Unbounded/Infeasible problems and primal/dual variables

(Roberto) #1

I am a bit confused by the documentation on dual variables (section 4.7, page 24 in the pdf). It says:

To associate the dual variable y with the inequality constraint Ax⪯bin this LP, we use the following syntax:

    variable x(n);
    dual variable y;
    minimize( c' * x );
    y : A * x <= b;

[…]In the unbounded case, x will contain an unbounded direction; i.e. , a point x satisfying

(1)   c' * x = -1,   A * x <= 0

[…]. In the infeasible case, […] y contains an unbounded dual direction; i.e., a point y satisfying

(2)   b' * y = −1,  A' * y = 0,  y >= 0.

But what about mutiple constraints:

y: A * x <= b;
z: C * x <= d;

From what I understand if the primal is unbounded then for all constraints (1) should hold. While if the dual is unbounded then there exist a constraint for which (2) holds. Is this correct?