[Ipopt] Which optimizer is best for an optimization problem with one variable being constrained by the rest variables?

[Peng Yu]

Sorry, there is a typo in my original email. "min" should be changed
to "max". I have several other changes. Here is the corrected problem.

max f(x), where x satisfies x_n > g(x_1, x_2,...,x_{n-1})

f(x) takes -Inf on the line of x_n = g(x_1, x_2,...,x_{n-1}) and is
undefined when x_n < g(x_1, x_2,...,x_{n-1}).

f is infinitely differentiable in the region of x_n > g(x_1,
x_2,...,x_{n-1}). g is piecewise infinitely differentiable in R^{n-1}.

The region defined by x_n > g(x_1, x_2,...,x_{n-1}) is not convex. I
am not sure if there is a transform of x_1,...,x_{n-1} to make the
region convex (BTW, Is there a way to find such a transform to make
the region convex?)

-- 
Regards,
Peng