Abstract
To solve the linear program (LP): minimizec T l subject toA l+b≥0, for ann×d-matrixA, ann-vectorb and ad-vectorc, the positive orthantS and the planeE(t) are defined by S={(x1,x)εℝn+1 ¦(x1,x)⩾0}, E(t)={(x1,x)εℝn+1¦x1=−cc l+t, x=Al+b}.
First a geometric algorithm is given to determine d(E(t),S) for fixedt, where d(·,·) denotes euclidean distance. This algorithm is used to construct a second algorithm to find the minimalt with E(t) ∩S ≠ ∅, and thus solve LP. It is shown that all algorithms are finite.
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Betke, U., Henk, M. Linear programming by minimizing distances. ZOR - Methods and Models of Operations Research 35, 299–307 (1991). https://doi.org/10.1007/BF01417518
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DOI: https://doi.org/10.1007/BF01417518