Linear programming by minimizing distances

AbstractTo solve the linear program (LP): minimizecTl 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=−ccl+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.