I believe the approach taken in 7.3.1 to deriving the dual problem for a linear program is incorrect.
I believe the Lagrangian is successfully derived in equation 7.41. However in this case one cannot derive the dual Lagrangian by differentiating the primal Lagrangian and setting equal to zero. This is because the primal Lagrangian is unbounded from below and doesn't have a minimum with respect to x. In deriving the dual Lagrangian the dual variables should be treated as constants and therefore setting the derivative to zero as in equation 7.42 doesn't make sense.
I believe the correct approach would be similar to that taken on p219 of Boyd and Vandenberghe's textbook. where the dual Lagrangian is negative infinity almost everywhere.
Hope this helps and thank you for making a great resource freely available.
I believe the approach taken in 7.3.1 to deriving the dual problem for a linear program is incorrect.
I believe the Lagrangian is successfully derived in equation 7.41. However in this case one cannot derive the dual Lagrangian by differentiating the primal Lagrangian and setting equal to zero. This is because the primal Lagrangian is unbounded from below and doesn't have a minimum with respect to x. In deriving the dual Lagrangian the dual variables should be treated as constants and therefore setting the derivative to zero as in equation 7.42 doesn't make sense.
I believe the correct approach would be similar to that taken on p219 of Boyd and Vandenberghe's textbook. where the dual Lagrangian is negative infinity almost everywhere.
Hope this helps and thank you for making a great resource freely available.