 Solution

For $C=\{x:Ax\ge b\}$, primal projected gradient descent isProjection onto $C$ is itself a constrained quadratic program. The dual method only projects componentwise onto ${\mathbb{R}}_{+}^m$ and uses the fixed matrix $A{Q}^{-1}{A}^T$, so its iterations can be substantially cheaper, especially when $Q^{-1}$ can be prefactored and the number of constraints is moderate.