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

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

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

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

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