MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Definitions and Results
covered by
Test # 1
Test #1 covers Homework sets #1#5 and Lectures #1#9, both problems and
theory. Here are some topics that one should review in detail.
Def

MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Test # 1
Due: 02/22/16
I. (a) Construct the dual problem of the primal LP:
max z = 2x1 x2 10x3 x4 ,
subject to:
x1 + 2x3 x4 3
x2 + 3x3 + 2x4 2
x1 , x 2 , x 3 , x 4 0
(b) State the S

MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Homework Week # 7
Due: 03/03/16
~
1. Consider
the LPmax ~c ~x such
that A~x b and ~x 0, where
1 1
2
1
1 , ~b = 2 , ~c =
A = 1
,
2
1
1
4
(a) Sketch the feasible set, and notice tha

MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Homework Week # 6
Due: 02/25/16
1. Show that (x1 , x2 , . . . , xn ) is a convex combination of two other points
of the feasible set for the Standard Form if and only if
(x1 , x2 ,

MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Homework Week # 8
Due: 03/17/16
1. Let A be an integral m n matrix. Using properties of determinants
prove that A is totally unimodular if and only if then augmented matrix
[A | Im

MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Homework # 10
Due: 03/31/16
1. Using the Separating Hyperplane Theorem prove the following variant of
Farkas Lemma:
Either
cfw_~x Rn |A~x ~b, ~x 0 =
6
or
cfw_~y Rm |~y A 0, ~y 0, ~

MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Homework # 9
Due: 03/24/16
1. Solve the integer program (IP) given below, by the Branch and Bound
method. This time first branch on x1 . (In class we branched on x2 .)
zIP = max 5x1

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MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Test # 1
Due: 02/22/16
I. (a) Construct the dual problem of the primal LP:
max z = x1 3x2 + 8x3 ,
subject to:
x1 + 2x3 4
x2 + 3x3 6
x1 5x2 1
x1 , x 2 , x 3 0
(b) State the Weak Dual

MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Test # 2
04/21/16
1. Consider the integer programming problem (IP):
Max
f (x1 , x2 )
such that 6x1 4x2
x1 + x2
x1 , x2
=
2x1 x2
15
5
0, integers.
(a) Write the (LP)-relaxation of th

MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Homework Week # 3
Due: 02/04/16
1. Which of the following sets are convex and which are not? Give proofs or
explanations to support your answers.
(a) cfw_(x1 , x2 ) | x21 + x22 1
(b

MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Homework Week # 4
Due: 02/11/16
1. Prove that if ~a Rn , ~a 6= ~0 and b R, then H~a,b 6= , i.e. there is some
~x Rn such that ~a ~x = b.
2. Every affine subspace A Rn with dim A = n

MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Homework Week # 1
Due: 01/21/16
1. The coach of a swim team needs to form a 200-yard medley relay
team to send to the state championship. Since most of his best swimmers
are very fa

MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Homework Week # 5
Due: 02/18/16
1. Let ~z1 , . . . ~zm Rn . Let
( m
)
m
X
X
A=
ti~zi |
ti = 1, 1 i m, ti 0 .
i=1
1=1
Prove that A is a convex set.
2. Suppose K Rn is a convex set.
(

MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Homework Week # 12
Due: 04/14/16
1. Let f, g : Z+ R+ be two functions.
(a) Explain why f = o(g) = f = O(g).
(b) Show that the converse is NOT true, i.e. give an example where f =
O(

MATH 7234
Optimization and Complexity
Spring 2016
Northeastern University
Homework Week # 11
Due: 04/07/16
1. Use the KKT conditions to find an optimal solution to:
Max
f (x, y) = x + 2y y 3
such that x + y 1
x, y
0.
2. Consider the function f (x, y, z)