Solving constrained optimization problems with Lagrange multipliers
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 the following exercise.
Part 3: CAS Computation
This portion of the activity guides you through the solution of the constrained
optimization problem boxed above 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?)
Notes from grader
