Math 3298 - Hampton - Lab 10 - Due 12/13

Newton's method in higher dimensions (from a Lagrange multiplier problem)

Suppose we wish to maximize a somewhat complicated function such as:

f[x_,y_]:=Exp[-y^2-x^4+x+1]-y^2*x^2-(x-1)^4

...over the solutions to a somewhat complicated constraint g=0 such as:

g[x_,y_]:=x^5-x y^4-y*x+1

Real-world problems can be much, much worse than this, but already it is impossible to find an explicit solution symbolically.  The simplest way to handle these situations is to use Newton's method (or some variant of it; here we will stick to the vanilla version).

Exercises

1. Make a combined contour plot of f[x,y] and the constraint g[x,y]==0.  This should give you some idea of where the maxima might be.  How many local maxima do you expect there are on the constraint curve?

2. Use the Lagrange multiplier method to get three equations for the maxima and minima of f[x,y] on g[x,y]==0.  Eliminate the Lagrange multiplier to obtain two equations in x and y, h[x,y] == g[x,y] == 0.  Construct the equations for the tangent planes to h[x,y] and g[x,y] at a point (x1,y1) (your initial guess).  These tangent planes each intersect z = 0 in a line.  These two lines intersect at (x2,y2).  Use Mathematica to find (x2,y2) as a function of (x1,y1).  

3. Use the function from part 2 to find some maxima.  What is the global maximum for this problem?  Were you correct in your guess from part 1?