A new nonlinear dynamical system that leads to eigenvalues

Japan J. Indust. Appl. Math.

1994

Self Cite

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

Section: W Lax Pair For Karmarkar's Dynamical Systemmentioning

confidence: 99%

Section: W Lax Pair For Karmarkar's Dynamical Systemmentioning

confidence: 99%

“…Substituting (15) into (13), we see that if c U = cj -1 then fj(t) satisfy (16) Integrating (16) gives …”

Section: Xj(t) = C~j Dt 'mentioning

confidence: 99%

Section: Fixed Point Analysis Of Karmarkar's Dynamical Systemmentioning

confidence: 99%

See 2 more Smart Citations

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

Japan J. Indust. Appl. Math.

1994

Self Cite

“…(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.…”

Section: §L Introductionmentioning

confidence: 99%

Neurodynamics and nonlinear integrable systems of Lax type

Japan J. Indust. Appl. Math.

1994

Self Cite

It is shown that an averaged learning equation in neurodynamics is an integrable gradient system having a Lax pair representation.In this decade, applied analysis based upon the theory of nonlinear integrable systems has been gradually developed. Though the theory has its origin mostly in classical mechanics, this methodology supplies a new mathematical tool which can be widely applied to: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. The motivation of applied analysis of integrable systems is explained in a review article [20]. Although it looks accidental, the dynamical theoretic approaches to the variety of interesting problems turn out to have properties in common. Namely, the dynamical systems which have emerged are integrable systems of Lax type. Rich information about integrable systems successfully helps us to solve the problems being considered as well as the dynamical systems themselves. Solvability of the original nonlinear problems may be related to integrability of the resulting dynamical systems at a deep level.In the following sections, we consider (vii) neurodynamics as a new topic of applied analysis of nonlinear integrable systems. We here take the position that we regard the problem of learning as a challenging problem in nonlinear dynamical systems. First we discuss a stochastic learning equation of Hebb type with a nonlinear *

On Explicitly Solvable Gradient Systems of Moser–Karmarkar Type

Journal of Mathematical Analysis and Applications

Faybusovich

1997

A generalization and a classification of explicitly solvable gradient systems which appear in linear programming and neurodynamics are studied, It is shown that nonlinear dynamical systems of Moser᎐Karmarkar type generically take both a Lax representation and a gradient form. A related optimization problem on a simplex is discussed. An integration procedure and a complete description of the invariant manifolds and the phase portrait of the dynamical systems of Moser᎐Karmarkar type are also investigated. ᮊ 1997 Academic Press