So I need to find the min and max values of f(x,y,z) = x^2 + 2y^2 + 3z^2 given the constraints x + y + z = 1 and x - y + 2z =2. I've gotten as far as (2x, 4y, 6z) = (u,u,u) + (m,-m,2m). I'm stuck trying to solve this system of equations. Any hints?
Write them out: 2x = u+m, 4y = u-m, 6z = u+2m. Solving for x, y and z as functions of u and m is quite easy. Now you need two more equations to determine u and m. Can you guess what they are? RGV
whilst lagrange is a good idea, and the comments above from ehild & Ray are good ideas about where to head - i think a geometric method would be a tiny bit quicker here... though its upto preference i guess, and all the same at the end of the day the intersection of two planes is a line. If you find the equation of that line it reduces to optimising a single variable function.
The use of Lagrange multipliers is crucial in achieving robust, efficient numerical constrained optimization algorithms. Often, the convergence speed of a recursive search algorithm can be increased by orders of magnitude by updating not only the estimates of x,y,z,... but also by intelligently updating the estimates of Lagrange multipliers. In many problems involving inequality constraints, one proceeds by assuming some of the constraints are equalities---the "active set---and ignoring the others; knowing the signs of the corresponding Lagrange multipliers is crucial in checking whether one has the correct active set. Some of the very best constrained optimization codes combine Lagrange multiplier and "penalty" methods---the so-called method of multipliers---to deal with difficult constrained optimization problems. Finally, solving the Lagrangian equations numerically is one of the standard methods to solve constrained optimization problems numerically. So, rather than being an unnecessary distraction, Lagrange multipliers play an extremely important part in modern optimization theory and practice. They should be known by every serious student who will ever do optimization in his/her professional life. That said, it is often the case that the material is not well-taught, and the actual geometric content of the method is perhaps not spelled out the way it ought to be, but that may be more the fault of textbook choice rather than of the material itself. Of course, if the method just appears as a small section in a Calculus course, the crucial role of the method in optimization practice may be missed or not sufficiently emphasized. RGV