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?