Homework 1, Optimization 550.362, Spring 2015
Problem 1: Use a Taylor expansion with remainder term to show that for all x : 0 x <
2
it
holds that tan(x) x. (Hint: Expand about the value 0, and consider the possible values that
the secant function can tak
Homework 2, 550.362 Optimization, Spring 2015
HOMEWORK RULES: You may discuss ideas for the homework with other students in the class,
but when it comes time to write up your solutions you must do it alone, and your write-up must
reect your understanding.
Homework 3, 550.362 Optimization, Spring 2015
Problem 1: Consider Rosenbrocks Function f (x, y) = (1 x)2 + 100(y x2 )2 .
a) Using ad-hoc/elementary reasoning, identify a global min and argue that this min is unique.
b) Conrm that this function is not conv
550.362 Optimization II, Exam 1, Spring 2015
Problem 1: (10 points) Consider the following problem:
min
x
1 x2 y 2 0
s.t.
a) Find a KKT point and show that it is a KKT point. b) Argue that it is not a local minimizer.
Solution: Here
f(
x
y
active, and its
550.362 Optimization II, Exam 1, Spring 2016
Problem 1: (10 points) Find a KKT point, and show that it is a KKT point, for the following:
min
x2
x21 + x22 1 0
s.t.
2x1
0
0
x1
x1
x2 ) = 1 , and g( x2 ) = 2x2 . At the point 1 the single constraint