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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!

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    Shouldn't the minimum always be $f(b)_1 + f(b)_2 + f(b)_3$ ? Looks like there is a missing $x$...2012-08-23
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    You have so many variables here...For which values are you seeking the solution (e.g. x or yi or b or mi)?2012-08-23
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    @EmmadKareem I am looking to find $x$ in the three functions (single value), that returns the smallest possible sum of $f(x)_1$+$f(x)_2$+$f(x)_3$2012-08-24
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    Looks like the optimal value of x: x1=x2=x3=b. this way, each f is minimized, and hence the sum is minimized.2012-08-24
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    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

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