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Topology and Convex Optimisation

In: Mathematical Methods in Economics and Social Choice

Author

Listed:
  • Norman Schofield

    (Washington University in Saint Louis)

Abstract

Chapter 3 covers Topology and convex optimization. In the Chap. 2 we introduced the notion of the scalar product of two vectors in ℜ n . More generally if a scalar product is defined on some space, then this permits the definition of a norm, or length, associated with a vector, and this in turn allows us to define the distance between two vectors. A distance function or metric may be defined on a space, X, even when X admits no norm. More general than the notion of a metric is that of a topology. This notion allows us to define the idea of continuity of a function as well as analogous ideas for a correspondence. We then introduce three powerful theorems, the Brouwer Fixed Point Theorem for a function, Michael’s Selection Theorem, and the Browder Fixed Point Theorem for a correspondence.

Suggested Citation

  • Norman Schofield, 2014. "Topology and Convex Optimisation," Springer Texts in Business and Economics, in: Mathematical Methods in Economics and Social Choice, edition 2, chapter 3, pages 77-133, Springer.
    • Handle: RePEc:spr:sptchp:978-3-642-39818-6_3
    DOI: 10.1007/978-3-642-39818-6_3
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