Deriving potential functions via a symmetry principle for nonlinear equations

Parameterized homotopy equations that define paths through the interior of the feasible region of a linear program are reformulated as gradient mappings, and a standard symmetry principle for nonlinear equations is then used to derive associated potential functions. These functions are the Lagrangians of weighted logarithmic barrier problems. Primal, dual and self-dual cases are considered, with emphasis on the situation when starting points are infeasible.