scholarly journals THE GLOBAL COMPLEXITY BOUNDS OF THE GRADIENT METHODS AND THE NEWTON-TYPE METHODS FOR THE UNCONSTRAINED MINIMIZATION PROBLEM

2013 ◽  
Vol 56 (0) ◽  
pp. 15-30
Author(s):  
Nobuo Yamashita ◽  
Kenji Ueda
Keyword(s):  
Minimization Problem ◽  
Gradient Methods ◽  
Unconstrained Minimization ◽  
Complexity Bounds ◽  
Unconstrained Minimization Problem

scholarly journals Stable Approximations of a Minimal Surface Problem with Variational Inequalities

1997 ◽  
Vol 2 (1-2) ◽  
pp. 137-161 ◽  
Author(s):  
M. Zuhair Nashed ◽  
Otmar Scherzer
Keyword(s):  
Dirichlet Problem ◽  
Variational Inequalities ◽  
Minimal Surface ◽  
Unconstrained Minimization ◽  
Trace Operator ◽  
Functions Of Bounded Variation ◽  
New Approach ◽  
Regularization Procedure ◽  
Unconstrained Minimization Problem ◽  
Stable Approximation

In this paper we develop a new approach for the stable approximation of a minimal surface problem associated with a relaxed Dirichlet problem in the spaceBV(Ω)of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional𝒥onBV(Ω)defined by𝒥(u)=𝒜(u)+∫∂Ω|Tu−Φ|, where𝒜(u)is the “area integral” ofuwith respect toΩ,Tis the “trace operator” fromBV(Ω)intoL i(∂Ω), andϕis the prescribed data on the boundary ofΩ. We establish convergence and stability of approximate regularized solutions which are solutions of a family of variational inequalities. We also prove convergence of an iterative method based on Uzawa's algorithm for implementation of our regularization procedure.

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