1. CS5321 Numerical Optimization
19 Interior-Point Methods for Nonlinear Programming
(Barrier Methods)
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2. Problem formulation cE is a vector of ci, iEx f ( ) s . t c E = I
cE is a vector of ci, iE
cI is a vector of ci, iI
s is the slack variables
Add barrier functions for inequality constraints. m i n x f ( ) X = 1 l s . t c E I
 is the barrier parameter, which may be reduced iteratively.
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3. KKT conditions The Lagrangian of the original problem is
y and z are the Lagrangian multipliers.
Let AE and AI denote the Jacobian of cE and cI.
The KKT condition of the barrier function is L ( x ; s y z ) = f m X i 1 l n T c E I r f ( x ) A T E y I z = S e c s
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4. Line-search algorithm
Solve KKT conditions by the Newton’s method
The primal-dual system
Update (x, s, y, z)+=(xpx,sps,ypy,zpz) and k+1
The fraction to the boundary rule: for (0,1) 2 6 4 r x L A T E I Z S 3 7 5 p s y z = f e c s = m a x f 2 ( ; 1 ] : + p ) g z
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5. Solving the primal-dual system
Let=S−1Z and rewrite the primal dual system as
The matrix become symmetric.
Eliminate ps and reform the problem
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6. Avoid singularity
Singularity may caused by (some elements goes to ) or ill-conditionness of xx2f and AI.
Use projected Hessian and modified 
Add diagonal shift 2 6 4 G A T E I 3 7 5
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7. Barrier param and step length
Barrier parameters {k} need converge to zero.
Static method k+1= kk for k(0,1)
Adaptive methods
In the linear programming, (chap 14)
Use merit function to decide whether a step is productive and should be accepted. (chap 18) k + 1 = s T z m Á ( x ; s ) = f m X i 1 l n + k c E I
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8. Trust-region SQP method
Two differences from the line search approach
Solve primal (x,s) and dual problem (y, z) alternately
Use scaling S−1 on ps.
The primal problem
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9. Solving the dual problem^ A = E I S
Define The dual problem is to solve
Using the least-square method
The solution cannot guarantee z to be positive.  replaced by a small positive number if zi0 ^ A T y z = r f ( x ) e y z = ( ^ A T ) 1 r f x e z i ! m n ( 1 3 ; = s )
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10. Scaling Let ps' = S−1ps. The problem can be rewritten as
Parameter rE and rI can be computed by solving r E = A ( x ) v + c ; I S s m i n k 2 . t : 8 V e
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11. Convergence theory
Theorem 19.1: Let {xk} be the iterative points of the basic algorithm and {k}0. Suppose f and c are continuously differentiable. The all limit points x* of {xk} are feasible. Moreover, if any x* satisfies LICQ, the KKT conditions are satisfied.
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