scholarly journals Cube Root Weak Convergence of Empirical Estimators of a Density Level Set

2020 ◽  
Author(s):  
Philippe Berthet ◽  
John H. J. Einmahl
Keyword(s):  
Weak Convergence ◽  
Level Set ◽  
Cube Root ◽  
Density Level ◽  
Empirical Estimators

scholarly journals The Combination Projection Method for Solving Convex Feasibility Problems

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 249 ◽  
Author(s):  
Songnian He ◽  
Qiao-Li Dong
Keyword(s):  
Hilbert Space ◽  
Positive Integer ◽  
Weak Convergence ◽  
Level Set ◽  
Projection Method ◽  
Real Hilbert Space ◽  
Convex Combination ◽  
Projection Methods ◽  
Feasibility Problem ◽  
Convex Feasibility

In this paper, we propose a new method, which is called the combination projection method (CPM), for solving the convex feasibility problem (CFP) of finding some x * ∈ C : = ∩ i = 1 m { x ∈ H | c i ( x ) ≤ 0 } , where m is a positive integer, H is a real Hilbert space, and { c i } i = 1 m are convex functions defined as H . The key of the CPM is that, for the current iterate x k , the CPM firstly constructs a new level set H k through a convex combination of some of { c i } i = 1 m in an appropriate way, and then updates the new iterate x k + 1 only by using the projection P H k . We also introduce the combination relaxation projection methods (CRPM) to project onto half-spaces to make CPM easily implementable. The simplicity and easy implementation are two advantages of our methods since only one projection is used in each iteration and the projections are also easy to calculate. The weak convergence theorems are proved and the numerical results show the advantages of our methods.

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