338 ACTIVITY 19: Optimization of Functions of Several Variables

WHY:

This activity is designed to review the theory and techniques for finding stationary points for functions of several variables and introduce some new techniques for deciding when the stationary points are actually optimal points. The concepts of Hessian matrix, Quadratic Form, and definiteness will be important during the rest of the course. Since many problems encountered in practice are nonlinear, the information learned today will be useful in your careers.

VOCABULARY:

Stationary Point
X is a stationary point for a function f(X) if all partial derivatives of f are zero at X.

Hessian Matrix
The Hessian matrix H associated with a function f(X) is the nxn matrix of second order partial derivatives of f. The ith row of H holds the partial derivatives of the ith component of the gradient vector. If f is twice continuously differentiable, then H is symmetric.

Quadratic Form
If A is a symmetric nxn matrix then the function f(X) = XAXT is called a Quadratic Form

Definiteness 
						(  positive definite
If  A  is a symmetric  nxn  matrix then  A  is  |  positive semidefinite
						|  negative definite
						(  negative semidefinite  
		  (  >  0
	if XAXT   |  >= 0   for all nonzero X.
		  |  <  0
		  (  <= 0   
	Otherwise,  A is said to be indefinite.


Optimal Point
A function f(X) has a relative minimum (maximum) at a stationary point Xo if H(Xo) is positive (negative) definite.

LEARNING OBJECTIVES:

  1. Discover how to compute the Hessian matrix.
  2. Discover how to find stationary points and determine if they are relative maxima, relative minima, or saddle points.
  3. Discover that the team is responsible for helping each member to learn.

PERFORMANCE CRITERIA:

  1. Quality of the answers to the Critical Thinking Questions.
  2. Level of understanding of each member of the team.

RESOURCES:

  1. Section 4.4, Strategic Mathematics
  2. 30 minutes

MODEL:

A manufacturing company operates its two available machines to polish its metal products. The two machines are equally efficient, although their maintenance costs are different. The company's daily maintenance and operating costs are given in dollars as:

f(x1, x2) = 60 - x1 - 1.2x2 + 0.05x12 + 0.1x1x2 + 0.07x22

where x1 and x2 are the hours of operation of the two machines. Find the number of hours to operate each machine in order to minimize daily cost.

First find the gradient vector and set it to 0 to find the stationary points:

grad f(x1, x2) = (-1 + 0.1x1 + 0.1x2, -1.2 + 0.1x1 + 0.14x2)

Solving the equations: 0.1x1 + 0.1x2 = 1 and 0.1x1 + 0.14x2 = 1.2 simultaneously

we get 0.04x2 = 0.2 or x2 = 5, and x1 = 5.

Now, to check that (5, 5) is a relative min, we compute the Hessian and verify that it is positive definite:

		    | 0.1  0.1 |
          H(5, 5) = | 0.1  0.14|.

Reducing H to REF we first divide h11 by 0.1, so d11 becomes 0.1. After putting column one in REF we get 


			| 1    1 |
			| 0  0.04|

Now divide h22 by 0.04, so d22 = 0.04. Since the diagonal matrix associated with the Hessian has positive entries along the diagonal, it is positive definite since: 


	      | 0.1    0 |[ h1  h2 ]
     [h1  h2] |  0  0.04 |
				
		=  0.1h12 + 0.04h22  
		> 0  when (h1, h2) <> (0, 0).

Thus the stationary point is a relative minimum.

CRITICAL THINKING QUESTIONS:

These will be provided in class.

SKILL EXERCISES:

  1. Problem 4.8 on page 164.

Math 338 Activity 19 -- Revised 11/14/98

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