An O(√nL)-Iteration Homogeneous and Self-Dual Linear Programming Algorithm

Abstract

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.

Disciplined Convex Programming

“…Nesterov and Nemirovsky proved that a polynomial-time barrier method can be constructed for any CP that meets certain technical conditions [117]. Other authors have shown that problems which do not meet those conditions can be embedded into larger problems that doeffectively making barrier methods universal [169,102,1711.…”

Conic Optimization Software

“…From the optimal solution of the larger problem one can recover an optimal solution of the original problem, or detect infeasibility [7][8][9][10][11][12];…”

On implementation of a self-dual embedding method for convex programming

“…In 1994, Ye, Todd and Mizuno [21] introduced the so-called homogeneous self-dual embedding technique to solve linear programs. The main feature of this technique is to construct an artificial problem by embedding a primal-dual problem pair.…”