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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. –  Rahul Nov 26 '10 at 19:05
    
@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. –  Graviton Nov 27 '10 at 6:59

1 Answer 1

This is not a complete answer. I'm assuming that you also have non-negativity constraints on all variables.

After looking at your question more carefully, I rearranged it to the following constraints: $$\begin{align} y_{i+1} - y_i & \in [g_{i_{\min}}, g_{i_{\max}}], \\ y_i & \le y_{i_{\max}} - h_i, \\ y_{i+1} - y_i & \in \left(1 + \frac{c}{h_i+b}\right)^{4/3}[w_{i_{\min}}, w_{i_{\max}}], \\ y_{i+1} - y_i & \ge R_{i_{\min}} \left(1 + \frac{c}{h_i+b}\right)^{4/3}\frac{1}{(h_i+a)^2},\end{align}$$ where $c = b - a$, $w_{i_{*}} = v_{i_{*}}^{1/3}$, and $R_{i_{\min}} = Q_{i_{\min}}^{1/3}$. Hopefully the abuse of notation in the third equation is understandable.

Now all the $y_i$ are on the left side of the constraints and the $h_i$ are on the right. This form should be easier to reason about. For example, I was initially concerned that the optimal $f_1$ might always be $0$. Note that all the right-hand sides get larger as the $h_i$ get smaller, so a solution with all $h_i = 0$ would only fail to be feasible if one of the lower bounds for $y_{i+1} - y_i$ became too large, i.e. incompatible with one of the upper bounds.

Also, since all the constraints except one involve only the differences between the $y_i$, you can decrease all $y_i$ by the same amount and still have a feasible solution, as long as none of them fall below zero. So when you get an optimal solution for $f_1$, you can subtract $\min\{y_j\}$ from $y_i$ and get the solution that minimizes $f_2$ for fixed $h_i$. That's not the same thing as only fixing $\sum h_i$, but it's something...

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I'm not sure how your proposal solve the problem; if I'm not mistaken, you are saying that we try to optimize for $f_i$ first, but how? –  Graviton Nov 27 '10 at 9:29
1  
@Ngu, can't you just use any standard nonlinear optimization software? Anyway, here's something else I thought of. You have two lower bounds on $y_{i+1}-y_i$ which depend on $h_i$; you also two upper bounds on the same quantity. All the $h_i$ seem to be effectively independent, so you can find the smallest $h_i$ that doesn't make any of the lower bounds exceed any of the upper bounds. I expect that should give you the optimal solution for $f_1$, but in case I overlooked something you may want to check it by computer as well. –  Rahul Nov 27 '10 at 18:35

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