Continuous Selections of Solutions for Locally Lipschitzian Equations

Arutyunov, A. V.; Izmailov, A. F.; Zhukovskiy, S. E.

doi:10.1007/s10957-020-01674-1

Cited by 12 publications

(2 citation statements)

References 17 publications

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“…Following a recent approach to the theory of implicit and inverse functions and to the theory of fixed and coincidence points (see [2][3][4]), the existence of problem solutions will be achieved by minimizing proper functions, which are shown to satisfy a Caristi-type condition. Recall that a function ϕ : X −→ [0, +∞] is said to satisfy a Caristi-type condition if there exists a constant κ > 0 such that…”

Section: Preliminariesmentioning

confidence: 99%

Conditions for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

Uderzo

2022

J Glob Optim

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In present paper, an analysis of the stability behaviour of ideal efficient solutions to parametric vector optimization problems is conducted. A sufficient condition for the existence of ideal efficient solutions to locally perturbed problems and their nearness to a given reference value is provided by refining recent results on the stability theory of parameterized set-valued inclusions. More precisely, the Lipschitz lower semicontinuity property of the solution mapping is established, with an estimate of the related modulus. A notable consequence of this fact is the calmness behaviour of the ideal value mapping associated to the parametric class of vector optimization problems. Within such an analysis, a refinement of a recent existence result, specific for ideal efficient solutions to unperturbed problem and enhanced by related error bounds, is discussed. Some connections with the concept of robustness in multi-objective optimization are also sketched.

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Section: Preliminariesmentioning

confidence: 99%

Conditions for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

Uderzo

2022

J Glob Optim

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“…Equation (1) is of particular interest when X and Y are not linearly topologically isomorphic spaces, for example, when , and In this case, global implicit and inverse function theorems were obtained in [3] assuming that the mapping f is smooth in x. A global inverse function theorem for locally Lipschitz mappings was derived in [4].…”

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confidence: 99%

On Global Solvability of Nonlinear Equations with Parameters

Arutyunov

Zhukovskiy

2021

Dokl. Math.

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We consider smooth mappings acting from one Banach space to another and depending on a parameter belonging to a topological space. Under various regularity assumptions, sufficient conditions for the existence of global and semilocal continuous inverse and implicit functions are obtained. We consider applications of these results to the problem of continuous extension of implicit functions and to the problem of coincidence points of smooth and continuous compact mappings.

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On implicit function theorem for locally Lipschitz equations

Arutyunov

Zhukovskiy

2022

Math. Program.

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Continuous Selections of Solutions for Locally Lipschitzian Equations

Cited by 12 publications

References 17 publications

Conditions for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

Conditions for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

On Global Solvability of Nonlinear Equations with Parameters

On implicit function theorem for locally Lipschitz equations

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