# Dual

### From Glossary

Another mathematical program with the property that its objective is always
a bound on the original mathematical program, called the
*primal*. Suppose the dual is . Then,
for all feasible in the primal and all in Y.
This immediately implies that if the primal is feasible, the dual cannot
be unbounded, and
vice versa: if the dual is feasible, the primal cannot be
unbounded. It also implies that if the dual is unbounded, the
primal must be infeasible (and vice versa). A dual provides a
sufficiency test for optimality, for if feasible
and can be found such that , it follows that is
optimal in the primal and is optimal in the dual. *Weak duality* is when this sufficient condition is all that can be
guaranteed. *Strong duality* is when a finite
optimal value to one problem ensures the existence of an optimal solution to
the other and that their optimal objective values are equal. There are
particular duals of significance.