duality

Every optimization problem (the primal) has an associated dual problem built from its Lagrangian, whose optimal value never exceeds the primal's (weak duality) and, under enough structure (e.g. Slater's condition, or any feasible bounded LP), equals it exactly (strong duality). The dual gives both a certificate of optimality (via complementary slackness) and, through its optimal multipliers, a sensitivity reading of how the primal optimum would move if a constraint shifted.