Lax pair and fixed point analysis of Karmarkar’s projective scaling trajectory for linear programming

Abstract

A Lax pair representation of a nonlinear dynamical system is presented which emerges in linear programming as a continuous version of Karmarkar's projective scaling algorithm. Karmarkar's continuous trajectory is also shown to be a gradient system. An expression of solution in a rational form is found by integrating the associated dynamical system. Fixed points of the trajectory also studied.Key words: Karmarkar's projective sca/ing algorithm, Lax pair form, gradient system, Toda lattice w I n t r o d u c t i o n There has been considerable interest in studying nonlinear dynamical systems for solving linear programming problems through interior-point algorithms since Karmarkar's idea [10] turned such problems into a problem in nonlinear ODEs. We here consider three nonlinear dynamical systems in linear programming.An algorithm introduced by Dikin [6] in 1967 is nowadays called the aŸ scaling algorithm. It is pointed out in [10] that the associated dynamical system, a continuous version of the algorithm, is essentially a gradient system with respect to a generalized Poincar› metric. Bayer and Lagarias [2, 3] and Faybusovich [7] showed that the dynamical system can be interpreted a s a completely integrable Hamiltonian system. The equations of motion ave linearized by a Legendre transformation. The affine scaling algorithm is simpler than other interior-point algorithms, however, it is believed [13] that it is n o t a polynomial-time algorithm.Karmarkar [10], in 1984, proposed an epoch-making polynomial-time linear programming algorithm called Karmarkar's algorithm of the projective scaling algorithm. A geometric structure of the algorithm was also discussed in [2,3,11,12]. Ir was shown in [11] that the tangent vector of the associated continuous trajectory can be interpreted as that of a geodesic on the interior of the feasible solution polytope. An integrability of rector fields of interior-point algorithms was also discussed by Iri in [9].One of the most important notions in the recent developments in integrable dynamical systems is their representation as an isospectral deformation of a certain linear operator L(x(t)) depending on a formal indeterminate x(t). The representation automatically exhibits first integrals as invariants of L(x(t)). See, for example, [1, p.59]. The isospectral deformation serves to express the dynamical system in a Lax pair form

Geometry and Dynamics of a Quantum Search Algorithm for an Ordered Tuple of Multi-Qubits

“…Due to the account given below, however, geometric studies were not made in [19] either on the search sequence in the ESOT generated by the Grover-type search algorithm or on the reduced search sequence in the QIS: Instead of geometric studies on the search sequences, it is the gradient dynamical system associated with the negative von Neumann entropy as the potential that is discussed in [19] on inspired by a series works of Nakamura [20][21][22] on complete integrablity of algorithms arising in applied mathematics. The result on the gradient system in [19] has drawn the author's interest to publish [23,24] on the gradient systems on the QIS realizing the Karmarkar flow for linear programming and a Hebb-type learning equation for multivariate analysis, while geometric studies on the search sequences were left undone.…”

“…The quotient space M 1 (2 n , ℓ)/ ∼ is realized as P ℓ defined by (21), where the projection of M 1 (2 n , ℓ) to P ℓ is given by…”

Neurodynamics and nonlinear integrable systems of Lax type

“…(ET))~ The asymptotics (19) imply that the Hebbian learning equation (2) with (1) yields the first principal component of the input data set. Another proof of (19) was given in [21] for the case where the covariance matrix E[XX -r] is positive semide¡…”

“…Another proof of (19) was given in [21] for the case where the covariance matrix E[XX -r] is positive semide¡…”

“…(i) linear prediction problems [13], (ii) the geometry of linear systems [10,14], (iii) maximum likelihood estimation [4,15], (iv) matrix eigenvalue problems [7,16,24], (v) the information geometry [17,18], (vi) linear programming problems [5,19], and so on. Here (i), (ii) and (vi) are essentially nonlinear problems being due to certain quotient structures and boundary conditions.…”