Chapter 3 Quadratic Programming

Optimization I; Chapter 3 63
where B ∈ lR n×n is symmetric positive definite, C ∈ lR m×n , A ∈ lR p×n , and
b ∈ lR n , c ∈ lR m , d ∈ lR p .
We write the matrices A and C in the form
A =
⎡
⎣
⎤
a 1
· ⎦ , a i ∈ lR n , C =
a p
⎡
⎣
⎤
c 1
· ⎦ , c i ∈ lR n . (3.30)
c m
The inequality constraints (3.29c) can be equivalently stated as
a T i x ≤ d i , 1 ≤ i ≤ m . (3.31)
The primal active set strategy is an iterative procedure:
Given a feasible iterate x (ν) , ν ≥ 0, we determine an active set
I ac (x (ν) ) ⊂ {1, ..., p} (3.32)
and consider the corresponding constraints as equality constraints, whereas the
remaining inequality constraints are disregarded. Setting
we find
p = x (ν) − x , b (ν) = Bx (ν) − b , (3.33)
f(x) = f(x (ν) − p) = 1 2 pT Bp − (b (ν) ) T p + g ,
where g := 1 2 (x(ν) ) T Bx (ν) − b T x (ν) .
The equality constrained QP problem to be solved at the (ν+1)-st iteration
step is then:
1
minimize
2 pT Bp − (b (ν) ) T p (3.34a)
over p ∈ lR n
subject to Cp = 0 (3.34b)
a T i p = 0 , i ∈ I ac (x (ν) ) ,
(3.34c)
We denote the solution of (3.34a)-(3.34c) by p (ν) . The new iterate x (ν+1) is then
obtained according to
x (ν+1) = x (ν) − α ν p (ν) , α ν ∈ [0, 1] , (3.35)
where α ν is chosen such that x (ν+1) stays feasible.
In particular, for i ∈ I ac (x (ν) ) we have
a T i x (ν+1) = a T i x (ν) − α ν a T i p (ν) = a T i x (ν) ≤ d i .