JIMO
An iterative method for general variational inequalities
Hongxia Yin
Journal of Industrial & Management Optimization 2005, 1(2): 201-209 doi: 10.3934/jimo.2005.1.201
Motivated by the observation that some reformulation based extragradient methods for general monotone variational inequalities in real Hilbert space may not generate a solution of the original problem, we propose an iterative method with line searches and prove its convergence for general pseudomonotone (monotone) variational inequality problems.
keywords: iterative method g−pseudomonotone convergence. Hilbert space General variational inequality problem
JIMO
Finding a stable solution of a system of nonlinear equations arising from dynamic systems
Carl. T. Kelley Liqun Qi Xiaojiao Tong Hongxia Yin
Journal of Industrial & Management Optimization 2011, 7(2): 497-521 doi: 10.3934/jimo.2011.7.497
In this paper, we present a new approach for finding a stable solution of a system of nonlinear equations arising from dynamical systems. We introduce the concept of stability functions and use this idea to construct stability solution models of several typical small signal stability problems in dynamical systems. Each model consists of a system of constrained semismooth equations. The advantage of the new models is twofold. Firstly, the stability requirement of dynamical systems is controlled by nonlinear inequalities. Secondly, the semismoothness property of the stability functions makes the models solvable by efficient numerical methods. We introduce smoothing functions for the stability functions and present a smoothing Newton method for solving the problems. Global and local quadratic convergence of the algorithm is established. Numerical examples from dynamical systems are also given to illustrate the efficiency of the new approach.
keywords: stable solutions saddle-node bifurcation Hopf bifurcation System of nonlinear equations stability functions smoothing Newton method.
JIMO
Semismooth reformulation and Newton's method for the security region problem of power systems
Liqun Qi Zheng yan Hongxia Yin
Journal of Industrial & Management Optimization 2008, 4(1): 143-153 doi: 10.3934/jimo.2008.4.143
In this paper, we investigate the reformulation of the steady state security region problem for electrical power systems. Firstly, a simple security region problem with one changeable parameter is reformulated into a system of semismooth equations, which is composed by the normal power flow equations and an additional piecewise smooth equation. Then the semismooth Newton method and the smoothing Newton method can be applied to solve the problem. Preliminary numerical results show that the method is promising. Finally, by using the smoothing technique, a more complicated security region problem, the Euclidean security region problem, is reformulated as an equality constrained optimization problem. These works provide a possibility to implement on-line calculation of the security region of electrical power systems.
keywords: Power system security region local superlinear convergence. global convergence smoothing technique semismooth Newton method

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