A smoothing inexact Newton method for variational inequalities with nonlinear constraints
Zhili Ge1,2, Qin Ni1, Xin Zhang3
1College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106 China.
Journal of Inequalities and Applications
|July 29, 2017
Summary
This study introduces a new smoothing inexact Newton method for solving nonlinear constrained variational inequalities. The method demonstrates effective performance and quadratic convergence for these complex mathematical problems.
Area of Science:
- Optimization
- Numerical Analysis
- Mathematical Programming
Background:
- Variational inequalities with nonlinear constraints represent a significant class of problems in applied mathematics and operations research.
- Existing methods often face challenges with convergence and computational efficiency for large-scale or complex constraint sets.
Purpose of the Study:
- To develop a novel smoothing inexact Newton method for solving variational inequalities featuring nonlinear constraints.
- To analyze the theoretical convergence properties of the proposed method.
- To assess the practical effectiveness of the method through numerical experiments.
Main Methods:
- Reformulation of the variational inequality problem into a system of parameterized smooth equations using the smoothed Fischer-Burmeister function.
- Application of an inexact Newton method where the linear system at each iteration is solved approximately.
- Theoretical analysis to establish global and local quadratic convergence under mild conditions.
Main Results:
- The proposed smoothing inexact Newton method effectively reformulates the problem into a solvable system of smooth equations.
- Theoretical convergence analysis confirms both global and local quadratic convergence under specific assumptions.
- Numerical results indicate the practical efficacy and efficiency of the developed method.
Conclusions:
- The smoothing inexact Newton method provides a robust and efficient approach for tackling variational inequalities with nonlinear constraints.
- The method's ability to achieve quadratic convergence makes it a promising tool for various applications.
- Further research can explore extensions to more complex problem classes or variations of the method.
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