How can I convert this Min function (bold highlihted) in objective problem into simple mathematical equation:

Min a[i]*(x[i]-y[i]) + b[i] min⁡(x[i],M-c[i])

s.t. c[i]-x[i]+y[i]≤M

Here x[i] and y[i] are decision variables.

asked 20 Feb, 05:58

fadiz's gravatar image

fadiz
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accept rate: 0%

edited 01 Mar, 13:41

fbahr's gravatar image

fbahr ♦
4.6k716

What are the bounds on the decision variables x and y? are they greater than or equal to zero?

Are a, and b greater than or equal to zero too?

(20 Feb, 09:36) Naveen Divak...

Both x and y are greater than or equal to zero.

a, and b are also non-negative inputs

(22 Feb, 04:45) fadiz

Replace \(\min(x_i, M-c_i)\) with a new variable \(z_i\). Also add a binary variable \(w_i\). Assuming that \(b_i>0\), constrain \(z_i\) as follows: \begin{align*} z_{i} & \ge x_{i}-Kw_{i}\\ z_{i} & \ge M-c_{i}-L(1-w_{i}) \end{align*} where \(K\) is a valid upper bound for \(x_i - M + c_i\) when \(M - c_i < x_i\) and \(L\) is a valid upper bound for \(M - c_i - x_i\) when \(M - c_i > x_i\).

This forces \(z_i\) to be either \(x_i\) or \(M - c_i\); the assumption that \(b_i > 0\) combined with the fact that the overall objective is minimization ensures that the solver will pick whichever choice is smaller (i.e., the minimum).

If \(b_i < 0\), you don't need the binary variable \(w_i\). Just constrain \(z_i\) to be less than or equal to both \(x_i\) and \(M - c_i\).

link

answered 21 Feb, 16:59

Paul%20Rubin's gravatar image

Paul Rubin ♦♦
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accept rate: 19%

edited 21 Feb, 17:02

Thanks Paul for such valuable information.

What if I have conditional variable such as z[i]:

Min a[i]*(x[i]-y[i]) + b[i] z[i]

where

if c[i] > M and x[i] ≥ c[i] - M

z[i] = M - c[i]

elseif c[i] > M and 0 ≤ x[i] ≤ c[i] - M

z[i] = x[i]

else

z[i] = 0

endif

where x[i] and y[i] are non-negative variables and a[i], b[i], c[i] and M are positive inputs.

Can it be converted into linear constraints?

(22 Feb, 09:13) fadiz

So in the first case you want z[i] to be negative, in the second case you want z[i] to be nonnegative, and if M - c[i] >= 0 you want z[i] to be zero?

(04 Mar, 19:55) Paul Rubin ♦♦
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Asked: 20 Feb, 05:58

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Last updated: 04 Mar, 19:55

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