scholarly journals Steepest-Descent Approach to Triple Hierarchical Constrained Optimization Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-19
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Projection Algorithm ◽  
Fixed Point Set

We introduce and analyze a hybrid steepest-descent algorithm by combining Korpelevich’s extragradient method, the steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to the unique solution of a triple hierarchical constrained optimization problem (THCOP) over the common fixed point set of finitely many nonexpansive mappings, with constraints of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and a convex minimization problem (CMP) in a real Hilbert space.

scholarly journals Hybrid Steepest-Descent Methods for Triple Hierarchical Variational Inequalities

2015 ◽  
Vol 2015 ◽  
pp. 1-22
Author(s):  
L. C. Ceng ◽  
A. Latif ◽  
C. F. Wen ◽  
A. E. Al-Mazrooei
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Steepest Descent ◽  
Solution Set ◽  
Steepest Descent Method ◽  
Common Element ◽  
Variational Inclusions

We introduce and analyze a relaxed iterative algorithm by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions, and the solution set of general system of variational inequalities (GSVI), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm for solving a hierarchical variational inequality problem with constraints of finitely many GMEPs, finitely many variational inclusions, and the GSVI. The results obtained in this paper improve and extend the corresponding results announced by many others.

scholarly journals Steepest-descent Ishikawa iterative methods for a class of variational inequalities in Banach spaces

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1557-1569
Author(s):  
Nguyen Buong ◽  
Nguyen Anh ◽  
Khuat Binh
Keyword(s):  
Iterative Method ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Uniformly Smooth ◽  
Differentiable Norm ◽  
Strictly Convex Banach Spaces

In this paper, for finding a fixed point of a nonexpansive mapping in either uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly G?teaux differentiable norm, we present a new explicit iterative method, based on a combination of the steepest-descent method with the Ishikawa iterative one. We also show its several particular cases one of which is the composite Halpern iterative method in literature. The explicit iterative method is also extended to the case of infinite family of nonexpansive mappings. Numerical experiments are given for illustration.

scholarly journals A modification of steepest descent method for solving large-scaled unconstrained optimization problems

2018 ◽  
Vol 7 (3.28) ◽  
pp. 72
Author(s):  
Siti Farhana Husin ◽  
Mustafa Mamat ◽  
Mohd Asrul Hery Ibrahim ◽  
Mohd Rivaie
Keyword(s):  
Unconstrained Optimization ◽  
Large Scale ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Search Direction ◽  
Exact Line Search ◽  
Scale Optimization

In this paper, we develop a new search direction for Steepest Descent (SD) method by replacing previous search direction from Conjugate Gradient (CG) method, , with gradient from the previous step,  for solving large-scale optimization problem. We also used one of the conjugate coefficient as a coefficient for matrix . Under some reasonable assumptions, we prove that the proposed method with exact line search satisfies descent property and possesses the globally convergent. Further, the numerical results on some unconstrained optimization problem show that the proposed algorithm is promising. 

scholarly journals Hybrid Iterative Scheme for Triple Hierarchical Variational Inequalities with Mixed Equilibrium, Variational Inclusion, and Minimization Constraints

2014 ◽  
Vol 2014 ◽  
pp. 1-22
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Solution Set ◽  
Steepest Descent Method ◽  
Projection Algorithm ◽  
Generalized Mixed Equilibrium Problem ◽  
Common Element ◽  
Variational Inclusions ◽  
Mixed Equilibrium

We introduce and analyze a hybrid iterative algorithm by combining Korpelevich's extragradient method, the hybrid steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of finitely many nonexpansive mappings, the solution set of a generalized mixed equilibrium problem (GMEP), the solution set of finitely many variational inclusions, and the solution set of a convex minimization problem (CMP), which is also a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solving a hierarchical variational inequality problem with constraints of the GMEP, the CMP, and finitely many variational inclusions.

scholarly journals Composite steepest-descent method for the triple hierarchical variational inequalities

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4403-4419
Author(s):  
Lu-Chuan Ceng ◽  
Jen-Chih Yao ◽  
Yonghong Yao
Keyword(s):  
Variational Inequality Problem ◽  
Real Hilbert Space ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Mild Conditions ◽  
Triple Hierarchical Variational Inequalities ◽  
Steepest Descent Algorithm ◽  
Triple Hierarchical Variational Inequality ◽  
Hierarchical Variational Inequalities

In this paper, we introduce and analyze a composite steepest-descent algorithm for solving the triple hierarchical variational inequality problem in a real Hilbert space. Under mild conditions, the strong convergence of the iteration sequences generated by the algorithm is established.

scholarly journals Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nguyen Buong
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Descent Method ◽  
Finite Family ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Point Problem ◽  
Lower Semi ◽  
The Family

<p style='text-indent:20px;'>In this paper, for solving the variational inequality problem over the set of common fixed points of a finite family of demiclosed quasi-nonexpansive mappings in Hilbert spaces, we propose two new strongly convergent methods, constructed by specific combinations between the steepest-descent method and the block-iterative ones. The strong convergence is proved without the boundedly regular assumptions on the family of fixed point sets as well as the approximately shrinking property for each mapping of the family, that are usually assumed in recent literature for similar problems. Applications to the multiple-operator split common fixed point problem (MOSCFPP) and the problem of common minimum points of a finite family of lower semi-continuous convex functions with numerical experiments are given.</p>

scholarly journals A new conjugate gradient method for acceleration of gradient descent algorithms

2021 ◽  
Vol 7 (1) ◽  
pp. 1-11
Author(s):  
Noureddine Rahali ◽  
Mohammed Belloufi ◽  
Rachid Benzine
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Optimization Problems ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Convergence Results ◽  
Wolfe Line Search ◽  
New Formula

AbstractAn accelerated of the steepest descent method for solving unconstrained optimization problems is presented. which propose a fundamentally different conjugate gradient method, in which the well-known parameter βk is computed by an new formula. Under common assumptions, by using a modified Wolfe line search, descent property and global convergence results were established for the new method. Experimental results provide evidence that our proposed method is in general superior to the classical steepest descent method and has a potential to significantly enhance the computational efficiency and robustness of the training process.

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