Prerequisites:
(Essential) Understanding of level surfaces, gradients, and directional derivatives.
(Desirable) Experience optimizing a function of two variables subject to a constraint
condition satisfied by a curve in the function's domain.
Introduction:
This activity will guide you through a graphical exploration of the method of Lagrange
multipliers for solving constrained optimization problems. The central ideas will be
illustrated with an example similar to the following exercise.
Guide / discussion questions:
(1) Open the exploratory tool. Expand the Choose objects folder from the control panel in the upper right hand corner. Select the Constraint 1 (plane) and Constraint 2 (sphere) checkboxes to add the solutions of the constraint equations to the main canvas. The graphic can be repositioned and reoriented using either a mouse (or touchscreen), or the Rotational controls. In this example, which type of intersection is formed by the plane and sphere?
(2) Select Constraint curve to add the circle of points simultaneously satisfying both constraint conditions to the graphic. How could you have determined that and intersect in a circle (rather than just a single point, or the empty set) without using a computer?
(3) Select Test point to add a distinguished point to the constraint circle. You can move this marked point to any position on the constraint circle using the Test point controls. At any given fixed position on the constraint curve, how many directions are possible for motion which does not leave the constraint curve? (More precisely, suppose is a regular parameterization for some portion of the constraint curve including point , and is a parameter value such that . How many possible directions could the velocity vector have?)
(4) Select the two Tangent to constraint checkboxes to add visual representations of the two “constraint sensitive” directions at the marked test point. What information about the objective function (the function we want to maximize subject to the constraints) would facilitate a determination of whether it will increase or decrease as the test point is moved along the constraint curve?
(5) Select Objective function gradient. What can you say about the directional derivative of at a point in a direction which makes an acute angle with ? What can you say about how will change as long as the direction in which is moved continues to form an acute angle with ?
(6) Select Objective function monitor. Using the Test point controls move the test point (point “”) around the constraint circle. View the left pane graphical representation of how the objective function values change (and/or the textual display on the left side of the top banner) to confirm your answers above. State a relationship between and the tangent line to the constraint curve at that precludes the possibility of attaining an extreme value at subject to the constraintes. Recast your observation as a necessary condition for to attain an extreme value at (subject to the constraints). Is this condition, in general, a sufficient condition?
(7) Define and so that the constraint conditions can be written as , . In particular, the constraint surfaces are level surfaces of the functions and . Give on the intersection of the two constraint surfaces, what's the relationship between and the constraint plane, and what's the relationship between and the constraint sphere?
(8) If attains an extreme value subject to the constraints , at point , must be parallel to and/or ?
(9) Select Normal to plane, Normal to sphere, and Span of constraint normals. Recast your necessary condition for to attain a constrained extreme value at (from step(6)) as a requirement on in terms of and . Formulate this condition in both geometric and linear algebraic terms.
(10) Record the coordinates (as precisely as you can) for the point where achieves its maximum subject to the constraints. What is the constrained maximum value of ?
(11) Select Obj fcn level surf's to display a few select level surfaces of the objective function . What relationship holds between level surfaces of and points where constrained extreme values of occur in this example? (For this step you're advised to unselect objects to reduce clutter, and unselect the Perspective camera to switch to an orthographic view.) Can you generalize this relationship? (Beware. This is a bit tricky.)
