Editing Quadratic programming (section)

==Lagrangian duality==
{{See also|Dual problem}}

The Lagrangian [[Dual problem|dual]] of a quadratic programming problem is also a quadratic programming problem. To see this let us focus on the case where {{math|1=''c'' = 0}} and {{mvar|Q}} is positive definite. We write the [[Lagrange multipliers|Lagrangian]] function as 
:<math>L(x,\lambda) = \tfrac{1}{2} x^\top Qx + \lambda^\top (Ax-b). </math>
Defining the (Lagrangian) dual function {{math|''g''(λ)}} as <math>g(\lambda) = \inf_{x} L(x,\lambda) </math>, we find an [[infimum]] of {{mvar|L}}, using <math>\nabla_{x} L(x,\lambda)=0</math> and positive-definiteness of {{mvar|Q}}:

:<math>x^* = -Q^{-1} A^\top \lambda.</math>

Hence the dual function is 
:<math>g(\lambda) = -\tfrac{1}{2} \lambda^\top AQ^{-1}A^\top \lambda - \lambda^\top b,</math>
and so the Lagrangian dual of the quadratic programming problem is

:<math>\text{maximize}_{\lambda\geq 0} \quad  -\tfrac{1}{2} \lambda^\top AQ^{-1} A^\top \lambda - \lambda^\top b.</math>

Besides the Lagrangian duality theory, there are other duality pairings (e.g. [[Wolfe duality|Wolfe]], etc.).