Abstract

Nonlinear variational inequalities in Banach spaces are considered. A suitable notion of condition number with respect to the right-hand side is introduced. A distance among variational inequalities is defined. Based on a new criterion about the Lipschitz property of set-valued mappings, it is shown that the distance to suitably restricted ill-conditioned variational inequalities is bounded from below by the reciprocal of the condition number. By using a similar upper bound of the companion paper "An upper bound for a condition number theorem of variational inequalities", we obtain a full condition number theorem for variational inequalities. The particular case of convex minimization problems is considered. Known results dealing with optimization problems are thereby generalized.

Author information

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

Keywords

  • Variational inequalities
  • Lipschitz set-valued mappings
  • condition number theorems

Mathematics Subject Classification

49J40, 49K40, 49J53, 90C31

References

  1. C. Baiocchi, A. Capelo: Disequazioni Variazionali e Quasivariazionali. Applicazioni a Problemi di Frontiera Libera. Volume 1, Pitagora, Bologna (1978).Google Scholar
  2. L. Blum, F. Cucker, M. Shub, S. Smale: Complexity and Real Computation, Springer, New York (1998).https://doi.org/10.1007/978-1-4612-0701-6Google Scholar
  3. P. Bürgisser, F. Cucker: Condition. The Geometry of Numerical Algorithms, Springer, Berlin (2013).https://doi.org/10.1007/978-3-642-38896-5Google Scholar
  4. A. L. Dontchev, R. T. Rockafellar: Implicit Functions and Solution Mappings, Springer, New York (2009).https://doi.org/10.1007/978-0-387-87821-8Google Scholar
  5. J. Renegar: Some perturbation theory for linear programming, Math. Programming 65 (1994) 73–91.https://doi.org/10.1007/BF01581690Google Scholar
  6. J. Renegar: Linear programming, complexity theory and elementary functional analysis, Math. Programming 70 (1995) 219–351.https://doi.org/10.1007/BF01585941Google Scholar
  7. T. Zolezzi: On the distance theorem in quadratic optimization, J. Convex Analysis 9 (2002) 693–700.https://doi.org/10.68381/jca09040Google Scholar
  8. T. Zolezzi: Condition number theorems in optimization, SIAM J. Optim. 14 (2003) 507–516.https://doi.org/10.1137/S1052623402411885Google Scholar
  9. T. Zolezzi: A condition number theorem in convex programming, Math. Programming Ser. A 149 (2015) 195–207.https://doi.org/10.1007/s10107-014-0745-5Google Scholar
  10. T. Zolezzi: On condition number theorems in mathematical programming, J. Optim. Theory Appl. 175 (2017) 597–623.https://doi.org/10.1007/s10957-017-1191-3Google Scholar
  11. T. Zolezzi: An upper bound for a condition number theorem of variational inequalities, submitted.Google Scholar

Suggested citation

T. Zolezzi. “A Lower Bound for a Condition Number Theorem of Variational Inequalities.” Journal of Convex Analysis 30 (2023), No. 4, 1379–1390. https://doi.org/10.68381/jca30062

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2023. Rights now held by Banach Press LLC.