Article ID: | iaor19971088 |
Country: | Netherlands |
Volume: | 70 |
Issue: | 3 |
Start Page Number: | 251 |
End Page Number: | 277 |
Publication Date: | Nov 1995 |
Journal: | Mathematical Programming (Series A) |
Authors: | Gill Philip E., Murray Walter, Saunders Michael A., Poncelen Dulce B. |
Keywords: | duality |
Many interior-point methods for linear programming are based on the properties of the logarithmic barrier function. After a preliminary discussion of the convergence of the (primal) projected Netwon barrier method, three types of barrier method are analyzed. These methods may be categorized as primal, dual and primal-dual, and may be derived from the application of Newton’s method to different variants of the same system of nonlinear equations. A fourth variant of the same equations leads to a new primal-dual method. In each of the methods discussed, convergence is demonstrated without the need for a nondegeneracy assumption or a transformation that makes the provision of a feasible point trivial. In particular, convergence is established for a primal-dual algorithm that allows a different step in the primal and dual variables and does not require primal and dual feasibility. Finally, a new method for treating free variables is proposed.