Optimal output from continuous inputs I have data consisting of a continuous dependent variable and several continuous independent variables. I need to figure out the optimal values of the independent variables that will maximize the dependent variable.  
This might be a pure optimization problem as opposed to a statistics problem, but I can't find any resources on the web that solve this problem.  
Edit: The data is already collected, so this is not a design problem.
Edit: To give some context, let the dependent variable be the amount of drug compound needed to kill an organism. The independent variables are various structural properties of the compound, which can be tweaked as desired. I want to find the "best" structure for the compound- the one that minimizes the dependent variable. These independent variables can be thought of as "knobs". The question is: What value should I set each knob to to minimize the dependent variable?
 A: A problem that comes to mind is that you are probably dealing with complex interactions between compounds, and unless applied carefully standard statistical techniques will likely give misleading answers in this situation.  I would also imagine there are some physical constraints on how the compounds can be varied that a purely statistical analysis wouldn't take into account.
For instance, suppose you fit a model that only included "main effects" such as a logistic regression model of the form $\log[p_i / (1 - p_i)] = \beta_0 + \sum_{i=1}^{n} \beta_i x_i$, where $p_i$ is the probability of death.  Then using only the output from this model you might reach absurd conclusions like "use a huge amount of compound $x_i$ because the coefficient is negative, and nothing else."  That would send your estimate close to zero, but it would probably be a terrible estimate and may not even be a feasible plan.
So if you were to fit a model, I would be particularly worried about interactions and use a model that can detect them automatically such as some popular machine learning algorithms like boosted trees.  Once you've fit this model you could then at least do an exhaustive search over the feasible set of values for the compounds and choose the combination with the smallest estimated probability.
