Convex programming in Hilbert space

@article{Goldstein1964ConvexPI,
  title={Convex programming in Hilbert space},
  author={Allen A. Goldstein},
  journal={Bulletin of the American Mathematical Society},
  year={1964},
  volume={70},
  pages={709-710}
}
  • A. Goldstein
  • Published 1 September 1964
  • Mathematics
  • Bulletin of the American Mathematical Society
This note gives a construction for minimizing certain twice-differentiable functions on a closed convex subset C, of a Hubert Space, H. The algorithm assumes one can constructively "project" points onto convex sets. A related algorithm may be found in Cheney-Goldstein [ l ] , where a constructive fixed-point theorem is employed to construct points inducing a minimum distance between two convex sets. In certain instances when such projections are not too difficult to construct, say on spheres… 

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References

SHOWING 1-3 OF 3 REFERENCES

Proximity maps for convex sets

The method of successive approximation is applied to the problem of obtaining points of minimum distance on two convex sets. Specifically, given a closed convex set K in Hilbert space, let P be the

MINIMIZING FUNCTIONALS ON HILBERT SPACE