Equality, Inequality, and Bound Constraints

Consider the problem with the quadratic objective function 

When equality and inequality constraints are both present, some additional problems arise vhth respect to the bounds-only case. The first problem is similar to the one highlighted in linear programming. Equality and inequality constraints can contain numerous anomalies. It is, therefore, indispensable to preprocess these constraints so as to remove the useless constraints and reduce the values of variable bounds. 

This search becomes efficient when variable bounds as well as inequality constraint bounds are present. 

The second problem is finding a feasible starting point. Several alternatives can be used. One very simple way is to start from a point that fulfills the variable bounds and solve the following linear programming problem  

In the BzzMath library, the BzzQuadraticProgramming class uses this technique to find the first feasible guess. 

---