The nonlinear geometry of linear programming. I. Affine and projective scaling trajectories

Abstract:

This series of papers studies a geometric structure underlying Karmarkar’s projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure studied is the set of trajectories obtained by integrating this vector field, which we call $P$-trajectories. We also study a related vector field, the affine scaling vector field, and its associated trajectories, called $A$-trajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. Affine and projective scaling vector fields are each defined for linear programs of a special form, called strict standard form and canonical form, respectively. This paper derives basic properties of $P$-trajectories and $A$-trajectories. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for $P$-trajectories and $A$-trajectories. It shows that projective transformations map $P$-trajectories into $P$-trajectories. It presents Karmarkar’s interpretation of $A$-trajectories as steepest descent paths of the objective function $\left \langle {{\mathbf {c}},{\mathbf {x}}} \right \rangle$ with respect to the Riemannian geometry $d{s^2} = \sum \nolimits _{i = 1}^n {d{x_i}d{x_i}/x_i^2}$ restricted to the relative interior of the polytope of feasible solutions. $P$-trajectories of a canonical form linear program are radial projections of $A$-trajectories of an associated standard form linear program. As a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program: This algorithm is essentially Karmarkar’s algorithm. These trajectories are studied in subsequent papers by two nonlinear changes of variables called Legendre transform coordinates and projective Legendre transform coordinates, respectively. It will be shown that $P$-trajectories have an algebraic and a geometric interpretation. They are algebraic curves, and they are geodesics (actually distinguished chords) of a geometry isometric to a Hubert geometry on a polytope combinatorially dual to the polytope of feasible solutions. The $A$-trajectories of strict standard form linear programs have similar interpretations: They are algebraic curves, and are geodesics of a geometry isometric to Euclidean geometry.