Consider the problem
V(q) = max ( - ||x|| n1/2 ) subject to [ w(x1) + ... + w(xn) ] / n ≤ q
where w is the function w(z) = |z-1| + 3|z-2| + ez-1 - 4 and q is a number near zero (to be explained below).
Observe that the problem is to find the admissible point which is closest to the origin. Note furthermore that w(1)=0 and that
w'(z)= sign|z-1| + 3 sign|z-2| + ez-1 so that
w'(1-)= -3
w'(1+)= -1
w'(2-)= e-2 (which is >0)
w'(1-)= -3
Now the "small" |q| should be interpreted as follows: you do not need to worry about the point (2,2,...,2)T and you need not worry about the admissible set degenerating to a singleton or to empty.
If you think the following is hard with general n, do it for n = 2. (Then you might try it for general n afterwards!)
Questions: In parts (a) to (c) you can take for granted that the solution is of the form x= k (1,1,...,1)T , some k > 0.
(a) For q>0 we can choose a k<1 (and we will choose k as close to zero as possible!). Find the limit V'(0+).
(b) For q<0 we must choose a k>1 (because (1,1,...,1)T is not admissible as w(1)=0). Find the limit V'(0-).
(c) What are the possible values of the Lagrange multiplier when q=0?
(Note that the usual uniqueness of the Lagrange multiplier fails when w is not differentiable.)
(d) Verify that x= (1,1,...,1)T does indeed solve the problem when q=0.