IE 5531 2013 Fall
Assignment 7 Solution
Problem 1
1. False. Consider the following counterexample:
f (x) = x
convex
g(x) = x2
convex
f (g(x) = x2 concave
2. True.
Proof Let h(x) = f (g(x),
h (x) = f (g(x)g (x)
h (x) = f (g(x)(g (x)2 + f (g(x)g (x)
Since f
IE5531 Assignment 1
Due in class (12pm), Sept 16th
Problem 1 (20pts). A company produces two kinds of products. A product of the rst type
requires 1/4 hours of assembly labor, 1/8 hours of testing, and $1.2 worth of raw materials.
A product of the second
Lecture 24: Final Review
Zizhuo Wang
University of Minnesota
Dec, 2013
Zizhuo Wang (University of Minnesota)
Engineering Optimization: Lecture 24
Dec, 2013
1 / 44
Final Exam
Final exam next Wednesday, Dec 11th, 10:15am -12:15pm
Two pieces of notes are all
IE5531 Assignment 5/Sample Midterm Exam
Due in class (12pm), Oct 23rd
Note: This homework is also the sample midterm. The solution is on Moodle.
However, please do it without looking at the solution rst. If you eventually did
it with the help of the solut
IE 5531 2013 Fall
Assignment 1 Solution
1.(a) Let x1 be the number of type 1 product, and x2 be the number of type 2 product.
maximize (9 1.2)x1 + (8 0.9)x2
1
s.t.
x + 1 x2 90
4 1
3
1
x1 + 1 x2 80
8
3
x1 , x2 0
(b) The standard form is as follows.
minimiz
IE 5531 Practice Final Exam
Prof. John Gunnar Carlsson
December 7, 2010
1
Integer programming
1. Use implicit enumeration to solve the integer program
minimize 7x1 + 3x2 + 2x3 + x4 + 2x5
4x1 + 2x2 x3 + 2x4 + x5
4x1 + 2x2 + 4x3 x4 2x5
s.t.
7
xi
Solution
Th
IE5531 Assignment 6
Due in class (12pm), Nov 4th
Problem 1 (20pts). Consider the function
f (x, y, z) = 2x2 + xy + y 2 + yz + z 2 6x 7y 8z 9
1. Use the rst-order necessary conditions, nd the candidate minimum points of f (x, y, z)
2. Verify using the seco
IE5531 Assignment 7
Due in class (12pm), Nov 13th
Problem 1 (25pts). Either prove or nd a counterexample for each of the following
statement (you can assume all the functions are second order continuously dierentiable):
1. If f (x) is convex, g(x) is conv
IE 5531 Final Exam - Dec 2013
Page 1 of 9
F INAL E XAM S OLUTION
IE 5531
Dec 11th, 2013
INSTRUCTIONS
a) Write ALL your answers in this exam paper.
b) Two pieces of notes are allowed. No computer or cell phone is allowed.
c) The exam time is 10:15am - 12:1
Lecture 16: More on KKT Conditions and Convexity
Zizhuo Wang
University of Minnesota
Nov, 2013
Zizhuo Wang (University of Minnesota)
Engineering Optimization: Lecture 16
Nov, 2013
1 / 29
Announcement
Homework 6 due today
Homework 7 due next Wednesday
Zizh
IE 5531 Sample Final Exam - Dec 2013
Page 1 of 8
S AMPLE F INAL E XAM S OLUTION
IE 5531
Dec, 2013
INSTRUCTIONS
a) Write ALL your answers in this exam paper.
b) Two pieces of notes are allowed. No computer or cell phone is allowed.
c) The exam time is 10:1
IE 5531 Sample Final Exam - Dec 2013
Page 1 of 9
S AMPLE F INAL E XAM
IE 5531
Dec, 2013
INSTRUCTIONS
a) Write ALL your answers in this exam paper.
b) Two pieces of notes are allowed. No computer or cell phone is allowed.
c) The exam time is 10:15am - 12:1
IE 5531 2013 Fall
Assignment 2 Solution
Problem 1
1. True. Set P lies in an ane subspace dened by m = n 1 linearly independent constraints of dimension one. Every solution of Ax = b is of the form x0 + x1 , where x0 is an
element of P and x1 is a nonzero
IE5531 Assignment 1
Due in class (12pm), Sept 25th
Problem 1 (25pts). Consider an LP in its standard form and the corresponding constraint
set P = cfw_x|Ax = b, x 0. Suppose that the matrix A has dimensions m n and that its
rows are linearly independent.
