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I have to develop an optimizer for a simulation. There are 7 reference values $ r_1, r_2,\ldots,r_7 $ (certain values which are expected to show up) and 7 corresponding actual values $ a_1,a_2,\ldots,a_7,$ the simulations's results. The deviations between reference values and actual values are listed as one single value, the sum of squares of all deviations. $ f(x)=\sum_{i=1}^{7}(r_i - a_i)^2=\sum_{i=1}^{7}e^2=||e||^2 $ where $x_1,x_2,\ldots,x_7 $ denote the input values.

Given this, I can calculate an arbitrary number of results $f(x)$ for many input vectors $x$, while the results are scalar values. Obviously I want to minimize the deviations. My minimization method makes use of the gradient. How can I calculate the partial derivatives to build up the gradient without knowing the function?

I only know the input values and the result, but the relationships between input values is completely unknown!

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    From your question, I'm not clear on whether $x = (a_1,...,a_7)$ or whether it also includes the 7 $r$ values as well? Also, what is the context of the problem: is it to find the "best" simulator, i.e. the simulator that gives the least deviation from reality over a wide range of inputs? If this is the problem, then I would expect you to be asking how to tune the simulator, i.e. optimise over a number of parameter variables. From your question, it seems you are trying to optimise over the inputs $x$, but that will only tell you the points over which the simulator is most accurate.2012-12-19

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You can assume:

$\nabla f(\vec x) = (\frac{\partial f}{\partial x_1},\cdots, \frac{\partial f}{\partial x_7}) \approx (\frac{f(x_1+h)-f(x_1)}{h},\cdots,\frac{f(x_7+h)-f(x_7)}{h})$

$h$ needs not to be a vector. Then, you can use the information given by the gradient to select the variable(s) to change. For example, adopting a steepest descent you may apply a change proportional to the negative of the (approximate) gradient of the function at the current point (see this Wikipedia entry, for example).

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    @bogus: If you use different values of $h$, you lose the benefits of being able to compare the components of the gradient to decide on the direction of steepest descent.2012-12-19