any ideas?
the problem is
Let c∈R, f:R^n →R and h:R^n →R. Suppose that X⊆R^n.
Consider the following two constrained optimisation problems:
I) Find x∈R^n to maximise f(x) subject to the constraint h(x)=c.
II) Find x∈X to maximise f(x) subject to the constraint h(x)=c.
a) Prove that if x* solves I and x*∈X, then x* solves II.
b) Suppose X=R^n(+). Provide a counter example to the following (false) claim: "if x* solves II and x(i)*>0 for each i∈{1,2,...,n}, then x* solves I.
thanks in advance
a) Suppose does solves I but does not solve II. Then there exists with and . Since is feasible to I, does not solve I which is a contradiction.
b) Think of a single variable function which has a peak at some positive value of x but has a larger peak for some negative value of x. Like this guy -x*(0.5x-1)*(x-2)*(x+1 - Wolfram|Alpha)