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Topic: find minimum of a function with abs and squares analytically
Replies: 3   Last Post: Dec 23, 2012 5:19 PM

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Posts: 1
Registered: 12/23/12
find minimum of a function with abs and squares analytically
Posted: Dec 23, 2012 12:08 PM
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I want to find the analytical minimum 'x_opt=argmin(x)' of the following function:

f(x) = alpha * |c + x| + beta * x ^ 2

where x is a real number (x is_element_of R), c is a real constant (c is_element_of R), alpha and beta are positive real constants (alpha is_element_of R+), beta is_element_of R+), |?| is the absolute value function and ^ is the power function.

Looks simple, but the absolute value function makes it somewhat tricky. As already mentioned, i want to find the solution to this minimization problem analytically, not numerically.

I managed to split the optimization up according to the three cases x < -c, x = -c, x > -c, and solve each case separately analytically (by setting the first derivative to zero).

For the three cases I have now the solutions x_opt = alpha/(2*beta) [for x < -c], x_opt = -c [for x = -c], and x_opt = -alpha/(2*beta) [for x > -c]. But how to 'combine' these solutions now to get the solution 'x_opt' (as a function of 'x') ?

So i would need a function 'phi(x)' which delivers me 'x_opt' for a given x, 'phi(x) = argmin f(x)'. How does phi(x) look like ?

thx in advance for any advice.

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