Math 271B: Numerical Optimization
Lecture 1
Nonlinearly Constrained Optimization
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Monday, January 4th, 2016
Overview of Math 271B
Class web-page: http:/ccom.ucsd.edu/~peg/math271b
The class text, lect
Math 271B: Numerical Optimization
Lecture 9
Augmented Lagrangian Methods
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Monday, January 25th, 2016
shifted constraint
c(x) s = 0
c(x) = 0
x
x()
x()
penalty trajectory
Recap: the shifted penalty func
Math 271B: Numerical Optimization
Lecture 17
Choosing the SQP Hessian
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Friday, February 12th, 2016
Recap: SQP methods
Each iteration of an SQP method involves solving the QP
minimize
n
pR
fk + gkTp +
Math 271B: Numerical Optimization
Lecture 14
Merit Function SQP Methods
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Friday, February 5th, 2016
Recap: SQP methods
If
xbk = xk + pk
and ybk = yk + qk
are the unique minimizer and Lagrange multipli
Math 271B: Numerical Optimization
Lecture 15
Merit Function SQP Methods II
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Monday, February 8th, 2016
The `1 penalty function is:
P1 (x ; ) = f (x) + kc(x)k1
The step k is chosen so that the improvem
Math 271B: Numerical Optimization
Lecture 16
Merit Function SQP Methods III
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Wednesday, February 10th, 2016
Recap: SQP methods
Each iteration of an SQP method involves solving the QP
minimize
n
pR
fk
Math 271B: Numerical Optimization
Lecture 13
Sequential Quadratic Programming
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Wednesday, February 3rd, 2016
Recap: Following the penalty trajectory
Methods based on the quadratic penalty function can
Math 271B: Numerical Optimization
Lecture 10
Properties of the Augmented Lagrangian
Method
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Wednesday, January 27th, 2016
Recap: the augmented Lagrangian
The augmented Lagrangian function is given by
Math 271B: Numerical Optimization
Lecture 11
Augmented Lagrangian Methods and
Regularization
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Friday, January 29th, 2016
Algorithm. Classical augmented Lagrangian Method for NEP.
Choose x0 , y0E , 0 (
Math 271B: Numerical Optimization
Lecture 7
Exact Penalty Functions
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Wednesday, January 20th, 2016
Recap: Minimizing the penalty function
The Newton equations for minimizing P2 (xk ; ) are
H(xk , k )
Math 271B: Numerical Optimization
Lecture 5
The Quadratic Penalty Function
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Wednesday, January 13th, 2016
Recap: quadratic penalty function
For a fixed positive penalty parameter , the quadratic penal
Math 271B: Numerical Optimization
Lecture 6
Minimizing the Quadratic Penalty Function
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Friday, January 15th, 2016
Recap: Convergence of the penalty method
The quadratic penalty method is based on the
Math 271B: Numerical Optimization
Lecture 3
Second-Order Optimality Conditions
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Friday, January 8th, 2016
Recap: constraint qualification
The problem of concern is
minimize
f (x)
n
xR
subject to c(x)
Math 271B: Numerical Optimization
Lecture 4
The Method of Newton-Lagrange
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Monday, January 11th, 2016
Recap: Second-order conditions
If the CQ holds at x , then x is a local solution of NEP only if:
(
Math 271B: Numerical Optimization
Lecture 18
Second Derivative SQP Methods
Philip E. Gill
c 2016
http:/ccom.ucsd.edu/~peg/math271B
Wednesday, February 17th, 2016
The rapid rate of convergence Newton-Lagrange method is based
on (pk , ybk ) satisfying the e