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2 results

  • 13 - Optimisation in Two Variables

  • Martin Anthony, London School of Economics and Political Science, Norman Biggs, London School of Economics and Political Science
  • Book:
    Mathematics for Economics and Finance
    Published online:
    24 May 2024
    Print publication:
    30 May 2024, pp 123-136

  • Summary

    Optimisation of two-variable functions is motivated via the example of a firm producing two goods, where the concepts of complementary and substitute goods are discussed. The general idea of a critical point is discussed and it is explained that there can be different types of critical point: maxima, minima and saddle points. It is explained how the second partial derivatives may be used to determine what type of critical point one has. This is shown explicitly for the case in which the two-variable function is quadratic, and then stated in general. Examples involving profit-maximisation are given.

  • 12 - Abnormal Extremals and Second Variation

  • Andrei Agrachev, Scuola Internazionale Superiore di Studi Avanzati, Trieste, Davide Barilari, Université de Paris VII (Denis Diderot), Ugo Boscain, Centre National de la Recherche Scientifique (CNRS), Paris
  • Book:
    A Comprehensive Introduction to Sub-Riemannian Geometry
    Published online:
    28 October 2019
    Print publication:
    31 October 2019, pp 402-455

  • Summary

    This chapter is devoted to the analysis of thesecond-order conditions for length-minimizers andthe analysis of abnormal extremals. After a firstresult on the regularity of the squaredsub-Riemannian distance from a fixed point, showingthat Lipschitz regularity is lost at those pointsreached by only abnormal length-minimizers, we moveto second-order conditions, known as the Goh andgeneralized Legendre conditions.