Hilbert Geometry Research Articles

The nonlinear geometry of linear programming. III. Projective Legendre transform coordinates and Hilbert geometry

This paper studies projective scaling trajectories, which are the trajectories obtained by following the infinitesimal version of Karmarkar’s linear programming algorithm. A nonlinear change of variables, projective Legendre transform coordinates, is introduced to study these trajectories. The projective Legendre transform mapping has a coordinate-free geometric interpretation in terms of the notion of "centering by a projective transformation." Let H {\mathsf {H}} be a set of linear programming constraints { ⟨ a j , x ⟩ ≥ b j : 1 ≤ j ≤ m } \{ \langle {{\mathbf {a}}_j},{\mathbf {x}}\rangle \geq {b_j}:1 \leq j \leq m\} on R n {{\mathbf {R}}^n} such that its polytope of feasible solutions P H {P_{\mathsf {H}}} is bounded and contains 0 {\mathbf {0}} in its interior. The projective Legendre transform mapping ψ H {\psi _{\mathsf {H}}} is given by \[ ψ H ( x ) = ϕ H ( x ) m + ⟨ ϕ H ( x ) , x ⟩ , w h e r e ϕ H ( x ) = − ∑ j = 1 m a j ⟨ a j , x ⟩ − b j . {\psi _{\mathsf {H}}}({\mathbf {x}}) = \frac {{{\phi _{\mathsf {H}}}({\mathbf {x}})}} {{m + \langle {\phi _{\mathsf {H}}}({\mathbf {x}}),{\mathbf {x}}\rangle }},\quad {\mathsf {where}}{\phi _{\mathsf {H}}}({\mathbf {x}}) = - \sum \limits _{j = 1}^m {\frac {{{{\mathbf {a}}_j}}} {{\langle {{\mathbf {a}}_j},{\mathbf {x}}\rangle - {b_j}}}.} \] Here ϕ H ( x ) {\phi _{\mathsf {H}}}(x) is the Legendre transform coordinate mapping introduced in part II. ψ H ( x ) {\psi _{\mathsf {H}}}({\mathbf {x}}) is a one-to-one and onto mapping of the interior of the feasible solution polytope Int ⁡ ( P H ) \operatorname {Int} ({P_{\mathsf {H}}}) to the interior of its polar polytope Int ⁡ ( P H ∘ ) \operatorname {Int} (P_{\mathsf {H}}^\circ ) . The set of projective scaling trajectories with objective function ⟨ c , x ⟩ − c 0 \langle {\mathbf {c}},{\mathbf {x}}\rangle - {c_0} are mapped under ψ H {\psi _{\mathsf {H}}} to the set of straight line segments in Int ⁡ ( P H ∘ ) \operatorname {Int} (P_{\mathsf {H}} ^\circ ) passing through the boundary point − c / c 0 - {\mathbf {c}}/{c_0} of P H ∘ P_{\mathsf {H}} ^\circ . As a consequence the projective scaling trajectories (for all objective functions) can be interpreted as the complete set of "geodesics" (actually distinguished chords) of a projectively invariant metric geometry on Int ⁡ ( P H ) \operatorname {Int} ({P_{\mathsf {H}}}) , which is isometric to Hilbert geometry on the interior of the polar polytope P H ∘ P_{\mathsf {H}}^\circ .

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

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 Ptrajectories. 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-trajectones. 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 (c, x) with respect to the Riemannian geometry ds2 Z?= dx, dx, /x2 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 Hilbert 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. Received by the editors July 28, 1986 and, in revised form, September 28, 1987 and March 21, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 90C05; Secondary 52A40, 34A34. Research of the first author partially supported by ONR contract N00014-87-K0214. (D 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page

Curvature in Hilbert geometries. II

Curvature in Hilbert geometries. II

The ptolemaic inequality in Hilbert geometries

The ptolemaic inequality in Hilbert geometries

Stochastic Processes As Curves in Hilbert Space

Regular complex-valued random processes $x(t)$ with finite moments of second order are studied by methods of Hilbert space geometry. A representation formula (4) is given for the process $x(t)$ in terms of “past and present innovations”. The number N is called the complete spectral multiplicity of the process $x(t)$ and is the smallest number for which such a representation exists. It is shown that the multiplicity of $x(t)$ is uniquely determined by the corresponding correlation function and that one can always find a harmonizing process $x(t)$ which has the multiplicity prescribed in advance.