I have a problem which I am unable to solve. If we consider the following problem $\min f(x)$, $G(x) = b$;
where $f$ is in $C^2(R^n)$, and $G$ from $R^n$ to $R^m$ is a $C^2$-function, $G = (g_1,\ldots , g_m)^t$ (transpose) , and $b \in R^m$.
If $x^{*}$ be a local minimum for the problem when $b = 0$, $y^{*}$ be a corresponding Kuhn-Tucker multiplier and suppose that the gradients $g_j(x^{*}), j = 1, \ldots, m$ are linearly independent. If we suppose additionally that the second order sufficient conditions hold at $x^{*}$. Then,
- (a) What are the second order sufficient conditions?
- (b) Prove that for all $b$ with $\|b\|$ sufficiently small, there exist $x(b)$, $y(b)$ which satisfy the necessary first order conditions for a local minimum of the problem. The mappings $b \mapsto x(b)$ and $b \mapsto y(b)$ are $C^1$ in a neighborhood of $0$.
- (c) Prove that ${\nabla}\; b (f(x(b))) = -y(b)$ for the functions defined in (b).
I don't know how to solve this question, especially part (b) and (c). Please help me!