Projection or Reduced Direction Search Methods for Bound-Constrained Problems

Projection or Reduced Direction Search Methods for Bound-Constrained Problems 

A problem with an easier solution has the quadratic objective function 1 

The difference between vector projection on a set of linear constraints and the use of null space of constraints to satisfy the constraints was illustrated earlier in Chapter 8. In both cases, the original direction is transformed into a new direction that satisfies the selected linear constraints. 

The proposed method can be called the Projection Direction Search Method or Reduced Direction Search Method since whatever search direction is selected, the projection on a bound constraint corresponds to the removal of such a constraint from the same direction. 

The proposed method is valid for QP where the matrix G is symmetric positive definite and the constraints are variable bounds. 

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