Boundary behavior of interior point algorithms in linear programming. (English) Zbl 0675.90050

Megiddo, Nimrod; Shub, Michael

Math. Oper. Res. 14, No. 1, 97-146 (1989).

The authors discuss the boundary behaviour of N. Karmarkar’s projection method [Combinatorica 4, 373-395 (1984; Zbl 0557.90065)] for linear programming and some of its variants [cf., E. R. Barnes, Math. Program. 36, 174-182 (1986; Zbl 0626.90052); R. J. Vanderbei, M. S. Meketon and B. A. Freedman, Algorithmica 1, 395-407 (1986; Zbl 0626.90056)] subsequently given by various authors. It is shown that the continuous trajectories of the vector fields induced by these algorithms extend continuously to the whole closed polyhedron. The authors give conditions under which a vector field gives rise to trajectories that visit the neighbourhoods of all the vertices of the Klee-Minty Cube. The behaviour of the trajectories induced by these algorithms near the vertices is investigated and it is shown that all the trajectories have a unique direction of convergence to the optimum. The unique direction of convergence is also established for the discrete version of the projection method and its variant.

Reviewer: R.N.Kaul

MSC:

90C05

Linear programming

Keywords:

interior point algorithms; boundary behaviour; projection method; continuous trajectories; vector fields

Citations:

Zbl 0557.90065; Zbl 0626.90052; Zbl 0626.90056