Math 588 Class log
- (1/21/2015)
Passed out course outline
and emergency information,
optimization problem,
traveling salesman (TSP), linear programming LP and minimum spanning tree (MST) as examples of optimization problems,
definition neighborhoods,
exact neighborhoods
(Sections 1.2, 1.3 and 1.4)
- (1/23/2015)
Sections 1.5 and section 1.6,
Equivalence of various forms of linear problems
(roughly section 2.1)
- (1/26/2015)
Example of conversion to standard form,
assumptions (Ax = b has at least one solution
and A has rank m basically assumption 2.1),
Examples of feasible regions,
Notation A_B and x_B,
definitions:
bfs, basic variable, nonbasic variable,
degenerate bfs, basis (as a subset of [n]),
Proved two propositions :
Every basis has at most
corresponding bfs, a feasible solution
is basic if and only if A_K has rank |K|
(K is the set of indices j such that x_j > 0).
- (1/28/2015)
Proved fundamental theorem:
if LP is feasible and bounded,
then there exists an optimal solution and
if LP has an optimal solution, then there
exists an optimal basic feasible solution
(not stated as a theorem in book).
Prove equivalence of vertices, extreme points
of feasible set and basic feasible solutions
(Theorem 2.4 in book)
- (1/30/2015)
Section 2.4,
Theorem 2.7 on moving to a new bfs,
Proposition on determining if an LP is unbound,
relative cost of j_0.
- (2/2/2015)
Theorem 2.8 (optimality criterion),
Simplex example,
- (2/4/2015)
Theorem on Tableaus and simplex algorithm,
Two-phase simplex
- (2/6/2015)
Cycling and lexicographic simplex,
Bland's algorithm (not finished)
- (2/9/2015)
Finish Bland's algorithm,
intro to duality,
weak duality,
strong duality,
duality in general
- (2/11/2015)
The dual of the dual is primal,
Primal and Dual can both be infeasible,
complementary slackness,
example Konig-Egervary with linear programming
- (2/13/2015)
Farkas lemma from strong duality,
strong duality from Farkas lemma,
intro to complexity
- (2/16/2015)
Discussion on complexity,
start proof that simplex with
Dantzig's rule is not polynomial
- (2/18/2015)
Finished proof that
Dantzig's rule is not polynomial
(the argument given in class
is from Alexander Schrijver's book
"Theory of Integer and Linear Programming"
section 11.4 pages 139-141),
brief intro to the Ellipsoid method
- (2/20/2015)
The ellipsoid method
- (2/23/2015)
The ellipsoid method continued,
intro to matrix games
- (2/27/2015)
No class
- (3/2/2015)
Matrix games, minimax theorem
- (3/4/2015)
Integer linear programming,
TSP example
- (3/6/2015)
TU matrices,
- (3/9/2015)
shortest path problem as example
of linear program defined by TU matrix,
revised simplex
- (3/11/2015)
Max flow and revised simplex
- (3/13/2015)
Dantzig-Wolfe decomposition,
Primal-dual algorithm intro
- (3/16/2015)
Primal dual in general,
Primal dual and shortest path,
- (3/18/2015)
Primal dual and shortest path cont.,
Primal dual and max flow,
Floyd Fulkerson algorithm
- (3/20/2015)
Max flow = min cut,
Floyd-Warshall algorithm
- (3/30/2015)
Min cost flow (Section 7.1),
Algorithm cycle (Section 7.2),
Algorithm buildup (Section 7.3)
- (4/1/2015)
The alphabeta algorithm for Hitchcock
(section 7.4)
- (4/3/2015)
Section 11.2 The Hungarian method for
minimum weight matching from the
alphabeta algorithm
- (4/6/2015)
Transformation of min-cost flow
to Hitchcock (Section 7.5),
Introduction to minimum spanning trees,
Lemma 12.1.
- (4/8/2015)
Prim and Kruskal alg (Section 12.1)
The greedy alg (Section 12.3)
Definition of Matroids,
subset systems,
Characterization of Matroids Theorem(12.5),
- (4/10/2015)
Rank of a matroid,
examples of matroids
(linear, graphic and partition),
intersection of matroids,
bipartite matching as the intersection
of two matroids,
first technical lemma to prove
Edmonds max cardinality of intersection
of two matroids theorem
- (4/13/2015)
second technical lemma on matroid intersection,
and description of algorithm and beginning
of proof
- (4/13/2015)
Finished proof of algorithm for max
cardinality set in the intersection
of two matroids
- (4/15/2015)
ILP and its LP relaxation,
Branch and bound for ILP section 18.1,
and branch and bound in a more general
context 18.2
branch and bound for TSP
- (4/17/2015)
second algorithm for branch and bound and TSP
Dominance relation 18.4
- (4/20/2015)
Section 18.6, Dynamic programming
Example 18.8 Dynamic programming and the TSP
- (4/22/2015)
Gomory cutting plane algorithm
- (4/24/2015)
Gomory cutting plane algorithm proof
(the argument given in class
is from Alexander Schrijver's book
"Theory of Integer and Linear Programming"
section 21.8 pages 356-358),
Intro to approximation algorithm,
min vertex cut
- (4/27/2015)
Approximation algorithms,
Theorem 17.10, Theorem 17.1,
Theorem 17.2 (sketch),
Theorem 17.3, Theorem 17.4
- (4/29/2015)
Christofides algorithm for 1/2 approximation
algorithm for TSP Theorem 17.5,
Machine scheduling approximation algorithm
from section 8.3 of the book
Understanding and using linear programming
- (5/01/2015)
Machine schedule algorithm continued
- (5/04/2015)
The Network simplex method
- (5/06/2015)
Codes and the Delsarte bound