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I have the following constraint functions:

$$g_{i_{min}} \leq y_{i+1}-y_{i} \leq g_{i_{max}}$$

$$y_{i_{max}}-y_{i} \geq h_{i}$$

$$v_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{3} (h_{i}+a)^{4}}{(h_{i}+b)^{4}}\Biggr] \leq v_{i_{max}}$$

$$Q_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{3} (h_{i}+a)^{10}}{(h_{i}+b)^{4}}\Biggr]$$

where $i=1,2,3$

The variables for minimization are $y_i$ and $h_i$. Note that $y_i$ are continuous and $h_i$ are discrete.

All $g_{i_{min}}, g_{i_{max}}, Q_{i_{min}},v_{i_{min}},v_{i_{max}}, y_{i_{max}},a,b$ are constant.

There are two objectives functions to minimize against:

$$f_{1}=\sum_{i=1}^3 h_i \quad\quad \text{(1)}$$

$$f_{2}=-\sum_{i=1}^4 y_i \quad\quad \text{(2)}$$

The idea is that $(1)$ should always be minimized first, and in the case of a draw, minimize $(2)$.

Any idea how to tackle this question?

On a second thought, I think for my application, I can combine the two objective functions into a single function. So my question how is is there anyway to minimize a set of nonlinear constraint equations, which some of the variables are discrete, and some are continuous?

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    Are all variables and constants positive? If so, you could try simplifying the last two constraint functions by raising to the 3rd power and multiplying throughout by $(h_i + b)^2$. I don't know if that'll get you any closer to a solution, but the equations will look less intimidating, so you may get more people willing to look at it.2010-11-26
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    @Rahul, I've manipulated the equation a bit to get rid of fractional power. But as far as I can tell it is no less daunting then the previous version.2010-11-27

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