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I have two functions $f(x,y)$ and $g(x,y)$ whose sum/product (whichever is possible) is to be minimized. The values of $x,y$ can vary in the interval $0<x,y<1$ (hence none of them can have a value of $1$ or $0$). The range of function $f(x,y)$ is from $0$ to $1$ while function $g(x,y)$ can have any value in the real set.

For function of two variables I know one technique called the Biconvex optimization. I can apply this technique if both of the functions are Biconvex however one of my function is Biconvex (namely $g(x,y)$) while the other is not Biconvex hence I can not apply this technique here. Are there other techniques which I can use for these types of optimization problems? Any suggestions will be highly appreciated.

Thanks in advance.

Edit: After reading this post (Optimization of a function of two variables.) I am quite confused about my statement of Biconvex optimization please shed some light over it also.

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Frank Moses
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    It seems that you try to find a minimum of a function over an open set, therefore you have no guarantee that it exists. – Tanj Jun 23 '16 at 11:26
  • @Tanj and why is that? can you provide some reference? Is it ok to change the sets from $(0,1)$ to $[0,1]$. P.S.: In my intuitive thinking I know that $f(x,y)$ increases with increasing $x,y$ while $g(x,y)$ decreases with increasing $x,y$. The function themselves are quite complex to show analytically that this intuition is true. Any comment on this? Thanks – Frank Moses Jun 24 '16 at 03:08
  • This comes from the difference between a minimum and an infimum. In most of the cases, the "minimum" of your function will be attained at the boundary of your domain. Therefore, if you try to minimize over an open set there will be some issues. – Tanj Jun 24 '16 at 08:00

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