Abstract

Let Ω⊂RP\Omega\subset R^P be a nonempty closed and convex set and f:RP→RPf:R^P\to R^P be a function. The inverse variational inequality is to find x∗∈RPx^*\in R^P such that f(x∗)∈Ω,⟨f′−f(x∗),x∗⟩≥0,∀f′∈Ω.f(x^*)\in \Omega,\quad \langle f'-f(x^*),x^*\rangle\ge 0,\quad \forall f'\in \Omega. The purpose of this paper is to investigate the well-posedness of the inverse variational inequality. We establish some characterizations of its well-posedness. We prove that under suitable conditions, the well-posedness of an inverse variational inequality is equivalent to the existence and uniqueness of its solution. Finally, we show that the well-posedness of an inverse variational inequality is equivalent to the well-posedness of an enlarged classical variational inequality.

Author information

Contact details are reproduced from the original publication and may be historical.

Rong Hu

Dept. of Computational Science, Chengdu University of Information Technology, Chengdu, Sichuan, P. R. China

Ya-Ping Fang

Dept. of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China

fabhcn@yahoo.com.cn

Keywords

  • Inverse variational inequality
  • variational inequality
  • well-posedness
  • metric characterization

Mathematics Subject Classification

49J40, 49K40

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Suggested citation

R. Hu, Y.-P. Fang. “Well-Posedness of Inverse Variational Inequalities.” Journal of Convex Analysis 15 (2008), No. 2, 427–437. https://doi.org/10.68381/jca15030

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Published by Heldermann Verlag, 2008. Rights now held by Banach Press.