ECES 811 Homework 4 Solution
John MacLaren Walsh, Ph.D.
ECES 811, Spring Quarter, 2012
1. You shouldnt need a solution for this as it is a straightforward exercise.
2. You shouldnt need a solution for this as it is a straightforward exercise.
3. Consider
Polyhedral Representation Conversion
John MacLaren Walsh, Ph.D.
April 27, 2014
1
References
See Padberg book and work by K. Fukuda, D. Avis. Details will be added April 27, 2014.
2
Overview
We saw in the rst two lectures in the course that polyhedra have
The Structure of Polyhedra & Linear Programming
John MacLaren Walsh, Ph.D.
January 12, 2012 & January 19, 2012
1
References
Numerous books have been written about linear programming. I recommend the rst one below for a rigorous graduate
level treatment th
Numerical Optimization Methods I
John MacLaren Walsh, Ph.D.
1
Reference
Nonlinear Programming, 2nd Ed., Dimitri P. Bertsekas. Athena Scientic, 1999.
Gradient Methods for Unconstrained Optimization
Steepest Descent
Suppose we are currently at a point xk a
Convex sets, convex functions, & some of their properties. (Part I)
John MacLaren Walsh, Ph.D.
April 12, 2010
1
References
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge Univ. Press, 2004.
R. T. Rockafellar, Convex Analysis, Princeton Univ.
ECES 811 Homework 1 Solutions
John MacLaren Walsh, Ph.D.
ECES 811, Winter Quarter, 2012
x2
o
o
o
x1
Figure 1: The polyhedron P. The extreme points are circled in blue and the half-line faces of P (which
recede in the direction of the extreme directions, s
ECES 811 Homework 2 Solutions
John MacLaren Walsh, Ph.D.
ECES 811, Winter Quarter, 2012
1. Consider the linear program
min
(x1 ,x2 )P
x1 + 2x2 ,
x1 + x2 1
x2 1
P := (x1 , x2 )
x1 + x2 1
(1)
(a) Find the extreme points of the constraint set P.
The extreme
Nonlinear Programming, Part II: Duality and Geometric Multipliers
John MacLaren Walsh, Ph.D.
Consider the general family of optimization problems of the form
x
inf
f (x)
xX
hi (x) = 0, i cfw_1, . . . , I
gj (x) 0, j cfw_1, . . . , J
(1)
where X RN , f : X
ECES 811 Homework 3 Solutions
John MacLaren Walsh, Ph.D.
ECES 811, Spring Quarter, 2010
1. For each set below, determine whether or not it is convex and prove your answer
(a)
(x, y) x2 + y 2 1, x 0
This set is convex. It is the intersection of two convex
Introduction to Combinatorial Optimization
John MacLaren Walsh, Ph.D.
1
Reference
Matroid Theory, D. J. A. Welsh, Dover, 2010. (Academic Press, 1976)
Matroid Theory, 2nd Ed., J. Oxley, Oxford University Press, 2011.
Combinatorial Optimization: Algorith
Polyhedral Representation Conversion
John MacLaren Walsh, Ph.D.
April 27, 2014
1
Reading References
M. Padberg, Linear Optimization and Extensions, 2nd Ed., Springer, 1999. See 7.4.
D. Avis, lrs: A Revised Implementation of the Reverse Search Vertex Enu
The Structure of Polyhedra & Linear Programming
John MacLaren Walsh, Ph.D.
April 2, 2014 & April 9, 2014
1
References
Numerous books have been written about linear programming. I recommend the rst one below for a rigorous graduate
level treatment that can
ECES 811 Homework 1 Solutions
Yunshu Liu
ECES 811, Spring Quarter, 2014
0
1. Prove that x is an extreme ray of the (assumed line free) recession cone C = x RN |Hx 0 if
and only if there are N 1 linearly independent inequalities among Hx 0 that hold with e
Convex sets, convex functions, & some of their properties. (Part I)
John MacLaren Walsh, Ph.D.
April 16, 2014
1
References
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge Univ. Press, 2004.
R. T. Rockafellar, Convex Analysis, Princeton Univ.
ECES 811 Homework 3 Solutions
Yunshu Liu
ECES 811, Spring Quarter, 2014
1. For each set below, determine whether or not it is convex and prove your answer
(a)
(x, y) y x4 , 2 x 2
This set is convex. It is the intersection of two convex sets.
(b) cfw_(x, y
ECES 811 Homework 2 Solutions
Yunshu Liu
ECES 811, Spring Quarter, 2014
1. Consider the linear program
min
(x1 ,x2 )P
x1 + x2 ,
2x1 x2 8
6x1 + x2 2
P := (x1 , x2 )
x1 3x2 3
3x1 2x2 9
(1)
(a) Find the extreme points of the constraint set P.
3
The extreme p