Math 482 Class log
- (1/22/2014)
Passed out course outline
and emergency information,
linear program in general form,
diet problem example,
converting an LP from general form to canonical and standard forms.
- (1/24/2014)
Example of converting from general to standard form,
drawing the feasible region of a 2d LP in the plane,
introduction to the simplex method,
basic feasible solutions,
started
first example of the simple method,
passed out
homework 1.
- (1/27/2014)
Finish first example of the simplex method,
lexicographic ordering (section 14.2 in book),
basic feasible solutions and
their associated basis,
started discussion the
the basic easy case of the
simplex method in general
- (1/29/2014)
Completed proof of
the basic easy case of the
simplex method in general
- (1/31/2014)
Presented an
example
of the two phase simplex method and proved
the
two phase simplex method in general.
passed out homework 2.
- (2/03/2014)
Discussed the simplex method as a series
of pre-multiplications by special matrices with
handout
- (2/05/2014)
Finished discussion about the simplex method as a series
of pre-multiplications by special matrices
presented the revised simplex method
- (2/07/2014)
Discussed cycling in certain version of the simplex method,
and Bland's algorithm.
Started began proof Bland's algorithm
- (2/10/2014)
announced quiz on 2/14/2014,
passed out handout on Bland's algorithm,
definitions related to convexity: convex combination, convex set, hyperplane, halfspace,
polyhedron, extreme point, facets and vertex.
Started proof relating extreme points and vertices of the set of feasible solutions
of a LP to the basic feasible solutions of an LP.
- (2/12/2014)
Finished proof relating extreme points and vertices of the set of feasible solutions
of a LP to the basic feasible solutions of an LP,
Introduction to duality
- (2/14/2014)
Dual to the diet problem,
weak duality,
the dual of the dual is primal,
proved strong duality relying on the correctness of the simplex method,
Quiz 1
- (2/17/2014)
Short discussion on quiz (Bland's rule),
Reviewed duality,
dual of an LP in standard form,
dual of an LP in general form,
introduced dual simplex method
- (2/19/2014)
Test 1
- (2/21/2014)
dual simplex method handout,
Farkas Lemma as an application of duality (Theorem 3.5)
- (2/24/2014)
Matrix games
- (2/26/2014)
Finished Matrix games and minimax theorem (see handout
introduced complementary slackness
- (2/28/2014)
Proved complementary slackness (Theorem 3.4),
introduced shortest path problem (Section 3.4)
- (3/03/2014)
shortest path problem as an LP (Section 3.4),
dual of the shortest path problem and finding a optimal
dual solution using complementary slackness (Section 3.4)
began talking about the revised simplex method (Section 4.1)
passed out handout
on the revised simplex method
- (3/05/2014)
Continued discussion of the revised simplex method (Section 4.1) and
when through example in
handout
- (3/07/2014)
Networks and flows in networks, discussed revised simplex for the max-flow problem (Section 4.3),
announced quiz on 3/12/2014
- (3/10/2014)
Finished discussion of the revised simplex for the max-flow problem
with example (Section 4.3)
- (3/12/2014)
Primal dual simplex theory (Section 5.1 and Section 5.2)
- (3/14/2014)
More primal dual simplex (Section 5.1 and Section 5.2) and example with
handout
- (3/17/2014)
Example with
presentation
and finished discussion of primal dual theory (Section 5.3)
- (3/19/2014)
Test 2
- (3/21/2014)
Primal dual applied to shortest path (section 5.4)
- (3/31/2014)
Passed back test 2,
Finished primal dual applied to shortest path
with example (section 5.4)
- (4/2/2014)
Primal dual applied to max flow
(section 5.6 and section 6.1))
augmenting path, introduced s,t-cuts,
the capacity of a cut,
Theorem 6.1
- (4/4/2014)
Theorem 6.2 (min-cut,max-flow),
Ford Fulkerson algorithm,
Theorem 6.3 (Ford-Fulkerson terminates
with a maximum flow)j
- (4/7/2014)
If capacities integral Ford-Fulkerson
gives a integral flow,
If capacites are rational Ford-Fulkerson
terminates,
If capacities are real Ford-Fulkerson
may not terminate (Section 6.3),
Problem with the Ford-Fulkerson algorithm
(section 9.2)
Application to finding a maximum matching
in a bipartite graph,
- (4/9/2014)
Ford-Fulkerson gives the edge
version of Menger's Theorem,
described Floyd-Warshall algorithm,
Quiz 3,
- (4/11/2014)
finished Floyd-Warshall algorithm
and presented an example.
- (4/14/2014)
Introduction to Integer Linear Programming (ILP),
the travelling salesman problem (TSP) and the satisfiablity problem as
ILPs (section 13.1)
- (4/16/2014)
Recap of TPS and satisfiabitity problems as ILP,
Definition of unimodular and totally unimodular (TUM),
Theorems 13.1 and Theorem 13.2
- (4/18/2014)
Theorem 13.3 and a corollary,
Brief introduction to the Gomory Cutting Plane algorithm,
Quiz 4
- (4/21/2014)
Description and example of the Gomory Cutting plane algorithm
- (4/23/2014)
Test 3
- (4/25/2014)
Lexicographic dual simplex,
Proof of Gomory cutting plane algorithm
- (4/28/2014)
Finished proof of Gomory cutting plane algorithm,
passed back exam 3
- (4/30/2014)
Introduction to the
Minimum spanning tree problem,
Prim Algorithm and an example,
12.1
- (5/02/2014)
Matroids definition,
Definitions:
independent set,
dependent set,
basis,
cycle/circuit,
statement of theorem 12.5,:w
(section 12.5)
- (5/05/2014)
Test 4
- (5/07/2014)
Proof of Theorem 12.5,
Prove that that for any graph G=(V,E)
E and the subsets of E that form a forest
are a matroid.