IE5531 Assignment 4
Due in class (12pm), Oct 14th, Monday
Problem 1 (20pts). Consider the following linear program:
maximize
5x1 + 2x2 + 5x3
subject to 2x1 + 3x2 + x3 4
x1 + 2x2 + 3x3 7
x 1 , x2 , x3 0
1. What is the corresponding dual problem
2. Solve th
IE 5531 Fall 2013
Assignment 8 Solution
Problem 1
Before we use bisection method to solve the problem, we can check the number of solutions and
the intervals from looking at the graph. The graph shows that two solutions exist on the interval
[1, 2] and [5
IE5531 Assignment 3
Due in class (12pm), Oct 2nd
Problem 1 (20pts). Consider the following linear program:
maximize 500x1 + 250x2 + 600x3
subject to 2x1 + x2 + x3 240
3x1 + x2 + 2x3 150
x1 + 2x2 + 4x3 180
x 1 , x2 , x3 0
Use Simplex method to solve it. Fo
IE 5531: Engineering Optimization I
Lecture 6: Simplex method topics, duality
Prof. John Gunnar Carlsson
September 27, 2010
Prof. John Gunnar Carlsson
IE 5531: Engineering Optimization I
September 27, 2010
1 / 34
Administrivia
PS 1, problem b.ii: discount
IE 5531: Engineering Optimization I
Lecture 8: More duality, complementary slackness, dual simplex method
Prof. John Gunnar Carlsson
October 4, 2010
Prof. John Gunnar Carlsson
IE 5531: Engineering Optimization I
October 4, 2010
1 / 21
Administrivia
Midter
IE 5531: Engineering Optimization I
Lecture 8: Sensitivity analysis
Prof. John Gunnar Carlsson
October 6, 2010
Prof. John Gunnar Carlsson
IE 5531: Engineering Optimization I
October 6, 2010
1 / 24
Administrivia
Midterm 1 will be on 10/18/10
Lectures 1-9 c
IE 5531: Engineering Optimization I
Lecture 3: Linear Programming, Continued
Prof. John Gunnar Carlsson
September 15, 2010
Prof. John Gunnar Carlsson
IE 5531: Engineering Optimization I
September 15, 2010
1 / 49
Pop quiz
Write the region above in the form
IE 5531: Engineering Optimization I
Lecture 3: Linear Programming, Continued
Prof. John Gunnar Carlsson
September 15, 2010
Prof. John Gunnar Carlsson
IE 5531: Engineering Optimization I
September 15, 2010
1 / 29
Administrivia
Lecture slides 1, 2, 3 posted
IE 5531: Engineering Optimization I
Lecture 2: Linear Programming
Prof. John Gunnar Carlsson
September 13, 2010
Prof. John Gunnar Carlsson
IE 5531: Engineering Optimization I
September 13, 2010
1 / 30
Administrivia
Lecture slides 1,2 posted later today
ht
IE 5531: Engineering Optimization I
Lecture 5: The Simplex method, continued
Prof. John Gunnar Carlsson
September 22, 2010
Prof. John Gunnar Carlsson
IE 5531: Engineering Optimization I
September 22, 2010
1 / 27
Administrivia
Lecture slides 4,5 posted
htt
IE5531 Assignment 1
Due in class (12pm), Sept 21st
For those questions that ask you to write MATLAB codes to solve the problem. Please
print the code and attach it to the homework. You also need to write down what is the
optimal solution and the optimal v
IE 5531: Engineering Optimization I
Lecture 10: Introduction to nonlinear optimization
Prof. John Gunnar Carlsson
October 11, 2010
Prof. John Gunnar Carlsson
IE 5531: Engineering Optimization I
October 11, 2010
1 / 26
Administrivia
Practice midterm 1 post
IE 5531: Engineering Optimization I
Lecture 7: Duality and applications
Prof. John Gunnar Carlsson
September 29, 2010
Prof. John Gunnar Carlsson
IE 5531: Engineering Optimization I
September 29, 2010
1 / 30
Administrivia
PS 2 posted this evening
Prof. Joh
IE 5531: Engineering Optimization I
Lecture 12: Nonlinear optimization, continued
Prof. John Gunnar Carlsson
October 20, 2010
Prof. John Gunnar Carlsson
IE 5531: Engineering Optimization I
October 20, 2010
1 / 26
Administrivia
PS4 posted later today
Prof.
IE 5531: Engineering Optimization I
Lecture 23: Final exam review
Prof. John Gunnar Carlsson
December 13, 2010
Prof. John Gunnar Carlsson
IE 5531: Engineering Optimization I
December 13, 2010
1 / 71
Administrivia
Project due any time before the end of nal