scholarly journals Projected-Reflected Subgradient-Extragradient Method and Its Real-World Applications

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 489
Author(s):  
Aviv Gibali ◽  
Olaniyi S. Iyiola ◽  
Lanre Akinyemi ◽  
Yekini Shehu
Keyword(s):  
Extragradient Method ◽  
A Priori ◽  
Lipschitz Constant ◽  
Monotone Mapping ◽  
Subgradient Extragradient Method ◽  
Lipschitz Continuous ◽  
Real World Applications ◽  
Potential Applicability ◽  
Tomography Reconstruction ◽  
The Cost

Our main focus in this work is the classical variational inequality problem with Lipschitz continuous and pseudo-monotone mapping in real Hilbert spaces. An adaptive reflected subgradient-extragradient method is presented along with its weak convergence analysis. The novelty of the proposed method lies in the fact that only one projection onto the feasible set in each iteration is required, and there is no need to know/approximate the Lipschitz constant of the cost function a priori. To illustrate and emphasize the potential applicability of the new scheme, several numerical experiments and comparisons in tomography reconstruction, Nash–Cournot oligopolistic equilibrium, and more are presented.

scholarly journals Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems

2020 ◽  
Vol 53 (1) ◽  
pp. 208-224 ◽  
Author(s):  
Timilehin Opeyemi Alakoya ◽  
Lateef Olakunle Jolaoso ◽  
Oluwatosin Temitope Mewomo
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Extragradient Method ◽  
Lipschitz Constant ◽  
Monotone Mapping ◽  
Variational Inequality Problems ◽  
Step Size ◽  
Lipschitz Continuous ◽  
Convergence Results ◽  
Monotone Variational Inequality

AbstractIn this work, we introduce two new inertial-type algorithms for solving variational inequality problems (VIPs) with monotone and Lipschitz continuous mappings in real Hilbert spaces. The first algorithm requires the computation of only one projection onto the feasible set per iteration while the second algorithm needs the computation of only one projection onto a half-space, and prior knowledge of the Lipschitz constant of the monotone mapping is not required in proving the strong convergence theorems for the two algorithms. Under some mild assumptions, we prove strong convergence results for the proposed algorithms to a solution of a VIP. Finally, we provide some numerical experiments to illustrate the efficiency and advantages of the proposed algorithms.

An inertial modified algorithm for solving variational inequalities

2020 ◽  
Vol 54 (1) ◽  
pp. 163-178 ◽  
Author(s):  
Dang Van Hieu ◽  
Pham Kim Quy
Keyword(s):  
Hilbert Spaces ◽  
Lipschitz Constant ◽  
Inertial Effects ◽  
Convergence Properties ◽  
Pseudomonotone Operators ◽  
Lipschitz Continuous ◽  
Strong Pseudomonotonicity ◽  
Computational Performance ◽  
Speed Up ◽  
The Cost

The paper deals with an inertial-like algorithm for solving a class of variational inequality problems involving Lipschitz continuous and strongly pseudomonotone operators in Hilbert spaces. The presented algorithm can be considered a combination of the modified subgradient extragradient-like algorithm and inertial effects. This is intended to speed up the convergence properties of the algorithm. The main feature of the new algorithm is that it is done without the prior knowledge of the Lipschitz constant and the modulus of strong pseudomonotonicity of the cost operator. Several experiments are performed to illustrate the convergence and computational performance of the new algorithm in comparison with others having similar features. The numerical results have confirmed that the proposed algorithm has a competitive advantage over the existing methods.

scholarly journals Two Nonmonotonic Self-Adaptive Strongly Convergent Projection-Type Methods for Solving Pseudomonotone Variational Inequalities

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Chainarong Khunpanuk ◽  
Bancha Panyanak ◽  
Nuttapol Pakkaranang
Keyword(s):  
Variational Inequalities ◽  
Iterative Methods ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Lipschitz Constant ◽  
Primary Objective ◽  
Subgradient Extragradient Method ◽  
Pseudomonotone Operator ◽  
Step Size ◽  
Iterative Schemes

