The Linear Programming Feasibility Problem
In Chap. 8 we introduced linear programming optimization problems. Then, in Chap. 9, we rephrased PCFP as one such problem. By doing so, we could apply an interior-point method to the latter and obtain an algorithm solving PCFP with a complexity bounded by a low-degree polynomial in n,m and Open image in new window . A question conspicuously left open is the solution of the optimization problem itself. The main theorem of Chap. 9 provides a key step of this solution but leaves at least two aspects untouched: the initial feasible point is assumed to be given and there is no hint as how to deduce, at some moment of the process, the optimizers and optimal value.
In this chapter we deal with the first of the two aspects above.