how to minimize the barrier function?

Question

suppose we have a barrier function to minimize:
$$ \frac{1}{3}(x+1)^3+y-r(-\frac{1}{x+1}-\frac{1}{y})$$ r will be decrease till to nearly 0 after several iteration we take 0.0001 as initial value and initial feasible value $X_1=[-2,1]^T$
I tried steepest descent:$X_2=X_1+\lambda^*S_1$
lambda star is step size it should be find in that way:
$\lambda^*= minimize\ f(X_1+\lambda^*S_1)$ and find the optimal $X^*$
$S_1=-\Delta f(X_1)$
in this case:
$\frac{\delta f}{\delta x}=(x + 1)^2 + \frac{-0.0001}{(1 - x) ^ 2}$
$\frac{\delta f}{\delta y}=1-\frac{0.0001}{y^2}$
and
$minimize\ f(X_1+\lambda^*S_1)=\frac{1}{3}((-2+x*-1)+1)^3+(1+x*-1)-0.0001(-\frac{1}{(-2+x*-1)+1}-\frac{1}{(1+x*-1)})$
if u try to minimize the above function to find $\lambda$ it will be infinity because this function is not convex or concave function so it don't have global minimal

Is it steepest descent not works in this barrier function?

which unconstrained minimization method can handle this kind of problem?

since this barrier function don't have Hessian matrix method such as Newton's method it won't works in this case