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1 results

  • 7 - Geometry of Polyhedral Sets

  • from Part II - Separation Theorem, Extreme Points, Recessive Directions, and Geometry of Polyhedral Sets
  • Fatma Kılınç-Karzan, Carnegie Mellon University, Pennsylvania, Arkadi Nemirovski, Georgia Institute of Technology
  • Book:
    Essential Mathematics for Convex Optimization
    Published online:
    22 October 2025
    Print publication:
    26 June 2025, pp 118-134

  • Summary

    In this chapter, we (a) present an algebraic characterization of extreme points of polyhedral sets and extreme rays of polyhedral cones, (b) describe extreme points of several important polyhedral sets, including the Birkhoff--von-Neumann Theorem on extreme points of the polytope of doubly stochastic matrices, (c) establish the theorem on the structure of polyhedral sets stating that nonempty polyhedral sets are exactly the sets representable as sums of convex hulls of nonempty finite sets and conic hulls of finite sets, and vice versa, (d) extract from the latter theorem basic descriptive results of linear programming theory, and (e) present and justify the Majorization Principle.