1. CS5321 Numerical Optimization
14 Linear Programming: The Interior Point Method
7/15/2019

2. Interior point methods for LP
There are many variations of IPM for LP
Primal IPM, dual IPM, primal-dual IPM
Potential Reduction Methods
Path Following Methods
Short-step methods
Long-step methods
Predictor-corrector methods
We will cover the path following method and the long-step method
7/15/2019

3. The dual problem of LP minxz=cTx subject to Ax = b, x  0
The dual problem is (see chap 12)
maxbT subject to AT  +s = c, s  0
Weak duality
For any x and  that are feasible in the primal and the dual problems, bT  cTx.
Strong duality
If x* and * are solutions to the primal and the dual problems, bT* = cTx*.
7/15/2019

4. Optimality conditions of LP
minxz=cTx subject to Ax = b, x  0
The Lagrangian function
L(x,,s) = cTx − T(Ax−b) − sTx
The optimality conditions (KKT, see chap 12)
The last one is the complementarity condition A T + s = c x b ; i
7/15/2019

5. Primal-dual method The optimality conditions can be expressed as
X=diag(x), S=diag(s) and e=(1,1,…,1)T.
F is a nonlinear equation: R2n+m  R2n+m.
Solved by the Newton’s method: (J the Jacobian of F.)
(x, , s)+=(x, , s), F ( x ; s ) = @ A T + c b X S e 1 f o r J ( x ; s ) @ 1 A = F
7/15/2019

6. Jacobian of F Let . The Jacobian J (x, , s) of F(x, , s) is F ( x ;s ) = @ A T + c b X S e 1 r J = @ r x c s b X S 1 A T I
7/15/2019

7. Solving the linear systems
The Newton’s direction is obtained by solving
S is an artificial variable. Let
The left hand side is symmetric, but indefinite. @ A T I S X 1 x s = r c b D = S 1 2 X D 2 A T x = r c + X 1 S b s = X 1 r S x
7/15/2019

8. Centering In practice, rXS=XSe −e rather than rXS=XSe.
Duality measure:
Centering parameter   [0,1]:
Decreasing as approaching solutions
This is related to the barrier method (chap 15) = 1 n X i x s T
Centered
direction
Without centering
7/15/2019

9. The path following algorithm
Given (x0,0,s0) with (x0,s0)>0
For k = 0, 1, 2 …until (xk)Tsk< 
Choose k[0,1] and let k=(xk)Tsk/n.
Solve the Newton’s direction (xk, k, sk)
Set (xk+1, k+1, sk+1)=(xk, k, sk) + k(xk, k, sk), where k is chosen to make (xk+1,sk+1)>0
7/15/2019

10. Central path and neighborhood
Feasible set F0={(x,,s) | Ax=b, AT+s=c, (x,s)>0}
Point (x,,s)F0 is on the central path C if x(i)s(i)=  for all i=1,2,…,n.
An example of the neighborhood of the central path
 is the duality measure (two slides before)
Typically =10−3.
7/15/2019

11. Long-step path-following
Given (x0,0,s0)N− and , min, max.
For k = 0, 1, 2 …until (xk)Tsk< 
Choose k[min,max] and let k=(xk)Tsk/n.
Solve the Newton’s direction (xk, k, sk)
Set (xk+1, k+1, sk+1)=(xk, k, sk) + k(xk, k, sk), where k is chosen to make (xk+1, k+1, sk+1)N−.
7/15/2019

12. Complexity
The long step following method can converge in O(nlog(1/)) iterations (Theorem 14.4)
The most effective interior point method for LP is the predictor-corrector algorithm (Mehrotra 1992)
7/15/2019