I am stuck on a homework question, which consists of two parts. I would appreciate any help!
Consider the nonlinear programming problem \begin{equation} \begin{aligned} & \text{minimize } & & f(x) \\ &\text{subject to } & & g(x) \ge 0, \end{aligned} \end{equation}
where $f:\mathbb{R}^n \rightarrow \mathbb{R}$ and $g:\mathbb{R}^n \rightarrow \mathbb{R}^m$ are continuously differentiable. A barrier transformation for a fixed positive parameter $\mu$ gives the problem
\begin{equation} \begin{aligned} & \text{minimize } & & f(x) - \mu \sum_{i=1}^m \ln(g_i(x)). \\ \end{aligned} \end{equation}
a) Show that the first-order necessary optimality conditions for the barrier problem is equivalent to the system of nonlinear equations
\begin{equation} \begin{aligned} \nabla f(x) - \nabla g(x) \lambda = 0 & &\\ g_i(x)\lambda_i = \mu, & & i = 1, \dots, m, \end{aligned} \end{equation}
assuming that $g(x) \gt 0$ and $\lambda \gt 0$ is kept implicitly.
b) Let $x(\mu)$, $\lambda(\mu)$ be a solution to the primal-dual nonlinear equations of (a) such that $g_i(x(\mu)) \gt 0, \; i = 1, \dots, m$, and $\lambda(\mu) \gt 0$. Show that $x(\mu)$ is a global minimizer to the barrier problem if $f$ and $−g_i, \; i = 1, . . . , m,$ are convex functions on $\mathbb{R}^n$.
My progress:
I know that the first-order optimality conditions are
\begin{equation} \begin{aligned} &\nabla \mathcal{L}(x,\lambda) = 0, \text{ where } \mathcal{L}(x,\lambda) = f(x) - \mu \sum_{i=1}^m \ln(g_i(x)) - \mu \sum_{i=1}^m \lambda_i g_i(x) \\ &\lambda \ge 0 \\ &\lambda_i^\top g_i(x) = 0, \; i = 1,\dots,m. \end{aligned} \end{equation}
I have figured that somehow $\lambda_i = \frac{\mu}{g_i(x)}$. Then the first and second optimality conditions are satisfied. However, the third one seems to be satisfied only if $\mu = 0$. Have I interpreted something wrong here? For b), I do not know where to start except to try to prove that the barrier function is a convex function. Would also appreciate help here.
Thanks!