Interior Point Methods
It was mentioned earlier that the standard simplex method searches for an optimum to a linear program by moving along the surface of a convex polyhedron from one extreme point to an adjacent extreme point in a fashion such that the objective value is nondecreasing between successive basic feasible solutions. The simplex process then either terminates, in a finite number of steps, at a finite optimal solution or reveals that no such solution exists. In this chapter we shall examine a set of interior point methods for solving a linear program (e.g., projective and affine potential reduction, primal and dual affine scaling, and path following), so named because they converge to an optimal point (if one exists) by passing through the interior of a convex polytope in a fashion such that, at each iteration, a weighted least squares problem is solved.
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