The primary objective of this study is to introduce two novel extragradient-type iterative schemes for solving variational inequality problems in a real Hilbert space. The proposed iterative schemes extend the well-known subgradient extragradient method and are used to solve variational inequalities involving the pseudomonotone operator in real Hilbert spaces. The proposed iterative methods have the primary advantage of using a simple mathematical formula for step size rule based on operator information rather than the Lipschitz constant or another line search method. Strong convergence results for the suggested iterative algorithms are well-established for mild conditions, such as Lipschitz continuity and mapping monotonicity. Finally, we present many numerical experiments that show the effectiveness and superiority of iterative methods.

scholarly journals A strong convergence theorem for solving pseudo-monotone variational inequalities and fixed point problems using subgradient extragradient method in Banach spaces

2021 ◽  
Vol 7 (4) ◽  
pp. 5015-5028
Author(s):  
Fei Ma ◽  
◽  
Jun Yang ◽  
Min Yin
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Extragradient Method ◽  
Lipschitz Constant ◽  
Fixed Point Problems ◽  
Strong Convergence Theorem ◽  
Subgradient Extragradient Method ◽  
Step Size ◽  
Common Solution

<abstract><p>In this paper, we introduce an algorithm for solving variational inequalities problem with Lipschitz continuous and pseudomonotone mapping in Banach space. We modify the subgradient extragradient method with a new and simple iterative step size, and the strong convergence to a common solution of the variational inequalities and fixed point problems is established without the knowledge of the Lipschitz constant. Finally, a numerical experiment is given in support of our results.</p></abstract>

scholarly journals An Efficient Parallel Extragradient Method for Systems of Variational Inequalities Involving Fixed Points of Demicontractive Mappings

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1915
Author(s):  
Lateef Olakunle Jolaoso ◽  
Maggie Aphane
Keyword(s):  
Variational Inequalities ◽  
Hilbert Spaces ◽  
Line Search ◽  
Convex Combination ◽  
Extragradient Method ◽  
Convergence Result ◽  
Armijo Line Search ◽  
Systems Of Variational Inequalities ◽  
Demicontractive Mappings ◽  
The Cost

Herein, we present a new parallel extragradient method for solving systems of variational inequalities and common fixed point problems for demicontractive mappings in real Hilbert spaces. The algorithm determines the next iterate by computing a computationally inexpensive projection onto a sub-level set which is constructed using a convex combination of finite functions and an Armijo line-search procedure. A strong convergence result is proved without the need for the assumption of Lipschitz continuity on the cost operators of the variational inequalities. Finally, some numerical experiments are performed to illustrate the performance of the proposed method.

scholarly journals Model calculations of posterior reliability indicators for the proposal of the maintenance system

2018 ◽  
Vol 157 ◽  
pp. 08003 ◽  
Author(s):  
Jana Galliková ◽  
Vladimír Stuchlý ◽  
Roman Poprocký ◽  
Peter Volna
Keyword(s):  
A Priori ◽  
Weibull Model ◽  
Calculation Methods ◽  
Optimal Process ◽  
Model Calculations ◽  
Maintenance System ◽  
Reliability Indicators ◽  
Technical Practice ◽  
Reliability Methods ◽  
The Cost

Designing the content and scale of maintenance of machines and equipment by a priori and posterior reliability methods in considered crucial to reducing the cost of the machine's life cycle, maintaining high operational readiness and reducing the consequences of failures. In the presented paper, attention is paid to the analysis of the calculation methods of posterior reliability for calculation indicators of reliability and to the use of the specified Weibull model for reliability calculations. The obtained results are further developed for models of optimal process calculations to perform scheduled maintenance interventions. Calculations of the other RAMS (reliability, availability, maintainability and safety) indicators that are critical to the design of an optimal engineering design with regard to maintenance and which do not receive sufficient attention in technical practice are also assessed.

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