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An exterior plane algorithm for linear programming
  • G Sampath
G Sampath
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An algorithm that solves a linear program by using planes exterior to the feasible region is described. Given a linear program in standard form, a constraint derived from the given objective function is added to the constraint set, and the objective function is removed. The added constraint is designed to make the program infeasible. It is then incrementally and unidirectionally moved toward the solution vertex in cycles, which causes the constraint set to change each time. A cycle consists of two steps: moving the added constraint and testing the updated constraint set for feasibility. Ultimately the moving constraint reaches the solution vertex and the program becomes feasible, this signals both the solution of the program and termination of the algorithm. Because of unidirectionality, the algorithm is guaranteed to terminate. If fixed point arithmetic with B bit numbers is used and feasibility testing complexity with N variables is O(f(N)), then the worst-case complexity of the incremental exterior plane algorithm is 2^B O(f(N)). f(N) is O(N^3) with the simplex method and O(N^3.5) with many interior point methods. The algorithm can be substantially improved by using binary search, the resulting complexity is O(B f(N)). The algorithm is compared qualitatively with three types of LP algorithms: simplex, ellipsoid, and interior point.