FarkasDual


FarkasDual

Type: double
Modifiable: No

Together, attributes FarkasDual and FarkasProof provide a certificate of infeasibility for the given infeasible problem. Specifically, FarkasDual can be used to form the following inequality from the original constraints that is infeasible within the bounds of the variables:

<span>$</span>\lambda^tAx \leq \lambda^tb.<span>$</span>

This Farkas constraint is valid, because <span>$</span>\lambda_i \geq 0<span>$</span> if the <span>$</span>i<span>$</span>-th constraint has a <span>$</span>\leq<span>$</span> sense and <span>$</span>\lambda_i \leq 0<span>$</span> if the <span>$</span>i<span>$</span>-th constraint has a <span>$</span>\geq<span>$</span> sense.

Let

<span>$</span>\bar{a} := \lambda^tA<span>$</span>

be the coefficients of this inequality and

<span>$</span>\bar{b} := \lambda^tb<span>$</span>

be its right hand side. With <span>$</span>L_j<span>$</span> and <span>$</span>U_j<span>$</span> being the lower and upper bounds of the variables <span>$</span>x_j<span>$</span> we have <span>$</span>\bar{a}_j \geq 0<span>$</span> if <span>$</span>U_j = \infty<span>$</span>, and <span>$</span>\bar{a}_j \leq 0<span>$</span> if <span>$</span>L_j = -\infty<span>$</span>.

The minimum violation of the Farkas constraint is achieved by setting <span>$</span>x^*_j := L_j<span>$</span> for <span>$</span>\bar{a}_j > 0<span>$</span> and <span>$</span>x^*_j := U_j<span>$</span> for <span>$</span>\bar{a}_j < 0<span>$</span>. Then, we can calculate the minimum violation as

<span>$</span>\beta := \bar{a}^tx^* - \bar{b} =
\sum\limits_{j:\bar{a}_j>0}\bar{a}_jL_j + \sum\limits_{j:\bar{a}_j<0}\bar{a}_jU_j - \bar{b}<span>$</span>

where <span>$</span>\beta>0<span>$</span>.

The FarkasProof attribute provides <span>$</span>\beta<span>$</span>, and the FarkasDual attributes provide the <span>$</span>\lambda<span>$</span> multipliers for the original constraints.

These attributes are only available when parameter InfUnbdInfo is set to 1.

For examples of how to query or modify attributes, refer to our Attribute Examples.