nonconvex sets
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2020 ◽  
Vol 245 (2) ◽  
pp. 125-154
Author(s):  
G. E. Ivanov ◽  
M. S. Lopushanski

2019 ◽  
Vol 47 (6) ◽  
pp. 569-573 ◽  
Author(s):  
Refail Kasimbeyli ◽  
Masoud Karimi

2019 ◽  
Vol 35 (3) ◽  
pp. 379-384
Author(s):  
SHOKOUH SHAHBEYK ◽  
MAJID SOLEIMANI-DAMANEH ◽  
◽  

In this paper, limiting proper minimal points of nonconvex sets in Euclidean finite-dimensional spaces are investigated. The relationships between these minimal points and Borwein, Benson, and Henig proper minimal points, under appropriate assumptions, are established. Furthermore, a density property is derived and a linear characterization of limiting proper minimal points is provided.


Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 186 ◽  
Author(s):  
Chang Cong ◽  
Peibiao Zhao

Monetary risk measures are interpreted as the smallest amount of external cash that must be added to a financial position to make the position acceptable. In this paper, A new concept: non-cash risk measure is proposed and this measure provides an approach to transform the unacceptable positions into the acceptable positions in a nonconvex set. Non-cash risk measure uses not only cash but also other kinds of assets to adjust the position. This risk measure is nonconvex due to the use of optimization problem in L 1 norm. A convex extension of the nonconvex risk measure is derived and the relationship between the convex extension and the non-cash risk measure is detailed.


2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Menglong Su ◽  
Yufeng Shang ◽  
Wenzhuang Zhu

In this paper, we propose a parameter perturbation homotopy continuation method for solving fixed point problems on more general nonconvex sets with both inequality and equality constraints. By adopting appropriate techniques, we make the initial points not certainly in the set consisting of the equality constraints. This point can improve the computational efficiency greatly when the equality constraints are complex. In addition, we also weaken the assumptions of the previous results in the literature so that the method proposed in this paper can be applied to solve fixed point problems in more general nonconvex sets. Under suitable conditions, we obtain the global convergence of this homotopy continuation method. Moreover, we provide several numerical examples to illustrate the results of this paper.


2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Menglong Su ◽  
Yufeng Shang

In recent years, fixed-point theorems have attracted increasing attention and have been widely investigated by many authors. Moreover, determining a fixed point has become an interesting topic. In this paper, we provide a constructive proof of the general Brouwer fixed-point theorem and then obtain the existence of a smooth path which connects a given point to the fixed point. We also present a non-interior point homotopy algorithm for solving fixed-point problems on a class of nonconvex sets by numerically tricking this homotopy path.


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