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

Abstract

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 /1-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 .4-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 geodesies of a Riemannian geometry isometric to Euclidean geometry. Each /1-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 0 in Legendre transform coordinates, called the central P-trajectory, is part of a straight line, and is contained in the ^-trajectory through 0 , called the central A-trajectory.Each /"-trajectory is part of a real algebraic curve. The central ^-trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form (c, x) = p . It is also the set of minima of a parametrized family of logarithmic barrier functions. Power-series expansions are derived for the central .4-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.-trajectories are defined for a general linear program. Using this definition, it is shown that the limit point Xoo of a central ^-trajectory on the boundary of the feasible solution polytope P is the center of the unique face of P containing Xoo in its relative interior.Received by the editors October 8, 1986 and, in revised form, June 9, 1987 and March 25, 1988. 1980 Mathematics Subject Classification (1985. Primary 90C05; Secondary 52A40, 34A34.The first author was partially supported by ONR contract N00014-87-K0214. 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 ...

Hessian Riemannian Gradient Flows in Convex Programming

“…3.1). The use of Riemannian methods in optimization has increased recently: in relation with Karmarkar algorithm and linear programming see Karmarkar [29], Bayer-Lagarias [5]; for continuous-time models of proximal type algorithms and related topics see Iusem-Svaiter-Da Cruz [27], Bolte-Teboulle [6]. For a systematic dynamical system approach to constrained optimization based on double bracket flows, see Brockett [8,9], the monograph of Helmke-Moore [22] and the references therein.…”

Polynomial-time algorithms for linear programming based only on primal scaling and projected gradients of a potential function

“…The algorithm works by tracing the central trajectory (see Megiddo [9] and Bayer and Lagarias [3] ) with Newton steps. Other central trajectory pathfollowing algorithms with the O(iii L) iteration count have been developed since then, see…”

On the convergence of primal-dual interior-point methods with wide neighborhoods

“…Analyses of the central path have been done by several authors (see, for instance, Sonnevend [15], Megiddo [10], and Bayer and Lagarias [1]). Our objective is to approximately follow this path to an optimal solution.…”