Convex Function

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 !

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern