A constrained minimization problem is the primal problem. Its Lagrangian can also generate a second optimization problem whose solutions provide bounds on the primal optimum.

For constraints

$$g_i(x)\le0,$$

define

$$\mathcal L(x,\lambda)=f(x)+\sum_i\lambda_i g_i(x),\qquad \lambda_i\ge0.$$

For any nonnegative multipliers, the dual function

$$q(\lambda)=\inf_x\mathcal L(x,\lambda)$$

uses the infimum—the greatest lower bound—of the Lagrangian over $x$. This value is a lower bound on every feasible primal objective. Maximizing that bound gives the dual problem.

The difference between a primal feasible objective and a dual value is a duality gap. Weak duality guarantees that the dual cannot exceed the primal optimum in a minimization problem.

Under additional conditions, important convex problems satisfy strong duality, so the best primal and dual values coincide. In many regular problems, the optimal multipliers can also quantify how the optimum changes when constraint bounds are relaxed.