Constrained Optimization Algorithm

Hi all,

I am facing the following Optimization problem. I'll post an example
with a simple case.

I have 2 variables, x and y.

I want to maximize (f(x) + g(y))/2 while keeping x + y <= 10.

I know this is a NLP problem.
I have tried to use the Newton-Raphson algorithm but I dont' like it
because it highly depends on the initial chosen point.

I'll eventually have to expand this problem to thousands of variables
(as opposed to having only 2 variables: x and y).

Anyone know which I NLP algorithm I should use and where I can find
resources about that particular algorithm?

Thanks
.

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