The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories

Abstract:

Karmarkar’s projective scaling algorithm for solving linear programming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem. This paper studies the set of trajectories obtained by integrating this vector field, called $P$-trajectories, as well as a related set of trajectories, called $A$-trajectories. The $A$-trajectories arise from another linear programming algorithm, the affine scaling algorithm. The affine and projective scaling vector fields are each defined for linear programs of a special form, called standard form and canonical form, respectively. These trajectories are studied using a nonlinear change of variables called Legendre transform coordinates, which is a projection of the gradient of a logarithmic barrier function. The Legendre transform coordinate mapping is given by rational functions, and its inverse mapping is algebraic. It depends only on the constraints of the linear program, and is a one-to-one mapping for canonical form linear programs. When the polytope of feasible solutions is bounded, there is a unique point mapping to zero, called the center. The $A$-trajectories of standard form linear programs are linearized by the Legendre transform coordinate mapping. When the polytope of feasible solutions is bounded, they are the complete set of geodesics of a Riemannian geometry isometric to Euclidean geometry. Each $A$-trajectory is part of a real algebraic curve. Each $P$-trajectory for a canonical form linear program lies in a plane in Legendre transform coordinates. The $P$-trajectory through ${\mathbf {0}}$ in Legendre transform coordinates, called the central $P$-trajectory, is part of a straight line, and is contained in the $A$-trajectory through ${\mathbf {0}}$, called the central $A$-trajectory. Each $P$-trajectory is part of a real algebraic curve. The central $A$-trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form $\langle {\mathbf {c}},{\mathbf {x}}\rangle = \mu$. It is also the set of minima of a parametrized family of logarithmic barrier functions. Power-series expansions are derived for the central $A$-trajectory, which is also the central $P$-trajectory. These power-series have a simple recursive form and are useful in developing "higher-order" analogues of Karmarkar’s algorithm. $A$-trajectories are defined for a general linear program. Using this definition, it is shown that the limit point ${{\mathbf {x}}_\infty }$ of a central $A$-trajectory on the boundary of the feasible solution polytope $P$ is the center of the unique face of $P$ containing ${{\mathbf {x}}_\infty }$ in its relative interior. The central trajectory of a combined primal-dual linear program has a simple set of polynomial relations determining it as an algebraic curve. These relations are a relaxed form of the complementary slackness conditions. This central trajectory algebraically projects onto the central trajectories of both the primal and dual linear programs, and this gives an algebraic correspondence between points on the positive parts of the central trajectories of the primal and dual linear programs. Two Lagrangian dynamical systems with simple Lagrangians are shown to have $A$-trajectories as ${\mathbf {\dot q}}$-trajectories. The Hamiltonian dynamical systems associated to these Lagrangian systems are completely integrable.