Semidenite Programming (SDP)
(see sections 9.3 and 10.2 in the book)
Like linear programming but also allows a new kind of constraint:
n
j=1
Here,
xj Aj B
x1 , . . . , xn are the variables
A1 , . . . , An and B are symmetric matrices
chosen by the user

Summary of discussion of scaling invariance
In lecture we discussed the notion of an LP algorithm being scaling invariant.
Let us recall what this means in the context of the ipm we are considering, where
the focus is on a (geometric) primal problem (x L

Duality Theory
1
Conic Programming
Linear programming (LP) and semi-denite programming (SDP) are special
cases of conic programming (CP). The ingredients for a CP instance are a closed,
convex cone K Rn (if x1 , x2 K and t1 , t2 0 then t1 x1 +t2 x2 K), a

A Key Primal-Dual Interior-Point Method
1
Introduction
One of the most fruitful research topics pursued in optimization during the last
25 years has been interior-point methods (ipms), rst for linear programming
and later for general convex programming. I

Fourier-Motzkin Elimination
n
Let P R be a polyhedral set, that is, a set dened by linear equations and
inequalities. Let A be an m n matrix. Here we show that cfw_Ax : x P is a
polyhedral set, too.
The proof is constructive in that from the linear equat

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 4: SOLUTIONS
1. Given vectors a1 , a2 , . . . , am spanning Rn , consider the cone
K =
o
n
x Rn : aTi x 0 (i = 1, 2, . . . , m) .
Prove that a ray R+ d (for 0 6= d Rn ) in this cone is an extreme ray
if an

ORIE 6300
MATH PROGRAMMING I
Fall 2013
MIDTERM: SOLUTIONS
1. Given an mn matrix A and a vector b Rm , suppose the polyhedron
P = cfw_x Rn : Ax = b, x 0
is nonempty.
(i) Calculate the recession cone 0+P .
Now assume P is also bounded.
(ii) What is 0+P ?
(i

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 11: SOLUTIONS
1. Given vectors ai Rn and scalars bi R for i = 1, 2, . . . , m, suppose
the system in the vector x Rn ,
aTi x bi for i = 1, 2, . . . , m,
defines a bounded polyhedron and has a strictly feas

The Simplex Method and Its Geometry
Here we develop the simplex method, emphasizing a geometric perspective rather
than the usual matrix perspective. However, as we progress, we are careful to
relate the two perspectives, since the matrix one is necessary

Chek Beng Chua 11/5/2001
OR63O Notes
Maximum—Flow Minimum—Cut Theorem
Let G = (V, E) be a connected directed graph. Let s and t be distinct vertices in V. Let ue > 0
be given for all e E E.
An s—t flow is a vector of ﬂows x = (maker; satisfying
2 mono: Z

Second-Order Cone Programming (SOCP)
SOCPs are LPs except in that
an additional type of constraint is allowed:
Ax + b cT x + d
where
is the Euclidean norm
A is an m n matrix
b Rm , c Rn , d R
SOCPs are eciently solved by interior-point methods
an appr

Rules of the Game
(the game being the take-home portion of the nal exam)
All take-home problems are due at noon, Thursday, December 8.
Except for contacting me, DO NOT COMMUNICATE WITH
ANYONE ABOUT THE TAKE-HOME PART OF THE
FINAL EXAM until after all of

Here are six problems that exemplify the level of diculty of questions you might
see on the in-class nal exam. The problems here actually came from the nal
exam last year. There was one additional problem last year, which was composed
of numerous true-fal

Chek Beng Chua 10/29/2001
OR630 Notes
Algebraic Interpretation of the Network Simplex Method
Consider a transshipment problem on the connected directed graph G = (V, E) with V =
{v1,.,vn} and E : {el,.,em} (n < m). For each edge ej E E, let cj be the per

Homework Problems for ORIE 6300, Fall 2011
The list of problems will be expanded regularly. Problems will come due in the
order they are listed. You will be told at least one week ahead of any problems that
will be due.
In solving problems, you are always

