ORIE 320 Professor Bland 9/10/07 Convexity, Polyhedra, Extreme Points A set S IRn is convex if for all points x1 , x2 S the point (1 - )x1 + x2 S, for all 0 1. Can you give a geometric interpretation of this property? Note that the intersection of convex sets is also convex. Let 1 , . . . , n and be constants, with at least one of 1 , . . . , n nonzero. The set of all vectors x = (x1 , , xn ) such that 1 x1 + . . . + n xn is called a halfspace of IRn . It should be clear that every halfspace is convex. A polyhedron in IRn is the intersection of finitely many half-spaces. For example, the set of all feasible solutions (x1 , x2 )T for the original version of Grandma's Sausage Problem, or the set of all feasible solutions x = (x1 , x2 , x3 , x4 , x5 )T that satisfy Ax = b for the 3 5 matrix A and the 3 1 vector b from the version of Grandma's Sausage Problem in which the slack variables x3 , x4 , x5 were added. (See p.3 of the handout Systems of Linear Equations and <a href="/keyword/linear-optimization/" >linear optimization</a> .) A polyhedron P is bounded if there is some constant u such that for every x = (x1 , . . . , xn )T P we have |xj | u j = 1, . . . n. Let P be a polyhedron in IRn . A point x P is an extreme point of P if 1 1 x1 , x2 P and x = x1 + x2 x = x1 = x2 . 2 2 If a polyhedron is nonempty and bounded, it has extreme points. Does your intuition tell you why? In the Scientific American Excerpt we saw that if you are maximizing a linear function over such a polyhedron P , then it is enough to examine the extreme points of P . (It follows from linearity of the objective function, the function being maximized, and convexity of the polyhedron.) However, enumerating all of the extreme points would be a practical impossibility if n, the number of variables, and the number of halfspaces defining P were large. So we will look for an algorithm that examines extreme points selectively. It moves along a sequence of x1 , x2 , x3 , . . . of extreme points of P with each move increasing the value of the function being maximized, until it finds an extreme point that can be recognized to be optimal. We will need some efficient way to test whether the current extreme point is optimal, and, if it fails the test, an efficient way to determine how to move to a better extreme point. The handout Systems of Linear Equations and <a href="/keyword/linear-optimization/" >linear optimization</a> shows how our knowledge of systems of linear equations will provide the right mechanisms. Before we return to that handout, one more comment regarding how convexity of P helps us in testing optimality of a linear function over P . If x P , then convexity guarantees that if there are better solutions than x anywhere in P , then there must be better solutions arbitrarily close to P . In other words, if x is locally optimal, then it is globally optimal. This is no longer true when optimizing a linear function over a non-convex set. For example, it helps explain why placing integrality constraints on the variables of what would otherwise be a linear programming problem usually makes it much more difficult to compute optimal solutions.