I'm having some trouble understanding about the independent rows in a basic solution. In the book Introduction to linear optimization by B&T, the authors give the definition of a basic solution as follows enter image description here

where they claim that all equality constraints are active. However, in a later example, I noticed the following:

enter image description here

Apparently, all constraints in the later picture are inequality constraints, but the authors treat them as equality constraints and claim since they are active, we then have a basic solution.

So my question is: does inequality and equality matter when it comes to active constraints? Should I treat them differently?


All equality constraints are active for all feasible solutions.

The inequality constraints which are active for a solution are those which are satisfied (hold) with equality at that solution.

From the viewpoint of determining how many active constraints are linearly independent, throw all the equality constraints plus the active inequality constraints into one hopper, with no further distinguishment needed.


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