### Optimization: Extreme points

### Hessian convexity criterion

The *Hessian matrix* gives an excellent sufficient for a bivariate function to be convex. Before discussing it, we give a general result about symmetric #(2\times2)#-matrices. Recall that a #(2\times2)#-matrix is **symmetric** if the two non-diagonal entries are equal.

A point #u# of the plane will be viewed as a row vector, with #u^{\top}# as the corresponding column vector.

Positive semidefinite 2 by 2 matrices

The following three statements regarding a symmetric #2\times2#-matrix #H=\matrix{h_{11}&h_{12}\\ h_{12} &h_{22}}# are equivalent:

- For every row vector #u# of length two, #{u} H\,{u}^{\top}\ge0#.
- #h_{11}\ge0#, #h_{22}\ge0#, and #\det(H) =h_{11}\cdot h_{22}-h_{12}^2\ge0#.
- For all numbers #x#, #y#, we have #h_{11}x^2+2h_{12}x\cdot y + h_{22} y^2\ge0#.

A symmetric matrix with these properties is called **positive semidefinite**.

The inequality in the first statement is strict for all nonzero vectors #{u}# if and only if all other inequalities in statements 2 and 3 are strict. In this case, the matrix #H# is called **positive definite**.

In order to establish a convexity criterion for #f# in terms of second partial derivatives, we use the Hessian matrix \[ H_f = \matrix{ f_{xx} &f_{xy}\\ f_{yx}&f_{yy}}\] In fact, #H_f# is a bivariate function. For a point #v=\rv{v_1,v_2}# of #\mathbb{R}^2#, we will write \[\left.H_f\right|_{v} = \matrix{ f_{xx}(v) &f_{xy}(v)\\ f_{yx}(v)&f_{yy}(v)}\] for its value at #v#.

Since the boundary of a domain may have global minima of a function that are not stationary points, we restrict ourselves to **domains** consisting of the interior of the domain and points on the boundary of the interior. The interior of a domain consists of all points that are the center of a disk that is fully contained in the domain. Domains of which each point is the center of a disk that lies in the domain are called **open**. But we also allow that domains have point on the boundary of the **interior**: these are point #p# outside the interior with the property that each disks having center #p# has points in the interior of the domain. A typical example of an open domain is the positive quadrant consisting of all points #\rv{x,y}# such that #x\gt0# and #y\gt0#. If we all points on the boundary of this domain, we obtain the domain consisting of all #\rv{x,y}# scuh that #x\ge0# and #y\ge0#.

Hessian convexity criterionSuppose that #f# is a function on a convex domain #S# all of whose first and second partial derivatives exist and are continuous. Then #f# is convex on #S# if and only if the Hessian matrix \(\left.H_f\right|_{v}\) of #f# at each point #v# of #S# is positive semidefinite.

The requirement that the Hessian matrix be positive semidefinite on the domain of #f# can be verified by use of the above theorem *Positive semidefinite 2 by 2 matrices*. Putting together these results with the theorem *From stationary points to global extrema*, we find

Global minima of convex functions

Let #f(x,y)# be a twice differentiable bivariate function with continuous second order derivatives defined on an open convex domain #S# in #\mathbb{R}^2#.

- If for all #\rv{x,y}# in #S#, \[f_{xx}(x,y)\leq 0, f_{yy}(x,y)\leq 0, \text{ and } f_{xx}(x,y)\cdot f_{yy}(x,y)-(f_{xy}(x,y))^2\geq 0\] then every stationary point of #f# is a global maximum.
- If for all #\rv{x,y}# in #S#, \[f_{xx}(x,y)\geq 0, f_{yy}(x,y)\geq 0, \text{ and } f_{xx}(x,y)\cdot f_{yy}(x,y)-(f_{xy}(x,y))^2\geq 0\] then every stationary point of #f# is a global minimum.

Examples will be given shortly.

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