Q3: Consider the constrained nonlinear program $$ \begin{array}{11} \min & f\left(x_{1}, x_{2} ight)=\frac{1}{2} x_{1}^{2}+\frac{1}{2} X_{2}^{2}-x_{1}+x_{2} \ \text { s.t. } & x_{1}, x_{2} \leq 0 \end{array} $$ (a) Write down the log barrier function $B_{k}(\mathbf{x})$ with parameter $k$. (b) Calculate the stationary points $\mathbf{x}^{k}=\left(x_{1}^{k}, X_{2}^{k} ight) $ for $B_{k}(\mathbf{x})$ and hence find $\mathbf{x}^{*}=\lim _{k ightarrow \infty} \mathbf{x}^{k}$. (c) For each stationary point, write down an estimate $\boldsymbol{\mu}^{k}$ of the optimal Lagrange multiplier vector $\boldsymbol{\mu}^{*}$, and find $\boldsymbol{\mu}^{*}=\lim _{k ightarrow \infty} \boldsymbol{\mu}^{k} .$ CS.Vs. 1377