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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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    Which minimization method are you using? If your simulation runs reasonably fast, you may be able to approximate the gradient numerically using a suitable adaptation of the Newton's formula $\frac{f(x+h) - f(x)}{h}$ for a choice of a small number $h$. Nonetheless, I think in this case a derivative-free method may be more efficient and reliable.2012-11-26
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    Let's assume that I add $$10^-4$$ to each element of the input vector x. x is not a single value but a vector vector holding the input values, so we can as well write it as $$ f(x_1, x_2, \ldots,x_7)$$. So h is $$h=(10^{-4}, 10^{-4},\ldots, 10^{-4})$$ (7 elements). What does the fraction then look like, what's the h in the denominator?2012-11-28
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    I have given some details in an answer.2012-11-28
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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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