Constrained optimization exploratory activity

Solving constrained optimization problems with Lagrange multipliers:

Part 3: CAS Computation

Introduction:

This portion of the activity guides you through the solution of the constrained optimization problem boxed below using MATLAB^®. Similar procedures can be used in other computer algebra systems.

Guide:

(1) Open MATLAB^® and prime the Symbolic Toolbox with a symbol declaration.

>> syms x y z lambda mu

(2) Define the objective function, and the functions appearing in the constraint equations.

>> f =
>> g =
>> h =

(3) Define the Lagrangian for our example problem.

>> lagrangian = f - lambda*g - mu*h

(4) Encode the necessary condition for constrained extrema as a single vector equation. (Don't confuse the double equal sign == [equality] with the single equal sign = [assignment].)

>> eqn = ( gradient(lagrangian, [x y z lambda mu]) == [0; 0; 0; 0; 0] )

(5) Solve the system.

>> candidates = solve( eqn, [x y z lambda mu] )

(6) Extract the spacial coordinates from the solutions (and peek at numerical approximations).

>> [ candidates.x candidates.y candidates.z ]
>> vpa( ans )

(7) Evaluate at the candidate points.

>> subs( f, [x y z], [candidates.x(1) candidates.y(1) candidates.z(1)] )
>> vpa( ans )
>> subs( f, [x y z], [candidates.x(2) candidates.y(2) candidates.z(2)] )
>> vpa( ans )

(8) Where is the constrained maximum of achieved and what is its value? (Is your answer here consistent with your answer to question (10) from Part 2?)

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