IE 510, NONLINEAR PROGRAMMING, HOMEWORK 1
DUE TUESDAY, FEB 9TH, 2016
1.a Show that f (x, y) = x2 /y is convex over the region y > 0.
Q
1.b Show that f (x1 , . . . , xn ) = ni=1 xpi i is concave over the region cfw_x1 > 0, . . . , xn > 0.
Here p1 , . . . ,
IE 510, NONLINEAR PROGRAMMING, HOMEWORK 2
DUE TUESDAY, FEB 23RD, 2016
1. Bertsekas, problem 1.3.6
Solution:
1.a. If lim sup(ek )1/k , then for any > , we have that eventually (ek )1/k or
k
ek . This implies ek converges faster than linearly with convergen
IE 510, NONLINEAR PROGRAMMING, HOMEWORK 2
DUE MARCH 3RD, 2015
1. Give an example of a point-to-point mapping from X = [0, 1] to R which is closed but
not continuous.
Solution: Take f : [0, 1] [0, 1] which maps 0 to 0 and maps any x > 0 to 1/x.
Observe tha
IE 510, Nonlinear Programming, TR 2-3:20 PM, Spring 215, 203 Transportation
Building
Nonlinear programming is about optimizing not-necessarily-linear functions possibly subject to
constraints. It finds applicability in a variety of fields ranging from eco
IE 510, NONLINEAR PROGRAMMING, HOMEWORK 3
DUE TUESDAY, MARCH 15TH, 2016
1. The purpose of this problem is to guide you through the analysis of the projected
gradient descent method for strongly convex functions.
(a) Suppose f : Rn R is a twice continuousl
IE 510, NONLINEAR PROGRAMMING, HOMEWORK 2
DUE TUESDAY, FEB 23RD, 2016
1. Bertsekas, problem 1.3.6
2. Bertsekas, problem 1.3.7.
3. Bertsekas, problem 1.4.1.
4. Bertsekas, problem 1.4.8.a
5. Implement the steepest descent method
xt+1 = arg min f (xt f (xt )
IE 510, NONLINEAR PROGRAMMING, HOMEWORK 4 SOLUTIONS
9.1 We argue that each time we add a vector dk+1 to the collection cfw_d0 , . . . , dk it is Qorthogonal to all the previous vectors. We prove this by induction. The statement is
vacuously true when k =
IE 510, NONLINEAR PROGRAMMING, HOMEWORK 5
DUE MAY 5TH, 2015
1. Given A, b consider the problem of minimizing |Ax b| for all x Rn . Write this as
a linear program, take the dual, and simplify as much as possible.
2. Consider the problem of minimizing (1/2)
IE 510, NONLINEAR PROGRAMMING, HOMEWORK 1
DUE TUESDAY, FEB 10TH, 2015
1. A convex optimization problem is the minimization of a function f (x) subject to the
constraints g1 (x) 0, g2 (x) 0, . . . , gk (x) 0 where all functions f (x), g1 (x), g2 (x), . . .
IE 510, NONLINEAR PROGRAMMING, HOMEWORK 3
DUE APRIL 21ST, 2015
1. Consider the problem of minimizing |Ax b|22 subject to the constraint that Cx = d.
Here A Rmn is a matrix with linearly independent columns and C is a matrix with
linearly independent rows.
IE 510, NONLINEAR PROGRAMMING, HOMEWORK 5
DUE THURSDAY, APRIL 28TH, 2016
1. Bertsekas, problem 4.2.2, (a) & (b).
2. Bertsekas, problem 4.2.6.
3. Bertsekas, problem 4.2.8
4. Consider the problem of projection a vector y onto the l1 unit ball, i.e., onto th
IE 510, NONLINEAR PROGRAMMING, HOMEWORK 2
DUE MARCH 3RD, 2015
1. Give an example of a point-to-point mapping from X = [0, 1] to R which is closed but
not continuous.
2. You may find the pseudocode for the golden search method here:
http:/homepages.math.ui
IE 510, NONLINEAR PROGRAMMING, HOMEWORK 4
DUE THURSDAY, MARCH 31ST, 2016
1. Bertsekas, problem 3.1.1
2. Bertsekas, problem 3.1.12
3. Bertsekas, problem 3.2.4
4. Bertsekas, problem 3.3.2.
5. Let p be a positive vector
P whose components add up to 1. We def
IE 510, NONLINEAR PROGRAMMING, HOMEWORK 1
DUE TUESDAY, FEB 9TH, 2016
1.a Show that f (x, y) = x2 /y is convex over the region y > 0.
Solution: The Hessian is
2
=
2/y
2x/y 2
2
2x/y 2x2 /y 3
A matrix is nonnegatie definite if and only if all of its princip
IE 510, NONLINEAR PROGRAMMING, HOMEWORK 1
DUE THURSDAY, FEB 5TH, 2015
1. A convex optimization problem is the minimization of a function f (x) subject to the
constraints g1 (x) 0, g2 (x) 0, . . . , gk (x) 0 where all functions f (x), g1 (x), g2 (x), . . .