# Thread: Tricky constrained maximisation problem

1. ## Tricky constrained maximisation problem

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.

3. a) Suppose $x^*$ does solves I but does not solve II. Then there exists $\hat{x} \in X$ with $h(\hat{x}) = c$ and $f(\hat{x}) > f(x^*)$. Since $\hat{x}$ is feasible to I, $x^*$ does not solve I which is a contradiction.