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 , . .
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, w
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
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
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.
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+
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+
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, . . . ,
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
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 flo
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
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
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 pro
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,
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 problem
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 du
(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 mea
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 ord
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
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.
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.
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. (
ORIE 6300
MATH PROGRAMMING I
Fall 2013
ASSIGNMENT 6: SOLUTIONS
1. Use the two-phase revised simplex method to solve the following linear
program:
maximize x1 + x2 + x3 x4
subject to
2x2
+ x4 = 3
1
x 6
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.
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. F
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
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.
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. Sol