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Directional metric pseudo subregularity of set-valued mappings: a general model

Metric regularity relative to a cone

Metric Regularity of the Sum of Multifunctions and Applications

  • Authors: Van Ngai, Huynh , Tron, Nguyen Tron , Thera, Michel
  • Date: 2014
  • Type: Text , Journal article
  • Relation: Journal of Optimization Theory and Applications Vol. 160, no. 2 (2014), p. 355-390
  • Relation: http://purl.org/au-research/grants/arc/DP110102011
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  • Description: The metric regularity of multifunctions plays a crucial role in modern variational analysis and optimization. This property is a key to study the stability of solutions of generalized equations. Many practical problems lead to generalized equations associated to the sum of multifunctions. This paper is devoted to study the metric regularity of the sum of multifunctions. As the sum of closed multifunctions is not necessarily closed, almost all known results in the literature on the metric regularity for one multifunction (which is assumed usually to be closed) fail to imply regularity properties of the sum of multifunctions. To avoid this difficulty, we use an approach based on the metric regularity of so-called epigraphical multifunctions and the theory of error bounds to study the metric regularity of the sum of two multifunctions, as well as some related important properties of variational systems. Firstly, we establish the metric regularity of the sum of a regular multifunction and a pseudo-Lipschitz multifunction with a suitable Lipschitz modulus. These results subsume some recent results by Durea and Strugariu. Secondly, we derive coderivative characterizations of the metric regularity of epigraphical multifunctions associated with the sum of multifunctions. Applications to the study of the behavior of solutions of variational systems are reported. © 2013 Springer Science+Business Media New York.

On optimal control of a sweeping process coupled with an ordinary differential equation

  • Authors: Adam, Lukas , Outrata, Jiri
  • Date: 2014
  • Type: Text , Journal article
  • Relation: Discrete and Continuous Dynamical Systems - Series B Vol. 19, no. 9 (November 2014 2014), p. 2709-2738
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  • Description: We study a special case of an optimal control problem governed by a differential equation and a differential rate{independent variational inequality, both with given initial conditions. Under certain conditions, the variational inequality can be reformulated as a differential inclusion with discontinuous right-hand side. This inclusion is known as sweeping process. We perform a discretization scheme and prove the convergence of optimal solutions of the discretized problems to the optimal solution of the original problem. For the discretized problems we study the properties of the solution map and compute its coderivative. Employing an appropriate chain rule, this enables us to compute the subdifferential of the objective function and to apply a suitable optimization technique to solve the discretized problems. The investigated problem is used to model a situation arising in the area of queuing theory.

On coderivatives and Lipschitzian properties of the dual pair in optimization

  • Authors: López, Marco , Ridolfi, Andrea , Vera De Serio, Virginia
  • Date: 2012
  • Type: Text , Journal article
  • Relation: Nonlinear Analysis, Theory, Methods and Applications Vol. 75, no. 3 (2012), p. 1461-1482
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  • Description: In this paper, we apply the concept of coderivative and other tools from the generalized differentiation theory for set-valued mappings to study the stability of the feasible sets of both the primal and the dual problem in infinite-dimensional linear optimization with infinitely many explicit constraints and an additional conic constraint. After providing some specific duality results for our dual pair, we study the Lipschitz-like property of both mappings and also give bounds for the associated Lipschitz moduli. The situation for the dual shows much more involved than the case of the primal problem. © 2011 Elsevier Ltd. All rights reserved.

Calmness of efficient solution maps in parametric vector optimization

  • Authors: Chuong, Thai Doan , Kruger, Alexander , Yao, J. C.
  • Date: 2011
  • Type: Journal article
  • Relation: Journal of Global Optimization Vol. 51, no. 4 (2011), p. 677-688
  • Relation: http://purl.org/au-research/grants/arc/DP110102011
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  • Description: The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Fréchet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained. © 2011 Springer Science+Business Media, LLC.
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