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Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.
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Find an optimizer for $g(x,y)$ if it exists
By conditions 2 and 3, $f(t,t)=t$ for all $t>0$. By continuity, $f(0,0)=\lim_{t\to 0^+} f(t,t)=\lim_{t\to 0^+}t=0$. From this it follows that $g(0,0)=0$.
As you said, if $g$ has a maximizer, then the …