A Generalized Convexity and Variational Inequality for Quasi-Convex Minimization

1996 ◽  
Vol 6 (1) ◽  
pp. 212-226 ◽  
Author(s):  
Phan Thien Thach ◽  
Masakazu Kojima
Keyword(s):  
Variational Inequality ◽  
Generalized Convexity ◽  
Convex Minimization

scholarly journals Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-25
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Convex Minimization ◽  
Solution Set ◽  
Projection Algorithm ◽  
Common Element ◽  
Point Problem ◽  
Hierarchical Fixed Point ◽  
Mixed Equilibria

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and 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 infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), 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 to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others.

scholarly journals Iterative algorithm for singularities of inclusion problems in Hadamard manifolds

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Parin Chaipunya ◽  
Konrawut Khammahawong ◽  
Poom Kumam
Keyword(s):  
Variational Inequality ◽  
Iterative Algorithm ◽  
Convex Minimization ◽  
Variational Inequality Problems ◽  
Hadamard Manifolds ◽  
Inclusion Problems ◽  
Numerical Example ◽  
Minimization Problems ◽  
Convex Minimization Problems

AbstractThe main purpose of this paper is to introduce a new iterative algorithm to solve inclusion problems in Hadamard manifolds. Moreover, applications to convex minimization problems and variational inequality problems are studied. A numerical example also is presented to support our main theorem.

scholarly journals Algorithm for Solutions of Nonlinear Equations of Strongly Monotone Type and Applications to Convex Minimization and Variational Inequality Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Mathew O. Aibinu ◽  
Surendra C. Thakur ◽  
Sibusiso Moyo
Keyword(s):  
Variational Inequality ◽  
Nonlinear Equations ◽  
Real Life ◽  
Convergence Result ◽  
Convex Minimization ◽  
Variational Inequality Problems ◽  
Strongly Monotone ◽  
Life Problems ◽  
Engineering Physics ◽  
Monotone Type

Real-life problems are governed by equations which are nonlinear in nature. Nonlinear equations occur in modeling problems, such as minimizing costs in industries and minimizing risks in businesses. A technique which does not involve the assumption of existence of a real constant whose calculation is unclear is used to obtain a strong convergence result for nonlinear equations of p,η-strongly monotone type, where η>0,p>1. An example is presented for the nonlinear equations of p,η-strongly monotone type. As a consequence of the main result, the solutions of convex minimization and variational inequality problems are obtained. This solution has applications in other fields such as engineering, physics, biology, chemistry, economics, and game theory.

scholarly journals New Convergence Theorems for Maximal Monotone Operators in Banach Spaces

2021 ◽  
Vol 15 ◽  
pp. 8-13
Author(s):  
Siwaporn Saewan
Keyword(s):  
Variational Inequality ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Convex Minimization ◽  
Variational Inequality Problems ◽  
Convergence Theorems

The purpose of this paper is to introduce a new hybrid iterative scheme for resolvents of maximal monotone operators in Banach spaces by using the notion of generalized fprojection. Next, we apply this result to the convex minimization and variational inequality problems in Banach spaces. The results presented in this paper improve and extend important recent results in the literature.

On an Euler-Lagrange equation for color to grayscale conversion

2019 ◽  
Vol 2019 (1) ◽  
pp. 95-98
Author(s):  
Hans Jakob Rivertz
Keyword(s):  
Variational Inequality ◽  
Color Image ◽  
Lagrange Equation ◽  
New Method ◽  
Grayscale Image ◽  
Structure Tensors

In this paper we give a new method to find a grayscale image from a color image. The idea is that the structure tensors of the grayscale image and the color image should be as equal as possible. This is measured by the energy of the tensor differences. We deduce an Euler-Lagrange equation and a second variational inequality. The second variational inequality is remarkably simple in its form. Our equation does not involve several steps, such as finding a gradient first and then integrating it. We show that if a color image is at least two times continuous differentiable, the resulting grayscale image is not necessarily two times continuous differentiable.

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