I tried using Method -> "QuadraticProgramming" in FindMinimum and got some strange error messages, which I illustrate with a simple problem. The strangest was that a constraint is required.

In[21]:= FindMinimum[x^2 + y^2, {x, y}, 
 Method -> "QuadraticProgramming"]

During evaluation of In[21]:= FindMinimum::qpuncon: Method -> QuadraticProgramming cannot be used on an unconstrained optimization problem. >>

Out[21]= FindMinimum[x^2 + y^2, {x, y}, 
 Method -> "QuadraticProgramming"]

In[24]:= FindMinimum[{x^2 + y^2, 1 <= 2}, {x, y}, 
 Method -> "QuadraticProgramming"]

During evaluation of In[24]:= FindMinimum::lpmp2: Warning: Method -> CLP is specified for a non-machine-precision problem;the problem will be converted into machine precision. Used Method -> Simplex for non-machine-number problem. >>

Out[24]= {0., {x -> 0., y -> 0.}}

In[26]:= FindMinimum[{x^2 + y^2, 1 <= 2}, {x, y}, Method -> "Simplex"]

During evaluation of In[26]:= FindMinimum::bdmtd: Value of option Method -> Simplex is not Automatic, "Gradient", "ConjugateGradient", "InteriorPoint", "QuasiNewton", "Newton", "LinearProgramming", "QuadraticProgramming", or "LevenbergMarquardt". >>

Out[26]= FindMinimum[{x^2 + y^2, 1 <= 2}, {x, y}, Method -> "Simplex"]

In[25]:= FindMinimum[{x^2 + y^2, 1 <= 2}, {x, y}, Method -> "CLP"]

During evaluation of In[25]:= FindMinimum::bdmtd: Value of option Method -> CLP is not Automatic, "Gradient", "ConjugateGradient", "InteriorPoint", "QuasiNewton", "Newton", "LinearProgramming", "QuadraticProgramming", or "LevenbergMarquardt". >>

Out[25]= FindMinimum[{x^2 + y^2, 1 <= 2}, {x, y}, Method -> "CLP"]