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I have this huge and ugly function. $$ f\left(x,y\right)= $$
$$ \frac{\left(1-C^xW^y\right)*{(1-C)}^{\left(1-C\right)x}*{\left(1-W\right)}^{\left(1-W\right)y}*C^{Cx}*W^{Wy}}{\left[{\left(1-C\right)}^{\left(1-C\right)x}*{\left(1-W\right)}^{\left(1-W\right)y}*C^{Cx}*W^{Wy}\right]+{[(1-C)}^{Cx}*C^{\left(1-C\right)x}*W^y]+{[C^x*(1-W)}^{Wy}]} $$ Subject to this constraint: $$ Ax+By <= D $$.
$ C, W, A, B$ and $D$ are constants, and variables are $x$ and $y$.
This is only a simplified version of the real function. The real function contains more than two variables (so probably I won't be able to use variable substitution).
Is there any known method to solve (or estimate) this optimization problem (Multivariate nonlinear goal function with linear constraint)?

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1 Answer 1

Firstly, I suggest to apply variable transformation to simplify restricted domain. If your domain is box, or positive orthant, or something like that there are a lot of techniques to solve your problem, and ugliness of function to optimize doesn't matter.

Variable substitutions don't need to be explicit. You can construct it numerically by pseudo-inverting your constraint matrix simultaneously with optimization iterations.

I would like to refer the book Conn, Gould, Toint - Trust Region Methods.

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