) · Zbl 0606.58022  Cuninghame-Green, R.A, Describing industrial processes with interference and approximating their steady-state behaviour, Oper. res. quart., 13, 95-100, (1962)  Cuninghame-Green, R.A, Minimax algebra, (1979), Springer Berlin · Zbl 0399.90052  Collatz, L, Functional analysis and numerical mathematics, (1966), Academic Press New York · Zbl 0221.65088  Karp, R.M, A characterization of the minimum cycle Mean in a digraph, Discrete math., 23, 309-311, (1978) · Zbl 0386.05032  Golitschek, M.V, Optimal cycles in doubly weighted graphs and approximation of bivariate functions by univariate ones, Numer. math., 39, 65-84, (1982) · Zbl 0541.65009  Leizarowitz, A, Infinite horizon systems with unbounded costs, Appl. math. optim., 13, 19-43, (1985) · Zbl 0591.93039
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