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
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 contin
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
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 dimen
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 , x
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 can
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
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 bel
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 count
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
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 th
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 Summar
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 n
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 Wed
IE 5531 Midterm - Oct 2016
Page 1 of 12
M IDTERM E XAM
IE 5531
Oct 24th, 2016
INSTRUCTIONS
a) Write ALL your answers in this exam paper.
b) One piece of note is allowed. No computer or calculator with
IE 5531 Midterm - Oct 2016
Page 1 of 12
M IDTERM E XAM S OLUTION
IE 5531
Oct 24th, 2016
INSTRUCTIONS
a) Write ALL your answers in this exam paper.
b) One piece of note is allowed. No computer or calcu
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
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 continuo
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,
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
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
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
H
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 poste
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 im
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 tod
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 d
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)