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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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Ray Koopman

Posts: 3,383
Registered: 12/7/04
Re: find minimum of a function with abs and squares analytically
Posted: Dec 23, 2012 4:19 PM
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On Dec 23, 9:08 am, wrote:
> Hi,
> 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.

Plot alpha*|c+x| and beta*x^2 on the same graph.
One minimum is at -c, the other is at 0.
The minimum of the sum of the two functions
is between their respective minima.

x_opt = sgn(-c)*min(alpha/(2*beta),|c|)

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