Well-Posedness of Minimal Time Problems with Constant Dynamics in Banach Spaces

Colombo, Giovanni; Goncharov, Vladimir V.; Mordukhovich, Boris S.

doi:10.1007/s11228-010-0151-y

Cited by 18 publications

(9 citation statements)

References 14 publications

(31 reference statements)

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“…Furthermore, we give a representation formula for the Clarke (Fréchet or Mordukhovich) subdifferential ofû (·) atx in terms of the respective constructions for F , θ (·) and C. A similar result was obtained in [26] in the case θ ≡ 0 (see also [14]). …”

Section: Regularity Of the Value Functionsupporting

confidence: 61%

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Regularity of a Kind of Marginal Functions in Hilbert Spaces

Pereira

Goncharov

2014

Optimization in Science and Engineering

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and goncha@uevora.pt Dedicated to Panos M. Pardalos in honor of his 60th birthday Summary. We study well-posedness of some mathematical programming problem depending on a parameter that generalizes in a certain sense the metric projection onto a closed nonconvex set. We are interested in regularity of the set of minimizers as well as of the value function, which can be seen, on one hand, as the viscosity solution to a Hamilton-Jacobi equation, while, on the other, as the minimal time in some related optimal time control problem. The regularity includes both the Fréchet differentiability of the value function and the Hölder continuity of its (Fréchet) gradient.

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Section: Regularity Of the Value Functionsupporting

confidence: 61%

“…Recalling the characterization of the duality mapping through the subdifferential of the Minkowski functional (see (10)) we derive from (14) that the uniform rotundity of F w.r.t. U implies, in particular, the uniform continuity of the Fréchet gradient ∇ρ F 0 (·) on the set U .…”

Section: Preliminariesmentioning

confidence: 99%

Regularity of a Kind of Marginal Functions in Hilbert Spaces

Pereira

Goncharov

2014

Optimization in Science and Engineering

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“…with the sets of ε-normals N ε (·; Ω) defined for ε ≥ 0 by 7) where N ε (x; Ω) := ∅ ifx / ∈ Ω for convenience. When the set Ω is locally closed aroundx and the space X is Asplund, we can equivalently replace N ε (·; Ω) in (2.7) by the prenormal/Fréchet normal cone N (·; Ω) := N 0 (·; Ω).…”

Section: Tools Of Variational Analysismentioning

confidence: 99%

“…with the constant dynamicsẋ ∈ F described by a closed, bounded, and convex subset F = ∅ of a Banach space X and with the closed target set Ω = ∅ in X; these are our standing assumptions in this paper. We refer the reader to [7,8,9,15] and the bibliographies therein for various results on minimal time functions and their applications. When F is the closed unit ball IB of X, the minimal time function (1.1) becomes the standard distance function d(x; Ω) = inf x − ω ω ∈ Ω (1.2) generated by the norm · on X.…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

Applications of Variational Analysis to a Generalized Fermat-Torricelli Problem

Mordukhovich

Nam²

2010

J Optim Theory Appl

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In this paper we develop new applications of variational analysis and generalized differentiation to the following optimization problem and its specifications: given n closed subsets of a Banach space, find such a point for which the sum of its distances to these sets is minimal. This problem can be viewed as an extension of the celebrated Fermat-Torricelli problem: given three points on the plane, find another point such that the sum of its distances to the designated points is minimal. The generalized Fermat-Torricelli problem formulated and studied in this paper is of undoubted mathematical interest and is promising for various applications including those frequently arising in location science, optimal networks, etc. Based on advanced tools and recent results of variational analysis and generalized differentiation, we derive necessary as well as necessary and sufficient optimality conditions for the extended version of the Fermat-Torricelli problem under consideration, which allow us to completely solve it in some important settings. Furthermore, we develop and justify a numerical algorithm of the subgradient type to find optimal solutions in convex settings and provide its numerical implementations.

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“…Variational analysis and generalized differentiations of the minimal time function associated with a convex dynamics set, which contains the origin in its interior, in a Hilbert space was initially studied by Colombo and Wolenski in [8,9]. Since then, the minimal time function has been extensively studied by many researchers in various ways and for different purposes; see, e.g., [2,3,7,13,17,18,22,23,26,29,31]. In [17], He and Ng studied generalized differentiations of the minimal time function in Banach spaces.…”

Section: Introductionmentioning

confidence: 99%

The minimal time function associated with a collection of sets

Luong

2020

ESAIM: COCV

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We define the minimal time function associated with a collection of sets which is motivated by the optimal time problem for nonconvex constant dynamics. We first provide various basic properties of this new function: lower semicontinuity, principle of optimality, convexity, Lipschitz continuity, among others. We also compute and estimate proximal, Fr\'echet and limiting subdifferentials of the new function at points inside the target set as well as at points outside the target. An application to location problems is also given.

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Well-Posedness of Minimal Time Problems with Constant Dynamics in Banach Spaces

Cited by 18 publications

References 14 publications

Regularity of a Kind of Marginal Functions in Hilbert Spaces

Regularity of a Kind of Marginal Functions in Hilbert Spaces

Applications of Variational Analysis to a Generalized Fermat-Torricelli Problem

The minimal time function associated with a collection of sets

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