4.43
(a) Consider the minimization of c1 x1 + c2 x2 subject to the constraints: x2 3 x1 2x2 + 2, x1 , x2 0.
Find necessary and sufficient conditions on (c1 , c2 ) for the optimal cost to be finite.
The dual becomes a maximization of 3p1 + 2p2 subject to t
IE411 Supplemental Materials: Game Theory and
Assortment Optimization
Xin Chen
1.
Introduction
Game theory provides a powerful mathematical framework for modeling and analyzing systems with
multiple decision makers, referred to as players, with possibly c
Homework 6 solutions
Exercise 5.5
(a) c3 0, c5 0 corresponding to finite optimal value, or c3 0, c5 < 0, , , 0 corresponding to optimal value
.
(b) Another optimal BFS is x = [0, 7/4, 3/4, 1/2, 0] by letting x3 enter the basis and x1 exist.
(c) When c3 <
University of Illinois at Urbana-Champaign
Department of Industrial and Enterprise Systems Engineering
IE411 Optimization of Large-Scale Linear Systems
Fall 2016
Lectures: 153MEB TR 2-3:20pm
Instructor: Dr. Xin Chen
Office: 216C TB
Fax: 217-244-5705
Phone
Primal and Dual LP Problems
Economic theory indicates that scarce (limited) resources
have value. In LP models, limited resources are allocated,
so they should be, valued.
Whenever we solve an LP problem, we implicitly solve
two problems: the primal resou
Linear
LinearProgramming
Programming(LP)
(LP)
Optimization in Engineering Design
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Why
WhyTalk
TalkAbout
About Linear
Linear Programming?
Program
Computational Methods for
Management and Economics
Carla Gomes
Module 7a
Duality
Duality
Every maximization LP problem in the standard
form gives rise to a minimization LP problem
called the dual problem
Every feasible solution in one yields a bound on
Chapter 4:
Linear Programming
Presented by Paul Moore
What is Linear Programming?
What is Linear Programming?
Say you own a 500 square acre farm. On
this farm you can grow wheat, barley, corn or
some combination of the 3. You have a limited
supply of fert
Integer linear programming
Optimization problems where design variables have to be
integers are more difficult than ones with continuous variables.
The degree of difficulty is particularly damaging for large
number of variables:
With continuous variabl
Chapter 2
Linear
Programming
1
2.1 Introduction to
Linear
Programming
A Linear Programming model seeks
to maximize or minimize a linear
function, subject to a set of linear
constraints.
The linear model consists of the
following components:
A set of dec
Linear Programming
Overview
Formulation of the problem and example
Incremental, deterministic algorithm
Randomized algorithm
Unbounded linear programs
Linear programming in higher dimensions
Lecture 4:
Computational
1
Problem description
Maximize
c1x
Linear Programming-Based
Approximation Algorithms
Shoshana Neuburger
Graduate Center, CUNY
May 13, 2009
1 of 56
Linear Programming (LP)
Given:
m linear inequality constraints
n Non-negative, real-valued variables
Linear objective function
Goal:
Optimi
HOMEWORK 7
Due December 1
For students registered for 3 credits: (1)-(2) below.
For students registered for 4 credits: (1)-(3) below.
(1) Solve the following problem by the decomposition technique using two convexity constraints:
max
s.t.
3x1
2x1
x1
3x1
+
Homework 5 Solutions
Exercise 1
Express the problem as
Minimize
subject to
3x1 x2 3x3 +x4
2x1 +x2 + x3 +x4 12
4x3
3x1
5
x1 +x2
2
x3 +x4 4
x3 +x4 4
x1
x2
x4 0
x3
which is of the form 1
Minimize cx, subject to Ax b and x X,
12
21 1 1
, b = and X = X1 X
The Simplex Method
Back to LP
Consider
( LP ) min cT x
Ax b
x0
Assume
X x Ax b; x 0
From Representation Theorem
k
x j x u j d j
s.t.
j 1
j
j 1
k
j
1
j 1
j 0
u j 0
k
j 1,.,k (1)
j 1,.,
v LP min c x
Rewrite LP as
j 1
s.t. (1)
T
l
j
j
c d
j 1
T
uj
j
4.4
Let A be a symmetric square matrix. Consider the linear programming problem
c0 x
minimize
subject to Ax c
x0
Prove that if x satisfies Ax = c and x 0, then x is an optimal solution.
Suppose Ax = c and x 0. Lets formulate the dual:
p0 c
maximize
subjec
HOMEWORK 1
Due September 6
For students registered for 3 credits: 1.4, 1.8, 1.10, 1.11, 1.15, 1.17, (1) and (2).
For students registered for 4 credits: 1.5, 1.7, 1.10, 1.11, 1.12, 1.15, 1.17, (1), (2)
and (3).
(1) (Two-Person Zero-Sum Game) Consider the R
HOMEWORK 3
Due October 4
For students registered for 3 credits: 3.7, 3.12, 3.17, 3.19, 3.20 and 3.31.
For students registered for 4 credits: 3.7, 3.9, 3.12, 3.17, 3.18, 3.19, 3.20, 3.23,
and 3.31.
For all students, solve problem 3.17 using Excel Solver (i
HOMEWORK 2
Due September 20
For students registered for 3 credits: 2.1, 2.9, 2.10, 2.12, (1), (2).
For students registered for 4 credits: 2.1, 2.6, 2.9, 2.10, 2.12, 2.13, (1), (2) and
(3).
(1) Find all extreme points and extreme directions of the followin
Homework 1 Solutions
Exercise 1.4
minimize 2x1 + 3z1
s.t. z2 + z3 5
x2 10 z1
x2 + 10 z1
x1 + 2 z2
x1 2 z2
x2 z3
x2 z3
1
Exercise 1.8
Let us choose as our objective to minimize the maximum deviation denoted
by maxi |Ii Ii | of the actual illuminations f
Linear-Programming
Applications
Linear-Programming
Applications
Constrained Optimization problems occur
frequently in economics:
maximizing output from a given budget;
or minimizing cost of a set of required
outputs.
Lagrangian multiplier problems requi
HOMEWORK 1
Due September 6
For students registered for 3 credits: 1.4, 1.8, 1.10, 1.11, 1.15, 1.17, (1) and (2).
For students registered for 4 credits: 1.5, 1.7, 1.10, 1.11, 1.12, 1.15, 1.17, (1), (2)
and (3).
(1) (Two-Person Zero-Sum Game) Consider the R
HOMEWORK 6
Due November 17
For students registered for 3 credits: 5.5, 5.8, 5.13, and (1).
For students registered for 4 credits: 5.5, 5.8, 5.13, 5.14 and (1).
(1) Consider the following problem.
Minimize
Subject to
3x1
x1
x1
x1 ,
5x2
+ x2
x2
x2 ,
+
+
x
Exercise 4. 1
The dual involves three variables, p1. p2. p3. The dual problem is
maximize 3102+ 6pc,
subject to 2p; + 339; pg 2 i
3m + .92 ~22; S 1
Pi + 41372 +2133 5 0
Pi. . 2m + P3 = 0
201 S 0: Pa 2 0-
a) True. If the optimal cost was cx, there would ex
IE420 HW1 Wonhee Lee
Daily Return of AAPL from 28 Jan 2014 to 28 Jan 2015Q1.
Daily Return of AMAZON from 28 Jan 2014 to 28 Jan 2015
IE420 HW1 Wonhee Lee
IE420 HW1 Wonhee Lee
Q2.
AMAZON
Apple
IE420 HW1 Wonhee Lee
Q3.
From Q-Q plot of two stocks, we can see