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Alexander J. Zaslavski - One of the best experts on this subject based on the ideXlab platform.

  • Stability Results for Bolza Problems
    Discrete-Time Optimal Control and Games on Large Intervals, 2017
    Co-Authors: Alexander J. Zaslavski
    Abstract:

    In this chapter we study the structure of solutions of a discrete-time control system with a compact metric space of states X. This control system is described by a Nonempty Closed Set Ω ⊂ X × X which determines a class of admissible trajectories (programs) and by a a pair of objective functions which determines an optimality criterion. We show that the turnpike phenomenon and the structure of solutions on finite intervals in the regions close to the endpoints are stable under small perturbations of the objective functions and the Set Ω.

  • Optimal Control Problems with Singleton Turnpikes
    Stability of the Turnpike Phenomenon in Discrete-Time Optimal Control Problems, 2014
    Co-Authors: Alexander J. Zaslavski
    Abstract:

    In this chapter we study the structure of solutions of a discrete-time control system with a compact metric space of states X which arises in economic dynamics. This control system is described by a Nonempty Closed Set \(\Omega \subSet X \times X\) which determines a class of admissible trajectories (programs) and by a bounded upper semicontinuous objective function \(v:X\times X \to R^1\) which determines an optimality criterion. We show the stability of the turnpike phenomenon under small perturbations of the objective function v and the Set Ω.

  • Optimal Control Problems with Discounting
    Stability of the Turnpike Phenomenon in Discrete-Time Optimal Control Problems, 2014
    Co-Authors: Alexander J. Zaslavski
    Abstract:

    In this chapter we continue our study of the structure of approximate solutions of the discrete-time optimal control problems with a compact metric space of states X and with a singleton turnpike. These problems are described by a Nonempty Closed Set \(\Omega \subSet X \times X\) which determines a class of admissible trajectories (programs) and by a bounded upper semicontinuous objective function \(v:X\times X \to R^1\) which determines an optimality criterion. We show the stability of the turnpike phenomenon under small perturbations of the objective function v and the Set Ω in the case with discounting. The results of the chapter generalize the results obtained in [54] for the discounting case with a perturbation only on the objective function.

  • Turnpike Properties of Discrete-Time Problems
    Turnpike Phenomenon and Infinite Horizon Optimal Control, 2014
    Co-Authors: Alexander J. Zaslavski
    Abstract:

    In this chapter we study the structure of approximate solutions of an autonomous discrete-time control system with a compact metric space of states X. This control system is described by a bounded upper semicontinuous function \(v: X \times X \rightarrow R^{1}\) which determines an optimality criterion and by a Nonempty Closed Set Ω ⊂ X × X which determines a class of admissible trajectories (programs). We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals. When X is a compact convex subSet of a finite-dimensional Euclidean space, the Set Ω is convex, and the function v is strictly concave we obtain a full description of the structure of approximate solutions.

  • Stability of a Turnpike Phenomenon for a Discrete-Time Optimal Control System
    Journal of Optimization Theory and Applications, 2010
    Co-Authors: Alexander J. Zaslavski
    Abstract:

    We study the structure of solutions of a discrete-time control system with a compact metric space of states X which arises in economic dynamics. This control system is described by a Nonempty Closed Set ΩźX×X which determines a class of admissible trajectories (programs) and by a bounded upper semicontinuous objective function v:ΩźR1 which determines an optimality criterion. We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals. In the present paper, we show that these turnpike properties are stable under perturbations of the objective function v.

Nath Triloki - One of the best experts on this subject based on the ideXlab platform.

  • Differentiability of Distance Function and The Proximinal Condition implying Convexity
    2020
    Co-Authors: Nath Triloki
    Abstract:

    We establish a necessary and sufficient condition for the differentiability of the distance function generated by a Nonempty Closed Set K in a real normed linear space X under a proximinality condition on K. We do not assume the uniform differentiability constraints on the norm of the space as in Giles [16]. Hence, our result advances that of Giles [16]. We prove that the proximinal condition of Giles [16] is true for almost suns. The proximinal condition ensures convexity of an almost sun in some class of strongly smooth spaces under a differentiability condition of the distance function. A necessary and sufficient condition is obtained for the convexity of Chebyshev Sets in Banach spaces with rotund dual.Comment: 15 pages, to appear in The journal of Analysi

Triloki Nath - One of the best experts on this subject based on the ideXlab platform.

