Chapter 3 Quadratic Programming

Optimization I; Chapter 3 68
linear programming problem:
minimize e T z , (3.44a)
over (x, z) ∈ lR n × lR m+p
subject to c T i x + γ i z i = c i , 1 ≤ i ≤ m , (3.44b)
a T i x − γ m+i z m+i ≤ d i , 1 ≤ i ≤ p , (3.44c)
z ≥ 0 ,
(3.44d)
where
e = (1, ..., 1) T , γ i =
{ −sign(c
T
i ˜x − c i ) , 1 ≤ i ≤ m
1 , m + 1 ≤ i ≤ m + p
A feasible starting point for this linear problem is given by
{
x (0) = ˜x , z (0)
|c
i =
T i ˜x − c i | , 1 ≤ i ≤ m
max(a T i−m˜x − d i−m , 0) , m + 1 ≤ i ≤ m + p
. (3.45)
.(3.46)
Obviously, the optimal value of the linear programming subproblem is zero, and
any solution provides a feasible point for the original one.
Another technique introduces a measure of infeasibility in the objective functional
in terms of a penalty parameter β > 0:
1
minimize
2 xT Bx − x T b + βt , (3.47a)
over (x, t) ∈ lR n × lR
subject to c T i x − c i ≤ t , 1 ≤ i ≤ m , (3.47b)
− (c T i x − c i ) ≤ t , 1 ≤ i ≤ m ,
a T i x − d i ≤ t , 1 ≤ i ≤ p ,
t ≥ 0 .
(3.47c)
(3.47d)
(3.47e)
For sufficiently large penalty parameter β > 0, the solution of (3.47a)-(3.47e) is
(x, 0) with x solving the original quadratic programming problem.
3.5.2 Primal-dual active set strategies
We consider a primal-dual active set strategy which does not require feasibility
of the iterates. It is based on a Moreau-Yosida type approximation of the
indicator function of the convex set of inequality constraints
K := {v ∈ lR p | v i ≤ 0 , 1 ≤ i ≤ p} . (3.48)
The indicator function I K : K → lR of K is given by
{
0 , v ∈ K
I K (v) :=
+∞ , v /∈ K . (3.49)