0
$\begingroup$

I have the requirements to minimize the following:

$ (f(x)_1 + f(x)_2 + f(x)_3) $

where:

$ f(x)_1 = y_1 - (\exp(b+m_1) \times x) $ $ f(x)_2 = y_2 - (\exp(b+m_2) \times x) $ $ f(x)_3 = y_3 - (\exp(b+m_3) \times x) $

given the range of $x$:

$ a = 1.191206112 $ $ b = 1.321909214 $ $ x \in R \space|\space a \le x \le b $

Is there a way to estimate the value of $x$ that returns the minimized sum of the three functions? As you can see, my $y$ and $m$ values are specific to the function but the $b$ is constant across all three.

Currently, I am testing random values between $[a,b]$ and recording the smallest sum. This takes about 50,000 iterations before I start approaching the asymptote. There has got to be a better way!

  • 0
    But $x$ must fall between the range stated. At the moment I test all values between the range min/max until I find the best value.2012-08-25

1 Answers 1

2

What about finding zeroes of the derivative of the sum of your three functions to find a candidate for the minimum? Are you sure the $x$ does not appear in the argument of the exponential function? If this is really not the case and if the third function is really of the same form as the first two, then, since $\exp(...)$ is always positive, the minimum is at the right end of the interval for $x$. Am I missing something?