  • Differentiability of distance function and the proximinal condition implying convexity
    The Journal of Analysis, 2020
    Co-Authors: Triloki Nath
    Abstract:

    We establish a necessary and sufficient condition for the differentiability of the distance function generated by a Nonempty Closed Set K in a real normed linear space X under a proximinality condition on K . We do not assume the uniform differentiability constraints on the norm of the space as in Giles (Proc Am Math Soc 104: 458–464, 1988). Hence, our result advances that of Giles (Proc Am Math Soc 104: 458–464, 1988). We prove that the proximinal condition of Giles (Proc Am Math Soc 104: 458–464, 1988) is true for almost suns. The proximinal condition ensures convexity of an almost sun in some class of strongly smooth spaces under a differentiability condition of the distance function. A necessary and sufficient condition is obtained for the convexity of Chebyshev Sets in Banach spaces with rotund dual.

Teturo Kamae - One of the best experts on this subject based on the ideXlab platform.

  • Infinitesimal geometry and superstationary factors of dynamical systems
    Topology and its Applications, 2013
    Co-Authors: Teturo Kamae
    Abstract:

    Abstract Let A be a finite Set with # A ⩾ 2 and N = { 0 , 1 , 2 , … } . A Nonempty Closed Set Θ ⊂ A N is called a superstationary Set if for any infinite Set { N 0 N 1 ⋯ } ⊂ N , we have { ω ( N 0 ) ω ( N 1 ) ⋯ ; ω ∈ Θ } = Θ . That is, Θ remains invariant for any selective observations, say at N 0 N 1 ⋯  . More generally, let Σ be any infinite Set and Ω ⊂ A Σ be a Nonempty Set. Let χ be a nonprincipal ultrafilter on Σ and let Ω [ χ ∞ ] be the projective limit of Ω [ χ k ] , where Ω [ χ k ] is the value at the product ultrafilter χ k of the natural extension of the mapping S ↦ Ω [ S ] from S = ( s 1 , … , s k ) ∈ Σ k to a subSet of A k given by Ω [ S ] = { ω ( s 1 ) ⋯ ω ( s k ) ; ω ∈ Ω } . We prove that Ω [ χ ∞ ] is a superstationary Set, which we call a superstationary factor of Ω at χ. In the study of dynamical systems with time parameter, quantities which are sensitive to the time scaling, such as entropy have been of exclusive interest. On the contrary, superstationary factors represent properties depending only on time order, and not on the spacing of time. These properties are shown to reflect local aspects of the geometry behind it. We also discuss a stronger and constructive version of superstationary factors.

  • Behavior of various complexity functions
    Theoretical Computer Science, 2012
    Co-Authors: Teturo Kamae
    Abstract:

    For a Nonempty Closed Set @[email protected]?A^N with [email protected]?#A

  • Super-stationary Set, subword problem and the complexity
    Discrete Mathematics, 2009
    Co-Authors: Teturo Kamae, Hui Rao, Bo Tan, Yumei Xue
    Abstract:

    Let Ω⊂{0,1}NΩ⊂{0,1}N be a Nonempty Closed Set with N={0,1,2,…}N={0,1,2,…}. For N={N0

Hieu, Vu Trung - One of the best experts on this subject based on the ideXlab platform.

  • A regularity condition in polynomial optimization
    2020
    Co-Authors: Hieu, Vu Trung
    Abstract:

    Let $f$ be a polynomial of degree $d$ and $K$ be a Nonempty Closed Set in $\R^n$. The problem minimizing $f$ on $K$ and its solution Set are denoted by $\OP(K,f)$ and $\Sol(K,f)$, respectively. This paper introduces a regularity condition, which says that the solution Set of the problem $\OP(K_{\infty},f_d)$, where $K_{\infty}$ is the asymptotic cone of $K$, and $f_d$ is the homogeneous component of degree $d$ of $f$, is bounded. Under this condition, a Frank-Wolfe type theorem is obtained, i.e. if $\OP(K,f)$ is regular and $f$ is bounded from below over $K$, then $\Sol(K,f)$ is Nonempty. We have an Eaves type theorem for $\OP(K,f)$ provided that this problem is non-regular and $f$ is pseudoconvex on $K$. Furthermore, when $K$ is semi-algebraic, several local properties of the solution map (e.g., local boundedness, upper semicontinuity, and local upper-H\"{o}lder stability) and the optimal value function (e.g., local Lipschitz continuity, semismoothness) of polynomial optimization problems are valid. The genericity of the regularity condition is discussed at the end.Comment: 17 page