McMaster University
CS-4-6TE3/CES722-723
Assignment-1 Solution Set
CS-4-6TE3/CES722-723
Assignment 1 Solution Set
Christopher Anand, Shefali Kulkarni-Thaker
October 25, 2011
Consider the so-called Rosenbrock (banana) function:
f (x1 , x2 ) = 100(x2 x2 )2
Tutorial 1
Fundamentals
CS/SWE 4/6TE3, CES 722/723
September 14, 2010
Review of derivatives, gradients and Hessians:
The gradient extends the notion of derivative, the Hessian matrix that of second derivative.
Given a function f of n variables x1 , x2 ,
COMP SCI and SFWR ENG 4-6TE3, and CES 723
Midterm Exam Solution (Fall 2006)
Oleksandr Romanko
October 27, 2006
1. Consider the function:
(x1 2x2 )2 ex1 +x2
f (x1 , x2 ) =
(a) (i)
(ii)
Give the gradient and the Hessian of f (x1 , x2 ).
Give the second-orde
Tutorial 3
Fundamentals
CS/SWE 4/6TE3, CES 722/723
September 28, 2010
Rate of Convergence
Let 1 , 2 , ., n be a convergent sequence, the order of convergence is p if
|k+1 |
<
k |k |p
p = sup p : lim
If p = 1, then let
|k+1 |
k |k |
= lim
If p = 1 and 0 <
Tutorial 2
Fundamentals
CS/SWE 4/6TE3, CES 722/723
September 21, 2010
Checking if Symmetric Matrix is PD or PSD
by Computing its Eigenvalues
Denition Any number such that the equation Ax = x has a non-zero vector-solution x is
called an eigenvalue (or a c
Slater points and Lagrange/Wolfe duals
1 Slater point and Ideal Slater point
Consider a convex problem with two constraints (1) x1 0 and (2) x2 + x2 1. Both of them
1
2
are regular constraints, one linear and the other nonlinear. Point (0, 0)T is a Slate
Linearly Constrained NLO : Consider the following linearly constrained non-linear optimization
(NLO) problem:
(NLO)
min x1 + x2 + x1 x2
s.t. 3x1 + 2x2 = 5.
a. Reduce this problem to an unconstrained optimization problem.
b. Let us assume that the variable
SQP Example: Consider the following non-linear optimization (NLO) problem:
min (x2 x1 )4 + (x1 + x2 x3 )2
s.t. 2x1 + 3x2 + 5x3 = 10.
Formulate the SQP subproblem for the candidate point x = (2, 2, 0)T and the candidate Lagrange
multiplier y = 2.
Solution
Notes
Corporate Policy: Principles of action
Industry associations:
Focused on communicating an industry position for the benefit of its members (against blatant regard for
public opinion)
Evolved to become more complex- lobbying still an important aspect