# 2.8 Minimizing a Quadratic Polynomial

In this section, we consider how to minimize quadratic polynomials. This problem is equivalent to that of maximizing a polynomial, since any maximum of a quadratic polynomial *p* occurs at a minimum of the quadratic polynomial –*p*.

Recall from elementary calculus that any minimum on of a differentiable function *f* : → occurs at a point *x* at which *f* ′(*x*) = 0. Generalizing to multiple dimensions, any minimum on ^{n} of a differentiable function *f* : ^{n} → occurs at a point * x* at which

*f*(

*) = 0.*

**x**A quadratic polynomial *p*: →

[2.89]

has a unique minimum if and only if *c* is positive, in which case the minimum occurs at the point

[2.90]

This generalizes in an intuitive manner to multiple dimensions. A quadratic polynomial *p*: →

[2.91]

has a unique minimum if and only if ** c** is positive definite, in which case the minimum occurs at the point

[2.92]

Compare [2.92] with [2.90]. This is another situation in which positive definite matrices play a role analogous to positive numbers. To understand our result more intuitively, consider Exhibit 2.9 It shows graphs for three quadratic polynomials of the form [2.91] from ^{2} to . The first one has a positive definite matrix ** c**. Both of its eigenvalues are positive, and the polynomial has a minimum. The second polynomial has a matrix

**with mixed eigenvalues. One is positive and the other is negative. The polynomial has a saddle shape, so it has neither a maximum nor a minimum. The third polynomial has a negative definite matrix**

*c***. Both of its eigenvalues are negative, and the polynomial has a maximum.**

*c***. It achieves a minimum. The second has matrix**

*c***with mixed eigenvalues—one is positive and the other is negative. It achieves neither a maximum nor a minimum. The third has a negative definite matrix**

*c***. It achieves a maximum.**

*c*###### Exercises

Similar to the graphs of Exhibit 2.9, sketch a graph for a quadratic polynomial from ^{2} to that has a positive semidefinite matrix ** c**. (Hint: Depending upon your solution, your polynomial will either have no minima or infinitely many.)

Solution

Repeat Exercise 2.11 assuming ** c** is negative semidefinite.

Solution

Consider the quadratic polynomial *p*: ^{3} → :

[2.93]

- Express the polynomial in matrix form [2.91]. Make sure your matrix
is symmetric.*c* - Apply the Cholesky algorithm to determine if your matrix
is positive definite.*c* - Solve for the point
indicated by [2.92].*x* - What is the polynomial’s value at the point
obtained in item (c)?*x* - Is your solution a maximum, minimum, or saddle point?