Solution concepts
In the last 50 years, many great algorithms have been developed for numerical optimization and these algorithms work well, especially in case of quadratic functions. As we have seen in the previous section, we only have quadratic functions and constraints; so these methods (that are implemented in R as well) can be used in the worst case scenarios (if there is nothing better).
However, a detailed discussion of numerical optimization is out of the scope of this book. Fortunately, in the special case of linear and quadratic functions and constraints, these methods are unnecessary; we can use the Lagrange theorem from the 18th century.
Theorem (Lagrange)
If 
and 
, (where 
) have continuous partial derivatives and 
is a relative extreme point of f(x) subject to the 
constraint where 
.
Then, there exist the coefficients 
such that 
In other words, all of the partial derivatives of the function 
are 0 (Bertsekas Dimitri P. (1999)).
In our case, the condition is also sufficient. The...
