Lecture 6: The Geometry of Linear Programs
Murty Ch 3
1
Linear Combinations Let S = cfw_v 1, . . . , v n be a set of vectors in linear combination of S is any vector 1v 1 + . . . + nv n where 1, . . . , n are scalars. Equivalently If v 1, . . . , v n are

Lecture 9: Multiobjective Programming
Reading: Sections 1.4, 17.1
1
Multiple Objective Programming Frequently analysts are faced with an optimization problem with a set of objectives, often inconsistent with each other. A multiple objective LP is an LP of

Lecture 13:
Interior Point Methods for LPs
1
A Quick Guide to Dierentiable Properties
of Functions
Let f : Rn R be a dierentiable function on n variables. The set of partial derivatives of f at a point
x Rn is described by the gradient :
]
[
x
f ()
x
f ()

Lecture 14:
Quadratic Programs
1
Quadratic Forms
A function f in n variables is quadratic if it is in polynomial form with terms of degree 2 or less. In particular,
quadratic functions have the form
f (x) =
n
n
mij xixj +
i=1 j=1
T
n
cj xj
j=1
= x M x + c

Lecture 11: Integer Linear Programs
1
Integer Linear Programs Consider the following version of Woody's problem: x1 = chairs per day, x2 = tables per day max z = 35x1 + 60x2 profits 14x1 + 26x2 190 pine 15x2 60 mahogany 8x1 + 3x2 92 labor x1 0 x2 0 Suppos

Lecture 14: Nonlinear Programming with Constraints
1
Solving the Box Problem Recall the NLP associated with making a box: max z = lwh subject to w+h 3 17lw + 3(lw + 2lh + 2wh) 108 Putting it in the form (N LP ) max f (x) gi(x) bi, i = 1, . . . , m.
we get

Lecture 13: Optimizing over Differentiable Functions
1
Nonlinear Programs Recall the general form of a mathematical program
(M P )
max f (x) min xX
where f : n is the objective function and X is a given feasible set in n. In nonlinear programs nonline

STOR 614
Solutions to Homework Assignment No. 3
1. Solution: Only if part. Suppose that x and y are adjacent, so that
they share n 1 linearly independent active constraints. Suppose that
constraints are
Ax = b,
xN (i) = 0, i = 1, , n m 1,
where N (1), N (

Degeneracy in Linear Programming
I heard that
todays tutorial is
all about Ellen
DeGeneres
Sorry, Stan. But the
topic is just as
interesting. Its
about degeneracy in
Linear Programming.
Photo removed due
to copyright
restrictions.
Degeneracy? Students at

STOR 614 Handout
Representations of polyhedrons
Polyhedral cones: pointed or not
Recall from the applications of duality lecture that a subset S of Rn is a
cone if x S for any x S and any real number 0. Note that this
definition implies that 0 S for any n

STOR 614, Spring 2015
Exam 1
The exam starts at 2:30pm and ends at 3:20pm. It is closed book/notes. Calculators
are allowed, but please do not use a computer. Please write all answers in the blue
book.
1. (20 points) A subset P R5 is defined by P = cfw_x

STOR 614 Handout
Optimality conditions for constrained nonlinear programs
Consider a nonlinear programming problem
min f (x)
x
where the feasible set is given by
= cfw_x Rn | gi (x) = 0, i E ;
gi (x) 0, i I
(1)
with E and I being disjoint finite index s

STOR 614 Handout
The Simplex Method
* The simplex tableau
Consider the following LP, where A Rmn has linear independent rows.
max
s.t.
z = cT x
Ax = b,
x
0.
Given a basis cfw_xB(1) , , xB(m) , let AB(1) , , AB(m) be the basic columns,
and let cB = (cB(1)

STOR 614 Handout
The Two-phase Simplex Algorithm
To solve an LP in general form using the simplex method, the rst step is
to convert it into standard form. In some cases, the converted LP is also in
canonical form, and then one can apply the simplex metho

Notes 3: From Geometry to Simplex Method
IND E 513
IND E 513
Notes 3
Slide 1
Polyhedra in Standard Form
Definition
A polyhedron in standard form is given by cfw_x R n |Ax = b, x 0,
where A is an m n matrix and b is a vector in R m .
I
I
There are m equali

Examples Computing Affine Dimension
Equality system to affine hull: Suppose we want to determine the dimension of the affine set defined by system of equalities in 5 given in Lecture 2: 2x1 + 14x2 + 12x3 - 2x4 = 4 -x1 - 7x2 - 4x3 - 4x5 = -1 x1 + 7x2 + 3x4

Lecture 10:
The Geometry of Linear Programs
Murty Ch 3
1
Linear Combinations
Let S = cfw_v 1, . . . , v n be a set of vectors in m. A
linear combination of S is any vector
1 v 1 + . . . + n v n
where 1, . . . , n are scalars. Equivalently
If v 1, . . . ,

Lecture 8: Parametric Analysis
Reading: Chapter 8, Sections 8.18.12
1
Tradeo Curves/Parametric Analysis Lets revisit the analysis of sensitivity of the optimal solutions to a change in the amount of pine available from b1 = 120 to 120 + . Using the method

STOR 614: Assignment#3 Solutions
1. (a) The original ith -goal equality can be represented by
zi si = gi .
The equation for this row in in the current tableau is
ai1 x1 + . . . + ain xn + si = si .
(since si is necessarily basic in this tableau), and it f

STOR 614 Assignment #6 Solutions
1. (a) The dual to (P S) is
min w = yb
yA c
y 0
where
b1
b2
2
2
3
y = (y1 , y+ , y , y ) b =
b2
b3
A11
A12
21
A
A22
A=
A21 A22
A31 A32
A12
A22
+A22
+A32
A13
A23
+A23
+A33
c = (c1 , c2 , c2 , c3 )
(b) Separating ou

STOR 614: Assignment#1 Solutions
1. (a) The standard max form:
(Ps )
max cx
Ax = b
x 0
has
A11 A12 A12 A13 In2
0
A21 A22 A22 A23
x=
0
0
A=
A31 A32 A32 A33
0 In3
x1
x2
+
x2
x3
s1
s2
1
b
b = b2
b3
c = (c1 , c2 , c2 , c3 , 0, 0)
(b) The canonical max f

STOR 614 Assignment #7: Solutions
1. (a) span(S): Let v 1 , . . . , v k be a set of points in span(S). Then each v i can be
written as a linear combination of u1 , . . . , ur :
i
v =
r
ij uj , i = 1, . . . , k.
j=1
Now any linear combination v 1 , . . . ,

Lecture 5 How They Really Do It: The Articial Phase I Method and The Revised Simplex Method
Reading: Section 2.4, Chapter 5
1
The Articial Variable Phase I Method Consider the problem of nding an initial basic feasible tableau for an LP in standard equali