Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems

Brockett, Roger W.

doi:10.1109/cdc.1988.194420

Cited by 187 publications

(223 citation statements)

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“…Let us recall that the number of stable fixed points of the Toda flow is 1 ( [7]). The unstable fixed points do not appear in the flow of Brockett's equation (1) ( [4]) and its generalizations ( [10]). …”

Section: M(s) = G-l(s)l~dig(s) 9 Su(n)mentioning

confidence: 99%

A new nonlinear dynamical system that leads to eigenvalues

Nakamura

1992

Japan J. Indust. Appl. Math.

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Section: M(s) = G-l(s)l~dig(s) 9 Su(n)mentioning

confidence: 99%

A new nonlinear dynamical system that leads to eigenvalues

Nakamura

1992

Japan J. Indust. Appl. Math.

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“…Brockett [5], in 1988, introduced a double-bracket Lax pair equation with quadratic nonlinearity which may solve a class of linear programming problems. The trajectory generically converges to: local minima of the objective function.…”

Section: Dl(x(t))_ [B(x(t)) L(x(t))]mentioning

confidence: 99%

“…The existence of Lax representation of double-bracket form plays an essential role in [16] 9 The equations (3) and (5) have cubic nonlinearity. It would be interesting to compare them to the double-bracket equations having quadratic nonlinearity in [5,15]. The malo J is an analogue of the momentum map in classical mechanics since J(K) is invariant under a torus action on K.…”

Section: W Lax Pair For Karmarkar's Dynamical Systemmentioning

confidence: 99%

“…It would be interesting and suggestive to explain the analogy between the dynamicM system (2) and the finite nonperiodic Toda lattice, a typical example of a completely integrable dynamical system of the Lax type [14]. [5,15] having quadratic nonlinearity.…”

Section: V(x) -2 (Ejl~ ~J)='mentioning

confidence: 99%

“…Hence the total number of the latter fixed points is 2 n -1. This gives the number of fixed points of the dynamical system (5). Using a result in [16] we can prove that n-fixed points of them corresponding to m = 1 lie on the vertices of the simplex and ave stable.…”

Section: Fixed Point Analysis Of Karmarkar's Dynamical Systemmentioning

confidence: 99%

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

Nakamura

1994

Japan J. Indust. Appl. Math.

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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

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Editorial for the special issue on extremum seeking control: Theory and applications

Benosman

KrstićMiroslav

GuayMartin

et al. 2021

Adaptive Control & Signal

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Real-world systems have to operate in uncertain and changing environments, thus the necessity to design feedback controllers which can adapt to such uncertainties. Controllers which are designed to satisfy this goal are called adaptive controllers. We can classify adaptive control into three main subclasses, for example, Reference 1: Classical model-based adaptive control, which mainly uses models of the controlled system; learning-based adaptive control, which uses both model-based and data-driven techniques to design modular adaptive controllers; and data-driven adaptive control, which is based on the interaction of the controller with the system. In this special issue, we shall focus on a specific class of data-driven adaptive controllers, namely, extremum seekers.Recently, there have been a lot of efforts in this direction of control. One of the reasons behind this increasing interest is that, following the pioneering ESC analysis paper, 2 the field of ESC has reached a certain maturity, which has led to good analysis and understanding of the main properties of the available algorithms. The idea behind ESC is to design a closed-loop dynamical system, such that its flow converges to the optimum of a given static or dynamic cost function, for example, References 3-5. This type of optimization algorithms is also referred to as continuous optimization, for example, Reference 6. The advantages of such optimization approaches comparatively to classical optimization algorithms is the fact that extremum seekers do not require a closed-form expression of the cost function. Moreover, they do not require multiple evaluation of the cost function at each point to estimate the gradient (or higher order derivatives) of the cost function, since they include some internal states which converge to an estimate of the derivatives of the cost over time. Besides, their formulation as a closed-loop control problem allows us to use numerous tools from dynamical systems theory and control theory, for example, time-scale separation, averaging theory, Lyapunov stability theory, to perform rigorous analysis of their convergence and robustness properties w.r.t. initial conditions changes and other hyperparameters tuning.In this special issue, we have included 10 papers, which could be divided into two main types. The first type, more theoretical, presents new extremum seeking methods designed for different dynamical systems, from systems with delays, and multiagent systems, to distributed systems. The second type of papers deals with real-world applications of extremum seeking controllers to challenging systems, ranging from microalgae cultivation control to heating and ventilation systems. These papers are compiled in a special virtual issue of the journal at the journal homepage. To access all of the papers, please follow the following link (https://onlinelibrary.wiley.com/toc/10991115/2021/35/7).For instance, Reference 7 presents a novel ESC for batch-to-batch optimal control of unknown time-varying systems. The authors show that the ...

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Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems

Cited by 187 publications

References 5 publications

A new nonlinear dynamical system that leads to eigenvalues

A new nonlinear dynamical system that leads to eigenvalues

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

Editorial for the special issue on extremum seeking control: Theory and applications

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