HW1, M588, 2015 SPRING
DUE: February 16 (M)
NAME:.
SCORE:.
Please write each solution on a separate (new) page.
(1) Show by an example that 2-change does not dene an exact neighborhood for the TSP.
(2) Does the fact that every basic feasible solution of a
Homework 2
M588, 2015 SPRING
DUE: March 4 (W)
NAME:.
SCORE:.
1. We say that a step in the simplex method is degenerate if the cost function stays xed.
In this question you show that it is sometimes necessary to take degenerate steps. That
is, show that th
LECTURE 2
Fri Jan 23, 2015
Convex sets
x, y Rn
(1) Let [x, y] = cfw_(1 )x + y : [0, 1]. Line segment from x to y.
(2) If z [x, y], z is convex combination of x and y. If z = x and z = y, then z is strict convex
combination of x and y.
(3) S Rn is convex i
LECTURE 5
Fri Jan 30, 2015
Simplex method
Notation: For LP mincfw_cT x, Ax = b, x 0, b [n] is a basis if |B| = m and AB nonsingular.
Assume B has an arbitrary ordering. B = cfw_K1 , . . . , Km , B(i) = Ki . AB = [AB(1) , . . . , AB(m) ].
= [n] \ B. B
=
LECTURE 6
Mon Feb 2, 2015
Simplex Method
Initial BFS x Rn , new BFS x Rn , cfw_j0 = B \ B. Ax = b = AB xB xB = A1
b, x
B = 0.
cfw_ B
xB(i)
. cT x =
w Rn , wB = A1
= 0. x = x + (ej w). = mini[m],wB(i) >0 w
B Aj0 , wB
B(i)
cT x + (cj cTB wB ), which yiel
LECTURE 3
Mon Jan 26, 2015
General form of linear program
Definition 3.1 (General form of linear program)
min
[ ]
x
cT (x, x ) Rn Rn )
x
[ ]
[ ]
x
x
A b, A = b, x 0
x
x
or
min
[ ]
x
cT (x, x ) Rn Rn )
x
A [ ]
b
A x
b
A x b
In |O
0
Example 3.
LECTURE 4
Wed Jan 28, 2015
Theorem 4.1 Consider the LP mincfw_cT x : Ax = b, x 0,
(i) If LP is feasible and bounded, then there exists an optimal solution.
(ii) If LP has an optimal solution, then it has an optimal BFS.
there are at most
(n)
m
BFS.
Proof.