Figure 15.11 Graphical solution to the LP problem in Example 15.15.
and
l 9 ' 1
(l0 552) 01 -i- 5202 + (30 - i552) a3 + (2 E62) a5 :2 b.
Substituting 52 z 4, we obtain
80,] + 4&2 + 120.3 I: b.
The solution is [8,4,12,0,O]T and the corresponding value of t
1. Determine whether the subset of R” is Closed and also whether it is
bounded. Circle the answers. (No work is requiredfor credit.)
(21) (4 points)
{Cruz/,2) l ZZI2+y2, 3:20, 231}
WM . E
! Closed l . Not Closed z Bounded i Not Bounded
(b) (4 points)
{(
3.2 Eigenvalues and Eigenvectors
Chenxu Wen
Definition 1. Let A be an n n matrix. A scalar (maybe a complex number) is
called an eigenvalue of A if there is a non-zero vector ~v satisfying
A~v = ~v .
Such a vector is called an eigenvector of A correspondi
Midterm Exam
University of California, Riverside
Department of Mathematics
MATH 120 NAME: AMW Kg!
Spring2017 (print)
In signing my name below I pledge that I have neither given nor received help on this exam.
Pledged: *3
Instructions: This exam ha
Homework 4
We always apply the simplex method in the tableau notation!
To receive full credit, justify your answer.
Question 1. Recall the fundamental theorem of linear programming.
Theorem. If a linear program has a finite optimal solution, then it has a
Homework 3
To receive full credit, justify your answer.
Question 1 (Extreme points). Graphically, determine the extreme points of
the following set. You do not need to justify your answer. (It is easy to see
them, but it requires some work to show why the
Homework 0: a brief review of Math 10A and Math 31
Only for this HW set, you do not justify your answer. However, you are
expected to know at least one correct way to solve the problems.
Question 1. Find all solutions to each of the following following sy
Midterm practice problem set solution
MATH 120
Name:
Student ID:
Your TA:
Instructions:
This exam has a total of xx points.
You have xx minutes. (You may not leave in the first 30 minutes.)
You must show all your work to receive full credit.
You may u
Homework 2
To receive full credit, justify your answer.
Question 1 (Definition). State the definition of the highlighted terms.
(a) A linear optimization problem/linear program.
Solution. A linear program is just another name for a linear optimization
pro
Homework 3
To receive full credit, justify your answer.
Question 1 (Extreme points). Graphically, determine the extreme points of the following set. You do not
need to justify your answer. (It is easy to see them, but it requires some work to show why the
Homework 6
Brief review of dualities
Linear program in standard form and its dual:
minimize
(P ) subject to
Maximize
z = cT x
dual
Ax = b (D) subject to
x 0.
w = by
AT y c
yi free.
Theorem 1 (Weak duality). If x is feasible for (P ) and y is feasible for
Notations
Chenxu Wen
We record a few notations from linear algebra that we will use in this course for
future reference.
1. We use capital letters to denote matrices, e.g. A, B, C, . Sometimes we write
the size of a matrix in the lower-right corner of the
3.1 Linear Transformation
Chenxu Wen
1
Definition of a linear transformation
Definition 1. A function f : Rm Rn is called a linear transformation if
1. f (a~x) = af (~x), for all ~x in Rm and for any scalar a;
2. f (~x + ~y ) = f (~x) + f (~y ) for all ~x
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Midterm Exam Two Sample
Monday, May 18
Mathematics 120 - Optimization
Professor Chadwick
(1) Consider the problem
minimize x1 x2
subject to 9x2 + x2 = 4
1
2
(a) (5 points) Find all points satisfying the Lagrange condition.
(b) (5 points) Use the second-or
Homework 6
Mathematics 120 - Optimization
Professor Chadwick
Due Friday, June 5
Problems taken from Chong, E.K.P. and Zak, S, An introduction to optimization, second edition,
available on iLearn.
15. Introduction to linear programming
(1) Exercises 15.1,
Math 120 Final Study Guide
Chapter 20: Problems With Equality Constraints
20.1 Introduction
Standard form
Feasible point
Maximize f(x) = - minimize f(x)
20.2 Problem Formulation
Regular point
Jacobian matrix
20.3 Tangent and Normal Spaces
Curve A curve C
Math 120 Optimization: Homework 2
Chenxu Wen
2 2 2
1. Let A = 2 2 2. Show that all leading principal minors of A are non2 2 0
negative, but A is not positive semi-definite. (Hint: Show that A is not positive
semi-definite by computing its eigenvalues.)
2.
Homework 4
We always apply the simplex method in the tableau notation!
To receive full credit, justify your answer.
Question 1. Recall the fundamental theorem of linear programming.
Theorem. If a linear program has a finite optimal solution, then it has
a
Homework 5
Brief review of dualities
Linear program in standard form and its dual:
minimize
(P ) subject to
z = cT x
Maximize
dual
Ax = b (D) subject to
x 0.
w = by
AT y c
yi free.
Theorem 1 (Weak duality). If x is feasible for (P ) and y is feasible for
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