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Minimum Norm Problems in Normed Vector Lattices

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Infinite Programming

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 259))

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Abstract

Let {xv}veI be a family of elements of a normed vector space N and let {cv}veI be a corresponding family of real numbers. Fan [3] studied the minimum norm problem:

$$Minimize \left\| f \right\|$$

subject to the linear inequality constraints

$$f({x_\nu }) \geqq {c_\nu },\nu \varepsilon I$$
((*))

.

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References

  1. Anderson E.J., “A Review of Duality Theory for Linear Programming over Topological Vector Spaces”, J. Math. Anal. Appl., vol. 97, pp. 380–392, 1983.

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  2. Braunschweiger C.C., “An Extension of the Nonhomogeneous Farkas Theorem”, Amer. Math. Monthly, vol. 69, pp. 969–975, 1962.

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  3. Fan K., “On Systems of Linear Inequalities”, Linear Inequalities and Related Systems (Annal of Mathematics Studies 38), Princeton: Princeton Univ. Press, 1956.

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  4. Hirano N., Komiya H. and Takahashi W., “A Generalization of the Hahn-Banach Theorem”, J. Math. Anal. Appl., vol. 88, pp. 333–340, 1982.

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  5. Klee V.L. Jr., “Separation Properties of Convex Cones”, Proc. Amer. Math. Soc, vol. 6, pp. 313–318, 1955.

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  6. Komiya H. and Takahashi W., “Systems of Linear Inequalities on Normed Linear Spaces”, Linear and Multilinear Algebra, vol. 13, pp. 267–279, 1983.

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  7. Namioka I., “Partially Ordered Linear Topological Spaces”, Mem. Amer. Math. Soc, vol. 24, 1957.

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  8. Neustadt L.W., “Optimization, a Moment Problem, and Nonlinear Programming”, J. SIAM Control, vol. A-2, pp. 33–53, 1964.

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  9. Schaefer, “Banach Lattices and Positive Operators”, Berlin/Heidelberg/ New York: Springer, 1974.

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Author information

Authors and Affiliations

  1. College of Commerce, Nihon University, 157 Tokyo, Japan

    H. Komiya

Authors
  1. H. Komiya
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Editor information

Editors and Affiliations

  1. Management Studies Group Engineering Department, Cambridge University, Mill Lane, Cambridge, CB2 1RX, UK

    Edward J. Anderson  & Andrew B. Philpott  & 

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© 1985 Springer-Verlag Berlin Heidelberg

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Komiya, H. (1985). Minimum Norm Problems in Normed Vector Lattices. In: Anderson, E.J., Philpott, A.B. (eds) Infinite Programming. Lecture Notes in Economics and Mathematical Systems, vol 259. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46564-2_17

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  • DOI: https://doi.org/10.1007/978-3-642-46564-2_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15996-4

  • Online ISBN: 978-3-642-46564-2

  • eBook Packages: Springer Book Archive

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