| # Strong Duality for QP ## Primal Problem Consider the QP where , , , , and . We assume for simplicity that (that is, is positive definite). ## Dual problem Let us form the dual of this problem. The Lagrangian is the function with values and the dual function is , with values The above problem is an unconstrained convex optimization problem, so optimal points are characterized by the condition that the gradient of the objective is zero: Since is invertible, we obtain the minimizer as , and the dual function as The dual problem reads The dual, just like the primal, is a QP. We note that the dual feasible set has a particularly simple form, in contrast to that of the primal, which is a generic polytope. ## Strong Duality Result We can apply Slater’s theorem to this QP, and obtain that a sufficient condition for strong duality to hold is that the QP is strictly feasible, that is, there exist such that . However, if , it can be shown that strong duality always holds.