ZHANG Li-li, LI Xing-si. New Smooth Gap Function for Box Constrained Variational Inequalities[J]. Applied Mathematics and Mechanics, 2013, 34(1): 27-37. doi: 10.3879/j.issn.1000-0887.2013.01.004
Citation: ZHANG Li-li, LI Xing-si. New Smooth Gap Function for Box Constrained Variational Inequalities[J]. Applied Mathematics and Mechanics, 2013, 34(1): 27-37. doi: 10.3879/j.issn.1000-0887.2013.01.004

New Smooth Gap Function for Box Constrained Variational Inequalities

doi: 10.3879/j.issn.1000-0887.2013.01.004
  • Received Date: 2012-06-20
  • Rev Recd Date: 2012-09-07
  • Publish Date: 2013-01-15
  • A new smooth gap function for box constrained variational inequality problem was proposed based on an integral global optimality condition. The smooth gap function was simple and had some good differentiable properties. The box constrained variational inequality problem could be reformulated as a differentiable optimization problem by using the proposed smooth gap function. Conditions under which any stationary point of the optimization problem was the solution to box constrained variational inequality problem were discussed. A simple frictional contact problem was analyzed to illustrate the application of this smooth gap function. Finally, numerical experiments confirmed the good theoretical properties of the method.
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  • [1]
    Billups S C,Dirkse S P,Ferris M C.A comparison of algorithms for large scale mixed complementarity problems[J].Computational Optimization and Applications,1997,7(1):3-25.
    [2]
    Harker P T,Pang J S.Finite dimensional inequality and nonlinear complementarity problems:a survey of theory, algorithms and applications[J].Mathematical Programming,1990,48(1/3):161-220.
    [3]
    Facchinei F, Pang J S.FiniteDimensional Variational Inequalities and Complementarity Problems[M].New York:Springer, 2003.
    [4]
    钟万勰,张洪武,吴承伟.变量变分原理及其在工程中的应用[M]. 北京:科学出版社,1997.(ZHONG Wan-xie, ZHANG Hong-wu, WU Cheng-wei. Parametric Variational Principle and Its Applications in Engineering[M].Beijing: Science Press,1997.(in Chinese))
    [5]
    何素艳, 李建宇, 李兴斯, 张洪武.工程力学中的互补问题: 模型[J].计算力学学报,2004,21(2): 185-190. (HE Su-yan, LI Jian-yu, LI Xing-si, ZHANG Hong-wu. Complementarity problems in engineering mechanics: models[J].Chinese Journal of Computational Mechanics,2004,21(2): 185-190.(in Chinese))
    [6]
    张培爱,何素艳,李建宇,李兴斯.工程力学中的互补问题(II): 算法[J]. 计算力学学报,2006,23(6): 696-705.(ZHANG Pei-ai, HE Su-yan, LI Jian-yu, LI Xing-si.Complementarity problems in engineering mechanics(II): algorithms[J]. Chinese Journal of Computational Mechanics,2006,23(6): 696-.(in Chinese))
    [7]
    SUN De-feng, Womersley R S. A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Gauss-Newton method[J].SIAM Journal of Optimization,1999,9(2):388-413.[8]Auslender A.Optimisation: Methodes Numeriques[M].Paris: Masson, 1976.
    [8]
    Fukushima M. Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems[J].Mathematical Programming,1992,53(3):99-110.
    [9]
    PENG Ji-ming. Equivalence of variational inequality problems to unconstrained minimization[J]. Mathematical Programming,1997,78(3):347-356.
    [10]
    乌力吉, 陈国庆. 箱约束变分不等式的一个简单光滑价值函数和阻尼牛顿法[J]. 应用数学和力学,2005,26(8):988-996.(Ulji, CHEN Guo-qing. New simple smooth merit function for box constrained variational inequalities and damped Newton type method[J].Applied Mathematics and Mechanics(English Edition),2005,26(8):1083-1092.)
    [11]
    HiriartUrruty J B. Conditions for global optimality[C]//Horst R, Pardalos P M.Handbook of Global Optimization, Dordrecht: Kluwer,1994: 1-25.
    [12]
    Erdelyi A.Asymptotic Expansions[M]. New York: Dover, 1965.
    [13]
    Henrici P.Applied and Computational Complex Analysis[M]. Vol 2. New York: Wiley, 1977.
    [14]
    Ellis R S. Entropy, Large Deviations, and Statistical Mechanics[M]. New York: SpringerVerlag, 1985.
    [15]
    Bonnaus J F, Shapiro A. Perturbation Analysis of Optimization Problems[M]. Springer, 2000.
    [16]
    Marcotte P. A new algorithm for solving variational inequalities with applications to the traffic assignment problem[J]. Mathematical Programming,1985, 33(3): 339-351.
    [17]
    Dirkse S P, Ferris M C. MCPLIB: a collection of nonlinear mixed complementarity problem[Z]. Computer Science Department, University of Wisconsin,1994.
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