kuratowski convergence
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Author(s):  
Khushboo Rai ◽  
Prof. C.S. Lalitha

This paper deals with scalarization and stability aspects for a unified set optimization problem. We provide characterizations for a unified preference relation and the corresponding  unified minimal solution in terms of a generalized oriented distance function of the sup-inf type. We  establish continuity of a function associated with the generalized oriented distance function and provide an existence result for the unified minimal solution. We establish  Painlevé-Kuratowski convergence of  minimal solutions of a family of   scalar problems  to the minimal solutions of the unified set optimization problem.


2020 ◽  
Vol 37 (04) ◽  
pp. 2040003
Author(s):  
Zai-Yun Peng ◽  
Jing-Jing Wang ◽  
Xian-Jun Long ◽  
Fu-Ping Liu

This paper is devoted to study the Painlevé–Kuratowski convergence of solution sets for perturbed symmetric set-valued quasi-equilibrium problems (SSQEP)[Formula: see text] via improvement sets. By virtue of the oriented distance function, the sufficient conditions of Painlevé–Kuratowski convergence of efficient solution sets for (SSQEP)[Formula: see text] are obtained through a new nonlinear scalarization technical. Then, under [Formula: see text]-convergence of set-valued mappings, the Painlevé–Kuratowski convergence of weak efficient solution sets for (SSQEP)[Formula: see text] is discussed. What’s more, with suitable convergence assumptions, we also establish the sufficient conditions of lower Painlevé–Kuratowski convergence of Borwein proper efficient solution sets for (SSQEP)[Formula: see text] under improvement sets. Some interesting examples are formulated to illustrate the significance of the main results.


2019 ◽  
Vol 35 (3) ◽  
pp. 393-406
Author(s):  
C. S. LALITHA ◽  
◽  

The main objective of this paper is to investigate the stability of solution sets of perturbed set optimization problems in the decision space as well as in the image space, by perturbing the objective maps. For a sequence of set-valued maps, a notion of gamma convergence is introduced to establish the external and internal stability in terms of Painlev´e–Kuratowski convergence of sequence of solution sets of perturbed problems under certain compactness assumptions and domination properties.


2012 ◽  
Vol 387 (1) ◽  
pp. 48-65 ◽  
Author(s):  
Zofia Denkowska ◽  
Maciej P. Denkowski

2008 ◽  
Vol 93 (2) ◽  
pp. 101-112 ◽  
Author(s):  
Maciej P. Denkowski ◽  
Rafał Pierzchała

2005 ◽  
Vol 5 (3) ◽  
Author(s):  
Filomena A. Lops

AbstractThe aim of this paper consists of introducing on a locally compact and σ-compact metric space a notion of set convergence, which generalizes the Hausdorff convergence, the local Hausdorff convergence and the Kuratowski convergence. We analyze the connections beetwen the three new notions: and. in particular, we prove a compactness result. As a first application of this convergence we give, on a sequence of sets, a condition which assures the lower semicontinuity of the Hausdorff measure with respect to this new convergence and we show that this condition is satisfied by any minimizing sequence of Mumford-Shah functional.


2004 ◽  
Vol 143 (1) ◽  
Author(s):  
Juan Ferrera ◽  
�ngeles Prieto

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