We present an O(√nL)-iteration homogeneous and self-dual linear programming (LP) algorithm. The algorithm possesses the following features: • It solves the linear programming problem without any regularity assumption concerning the existence of optimal, feasible, or interior feasible solutions. • It can start at any positive primal-dual pair, feasible or infeasible, near the central ray of the positive orthant (cone), and it does not use any big M penalty parameter or lower bound. • Each iteration solves a system of linear equations whose dimension is almost the same as that solved in the standard (primal-dual) interior-point algorithms. • If the LP problem has a solution, the algorithm generates a sequence that approaches feasibility and optimality simultaneously; if the problem is infeasible or unbounded, the algorithm will correctly detect infeasibility for at least one of the primal and dual problems.

We describe several adaptive-step primal-dual interior point algorithms for linear programming. All have polynomial time complexity while some allow very long steps in favorable circumstances. We provide heuristic reasoning for expecting that the algorithms will perform much better in practice than guaranteed by the worst-case estimates, based on an analysis using a nonrigorous probabilistic assumption.

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