Lecture 1: Introduction to Optimization
Zizhuo Wang
Department of Industrial and Systems Engineering
University of Minnesota
Sep, 2013
Zizhuo Wang
Engineering Optimization: Lecture 1
Course Logistics
Meeting times:
Mon/Wed 10:15am-12:00pm
Keller 3-115
My
IE5531 Assignment 8
Due in class (12pm), Nov 30th
Please attach the code (and the figures for Problems 1 and 2) for your problems.
Problem 1 (40pts). Write a computer code in MATLAB using the gradient descent method
to solve the optimization problem
minim
IE5531 Assignment 2
Due in class (12pm), Sep 28th
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. F
IE5531 Assignment 3
Due in class (12pm), Oct 5th
Problem 1 (20pts). Consider the following LP:
maximize 500x1 + 250x2 + 600x3
subject to 2x1 + x2 + x3 240
3x1 + x2 + 2x3 150
x1 + 2x2 + 4x3 180
x 1 , x2 , x3 0
Use the Simplex method to solve it. For each s
IE5531 Assignment 6
Due in class (12pm), Nov 9th
Problem 1 (25pts). Consider the function
f (x, y, z) = 2x2 + xy + y 2 + yz + z 2 6x 7y 8z 9
1. Use the first-order necessary conditions to find the candidate minimizer of f (x, y, z).
2. Verify using the se
IE5531 Assignment 5
Due in class (12pm), Oct 19th, Wednesday
Problem 1 (30pts). Consider the following linear optimization problem:
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 pro
IE 5531
Assignment 5 Solution
Problem 1
1. The dual problem is:
minimize
s.t.
4y1 + 7y2
2y1 + y2 5
3y1 + 2y2 2
y1 + 3y2 5
y1 , y2 0
2. The feasible region and optimal solution is given in the plot below.
3. The complementarity conditions are:
y1 (2x1 + 3x
IE 5531
Assignment 3 Solution
Problem 1
First, write the canonical form of the linear program:
minimize 500x1 250x2 600x3
s.t.
2x1 + x2 + x3 + s1 = 240
3x1 + x2 + 2x3 + s2 = 150
x1 + 2x2 + 4x3 + s3 = 180
xi , si 0
In the initial tableau, the basic set isc
IE 5531 2016 Fall
Assignment 2 Solution
Problem 1
1. False. Consider the following counterexample:
minimize 0
s.t.
x0
then the optimal solution set is unbounded.
2. False. Consider the following counterexample: minimize 0, s.t. x 0. Then any feasible
x is
IE 5531 2016 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
s.t.
(9 1.2)x1 + (8 0.9)x2
1
1
4 x1 + 3 x2 90
1
1
8 x1 + 3 x2 80
x1 , x2 0
(b) The standard form is as follows.
minimize
IE 5531
Assignment 6 Solution
Problem 1
1. By first order necessary condition f (x ) = 0, we have
4x + y 6 = 0
x + 2y 7 + z = 0
2z 8 + y = 0
Solving the above equations, we get a candidate point of the minimizer: ( 56 , 56 , 17
).
5
2. We want to show 2 f
IE5531 Assignment 7
Due in class (12pm), Nov 23rd
Problem 1 (25pts). Either prove or find a counterexample for each of the following
statement (you can assume all the functions are second order continuously differentiable):
1. If f (x) is convex, g(x) is
Lecture 13: Introduction to Nonlinear Optimization
Zizhuo Wang
University of Minnesota
Oct, 2013
Zizhuo Wang (University of Minnesota)
Engineering Optimization: Lecture 13
Oct, 2013
1 / 39
Exam Summary
Overall average/median: 68
Overall standard deviation
Lecture 11: More on Linear Programming
Zizhuo Wang
University of Minnesota
Oct, 2013
Zizhuo Wang ( University of Minnesota)
Engineering Optimization: Lecture 11
Oct, 2013
1 / 27
Midterm Exam
Midterm next Monday, Oct 21st, in class (10:15am -12:15pm)
One p
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
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
Lecture 12: Midterm Review
Zizhuo Wang
University of Minnesota
Oct, 2013
Zizhuo Wang ( University of Minnesota)
Engineering Optimization: Lecture 12
Oct, 2013
1 / 32
Midterm Exam
Midterm next Monday, Oct 21st, in class (10:15am -12:15pm)
One piece of note
Lecture 4: The Simplex Algorithm
Zizhuo Wang
Department of Industrial and Systems Engineering
University of Minnesota
Sep, 2013
Zizhuo Wang (Department of Industrial and Systems Engineering University of Minnesota)
Engineering Optimization: Lecture 4
Sep,
Lecture 2: Linear Programming
Zizhuo Wang
Department of Industrial and Systems Engineering
University of Minnesota
Sep, 2013
Zizhuo Wang
Engineering Optimization: Lecture 2
Logistics
Homework 1 posted on Moodle
Due next Monday, Sept 16th
TA oce hour this
Lecture 3: The geometry of Linear Programming
Zizhuo Wang
Department of Industrial and Systems Engineering
University of Minnesota
Sep, 2013
Zizhuo Wang
Engineering Optimization: Lecture 3
Logistics
Homework due next Monday (9/16)
Either drop it in class
Lecture 5: The Simplex Algorithm
Zizhuo Wang
University of Minnesota
Sep, 2013
Zizhuo Wang (University of Minnesota)
Engineering Optimization: Lecture 5
Sep, 2013
1 / 33
Announcements
Homework 2 posted
Due next Wednesday, September 25th
Zizhuo Wang (Unive
Lecture 6: The Simplex Tableau
Zizhuo Wang
University of Minnesota
Sep, 2013
Zizhuo Wang (University of Minnesota)
Engineering Optimization: Lecture 6
Sep, 2013
1 / 32
Recap
Last week, we showed an important fact about linear program
If there is an optima
Lecture 7: Duality Theory
Zizhuo Wang
Department of Industrial and Systems Engineering
University of Minnesota
Sep, 2013
Zizhuo Wang
Engineering Optimization: Lecture 7
Announcement
Homework 2 due today
Homework 3 posted, due next Wednesday (Oct 2nd)
Zizh
Lecture 10: Sensitivity Analysis
Zizhuo Wang
Department of Industrial and Systems Engineering
University of Minnesota
Oct, 2013
Zizhuo Wang
Engineering Optimization: Lecture 10
Announcement
Homework due on Oct 14th
My oce hour this week: Oct 10th, 9-10am
Lecture 8: Duality of LP
Zizhuo Wang
University of Minnesota
Sep, 2013
Zizhuo Wang ( University of Minnesota)
Engineering Optimization: Lecture 8
Sep, 2013
1 / 31
Announcement
Homework 3 is due on Wednesday
Next Mondays (Oct 7th) class is canceled, my oce
IE 5531
Assignment 7 Solution
Problem 1
1. False. Consider the following counterexample:
f (x) = x,
convex
g(x) = x2 ,
convex
2
f (g(x) = x , concave
2. True.
Proof Let h(x) = f (g(x). Assuming f (x) and g(x) are differentiable, take the first order
and s