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A note on convex cones and constraint qualifications in infinite-dimensional vector spaces

Abstract

In connection with mathematical programming in infinite-dimensional vector spaces, Zowe has studied the relationship between the Slater constraint qualification and a formally weaker qualification used by Kurcyusz. The attractive feature of the latter is that it involves only active constraints. Zowe has proved that, in barreled spaces, the two qualifications are equivalent and has asked whether the assumption of barreledness is superfluous. By studying cores and interiors of convex cones, we show that the two constraint qualifications are equivalent in a given topological vector spaceE iff every barrel inE is a neighborhood of the origin. Thus, whenE is locally convex, the two constraint qualifications are equivalent iffE is barreled. Other questions of Zowe are also answered.

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Additional information

This research was supported in part by the Office of Naval Research, and in part by the Sonderforschungsbereich 21, Institut für Operations Research, Bonn, Federal Republic of Germany. The author is indebted to Professor J. Zowe for some helpful comments.

Communicated by O. L. Mangasarian

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Klee, V. A note on convex cones and constraint qualifications in infinite-dimensional vector spaces. J Optim Theory Appl 37, 277–284 (1982). https://doi.org/10.1007/BF00934771

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Key Words

  • Convex cones
  • convex sets
  • cores
  • barreled spaces
  • constraint qualifications
  • Karush-Kuhn-Tucker conditions
  • Lagrange multipliers