the problem is
Let \(\displaystyle c\in{R}\), f:R^n ?R and h:R^n ?R. Suppose that X\(\displaystyle \subseteq{R}^{n}\).
Consider the following two constrained optimisation problems:
I) Find x\(\displaystyle \in{R}^{n}\) to maximise f(x) subject to the constraint h(x)=c.
II) Find x\(\displaystyle \in{X}\) to maximise f(x) subject to the constraint h(x)=c.
a) Prove that if x* solves I and x*\(\displaystyle \in{X}\), 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\(\displaystyle \in\) {1,2,...,n}, then x* solves I.
thanks in advance
Let \(\displaystyle c\in{R}\), f:R^n ?R and h:R^n ?R. Suppose that X\(\displaystyle \subseteq{R}^{n}\).
Consider the following two constrained optimisation problems:
I) Find x\(\displaystyle \in{R}^{n}\) to maximise f(x) subject to the constraint h(x)=c.
II) Find x\(\displaystyle \in{X}\) to maximise f(x) subject to the constraint h(x)=c.
a) Prove that if x* solves I and x*\(\displaystyle \in{X}\), 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\(\displaystyle \in\) {1,2,...,n}, then x* solves I.
thanks in advance
