Convex Function - Convex Functions Our final topic in this...

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Convex Functions Our final topic in this first part of the course is that of convex functions. Again, we will concentrate on the situation of a map f : R n R although the situation can be generalized immediately by replacing R n with any real vector space V . We will state many of the definitions below in this more general setting. We will also find it useful, and in fact modern algorithms reflect this usefulness, to consider functions f : R n R where R is the set of extended real numbers introduced earlier. Before beginning with the main part of the discussion, we want to keep a couple of examples in mind. The primal example of a convex function is x mapsto→ x 2 ,x R . As we learn in elementary calculus, this function is infinitely often differentiable and has a single critical point at which the function in fact takes on, not just a relative minimum, but an absolute minimum . Figure 1: The Generic Parabola The critical points are, by definition, the solution of the equation d dx x 2 = 2 x or 2 x = 0. We can apply the second derivative test at the point x = 0 to determine the nature of the critical point and we find that, since d 2 dx 2 ( x 2 ) = 2 > 0, the function is ”concave up” and the critical point is indeed a point of relative minimum. That this point gives an absolute 1
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minimum to the function, we need only see that the function values are bounded below by zero since x 2 > 0 for all x negationslash = 0. We can give a similar example in R 2 . Example 1.1 We consider the function ( x,y ) mapsto→ 1 2 x 2 + 1 3 y 2 := z, whose graph appears in the next figure. –2 –1 1 2 z –4 –2 2 4 y –2 –1 1 2 x Figure 2: An Elliptic Parabola This is an elliptic paraboloid. In this case we expect that, once again, the minimum will occur at the origin of coordinates and, setting f ( x,y ) = z , we can compute grad ( f ) ( x,y ) = x 2 3 y , and H(f(x , y)) = 1 0 0 2 3 . Notice that, in our terminology, the Hessian matrix H ( f ) is positive definite at all points ( x,y ) R 2 . The critical points are exactly those for which grad [f(x , y)] = 0 whose only solution is x = 0 ,y = 0. The second derivative test is just that det H(f(x , y)) = parenleftbigg 2 f x 2 parenrightbiggparenleftbigg 2 f y 2 parenrightbigg 2 f x y > 0 2
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which is clearly satisfied. Again, since for all ( x,y ) negationslash = (0 , 0) ,z > 0, the origin is a point where f has an absolute minimum. As the idea of convex set lies at the foundation of our analysis, we want to describe the set of convex functions in terms of convex sets. We recall that, if A and B are two non-empty sets, then the Cartesian product of these two sets A × B is defined as the set of ordered pairs { ( a,b ) : a A,b B } . Notice that order does matter here and that A × B negationslash = B × A !
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