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Journal of Optimization Theory and Applications

, Volume 163, Issue 3, pp 697–706 | Cite as

A Note on the Existence of Nonsmooth Nonconvex Optimization Problems

  • Kazufumi Ito
  • Karl KunischEmail author
Article

Abstract

Sufficient conditions for the existence of a solution to an abstract optimization problem in Banach spaces are given, which do not rely on convexity, regularity properties or a straightforward coerciveness assumption. Applications to sparsity-constrained optimization and to problems from mechanics are provided.

Keywords

Nonsmooth Nonconvex optimization Coerciveness 

Mathematics Subject Classification (2000)

49J 35Q93 

Notes

Acknowledgments

K. Ito research partially supported by US-ARO grant 49308-MA and US-AFSOR grant FA 9550-06-1-0241. K. Kunisch supported in part by the Fonds zur Förderung der wissenschaftlichen Forschung under SFB 32, “Mathematical Optimization and Applications in the Biomedical Sciences”.

References

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    Baiocchi, C., Buttazzo, G., Gastaldi, F., Tomarelli, F.: General existence theorems for unilateral problems in continuum mechanics. Arch. Ration. Mech. Anal. 100, 149–189 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
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    Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  3. 3.
    Penot, J.-P.: Noncoercive problems and asymptotic conditions. Asymptot. Anal. 49, 205–215 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsNorth Carolina State UniversityRaleighUSA
  2. 2.Institut für MathematikKarl-Franzens-Universität GrazGrazAustria

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