Non-Linear Programming:
The Karush-Kuhn-Tucker Theorem
1. The Big Picture
Consider an LP and its dual:
max bT y
s.t. AT y = c
y0.
min cT x
s.t. Ax b
We know the primal has an optimal solution i the dual has an optimal solution.
Moreover, we know that if x

(Computational) Complexity Theory
1
Introduction
Computational complexity theory or, simply, complexity theory1 is the
mathematical study of the eciency of algorithms. It aims to formalize what
it means for an algorithm to work quickly.
To create a fully

Chek Beng Chua 10/22/2001
OR630 Notes
Network Simplex Method
1 Directed Graphs
A directed graph G is a pair of sets (V, E), where V is a set of vertices and E is a subset of
edges7 each edge is an ordered pair (U721) of distinct vertices u and y. We denot

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 8: SOLUTIONS
1. (a) Find a closed subset of R2 whose convex hull is not closed.
(b) Consider a set S Rn . Prove that any point in the convex hull of
S is in fact a convex combination of n + 1 points in S.

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 12
Due: Tuesday December 10 in class.
Answer all questions, explaining your answers carefully, and acknowledging any source or person you consult.
1. For pairs 0 < (x, s) Rn Rn , prove that the primal-dual

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 2
Due: Tuesday 17 September in class.
Answer all questions, explaining your answers carefully, and acknowledging any source or person you consult.
1. Construct an example of an infeasible linear program in

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 8
Due: Thursday October 31 in class.
Answer all questions, explaining your answers carefully, and acknowledging any source or person you consult.
1. (a) Find a closed subset of R2 whose convex hull is not

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 11
Due: Tuesday November 26 in class.
Answer all questions, explaining your answers carefully, and acknowledging any source or person you consult.
1. Given vectors ai Rn and scalars bi R for i = 1, 2, . .

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 9
Due: Tuesday November 12 in class.
Answer all questions, explaining your answers carefully, and acknowledging any source or person you consult.
1. Find the ellipsoid of minimum volume containing the half

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 7
Due: Wednesday October 23 in recitation.
Answer all questions, explaining your answers carefully, and acknowledging any source or person you consult.
1. Consider the following linear program:
maximize
x1

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 1
Due: Tuesday 10 September in class.
Answer all questions, explaining your answers carefully, and acknowledging any source or person you consult.
1. Consider the following system:
2x1
x1
x1
2x1
+ 2x2 x3

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 5
Due: Tuesday 8 October in class.
Answer all questions, explaining your answers carefully, and acknowledging any source or person you consult.
1. Solve the following linear program
maximize
subject to
2x1

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 4
Due: Tuesday 1 October in class.
Answer all questions, explaining your answers carefully, and acknowledging any source or person you consult.
1. Given vectors a1 , a2 , . . . , am spanning Rn , consider

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 5: SOLUTIONS
1. Solve the following linear program
maximize
subject to
2x1
2x1
x1
x1
x1
+ 3x2
+ x2
+ x2
+ x2
,
x2
10
6
4
0
graphically. Then use the revised simplex method to solve the equivalent probl

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 2: SOLUTIONS
1. Construct an example of an infeasible linear program in the form
(
maximize cT x
subject to Ax b,
with an infeasible dual problem.
Solution.
The problem
maxcfw_1.x : 0.x 1, x R
has dual
min

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 10: SOLUTIONS
1. Given a convex set Z Rn , consider a convex function f : Z R, by
which we mean that the epigraph of f ,
n
o
(z, ) Rn R : f (z) ,
is convex. For any point z Z, we define a set
f (
z) =
n
o

ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 1: SOLUTIONS
1. Consider the following system:
2x1
x1
x1
2x1
+ 2x2 x3
x2 + 2x3
2x2 + x3
x2
1
2
2
2 .
(1)
(2)
(3)
(4)
Use Fourier elimination to find either a solution, or a nonnegative linear
combinat