Math 482 Class log
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(1 - Monday 8/22/2016)
Brief syllabus review,
Examples adapted from Gartner and Matouskek (GM) Section 1.1 -
unique optimum, infinite number of optimal solutions, infeasible, unbounded LPs.
Definitions: linear program, objectives function, constraint,
linear function, affine function, feasible solution, feasible region,
optimal solution/optimum, infeasible, unbounded.
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(2 - Wednesday 8/24/2016)
Section 2.1 in Papadimitriou and Steiglitz (PS),
General/Canonical/Standard forms, converting between forms,
introduction to simplex
Definitions:
general form, standard form, canonical form, slack variable, surplus variable
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(3 - Friday 8/26/2016)
intro and terms for simplex, standard form assumption, full rank assumption, A_B notation for matrices and
c_B^T notation for vectors,
definition of basis, simple simplex example and solved for a basis, basic solution, feasible basis, basic feasible solution
started full simplex presentation - will finish on Monday
Definitions:
full rank, solved for a basis, basis, basic solution, basic variables, nonbasic variables, basic feasible solution,
feasible basis
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(4 - Monday 8/29/2016)
finished simplex presentation, tableau notation, started unbounded simplex example
Definitions:
tableau, pre-multiplication matrix, relative cost of a variable/column, relative cost vector
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(5 - Wednesday 8/31/2016)
finished unbounded simplex example, started theorem 1
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(6 - Friday 9/2/2016)
Finished Theorem 1, started introduction to cycling
Definitions:
degenerate basic feasible solution
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(7 - Wednesday 9/7/2016)
Finished introduction to cycling and example,
Introduced lexicographic ordering (section 14.2 in GM),
Starting Theorem 14.1 (lex-simplex)
Definitions:
zero-level, degenerate basic feasible solution, cycling, lexicographically positive/negative/zero,
lexicographically greater than/equal to/less than
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(8 - Friday 9/9/2016)
Finished Theorem 14.1 (lex-simplex)
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(9 - Monday 9/12/2016)
Bland's anti-cycling algorithm, Two phase-simplex,
Definitions:
artificial variables, driving artificial variables out of the basis
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(10 - Wednesday 9/14/2016)
Fundamental Theorem, Introduction to Duality
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(11 - Friday 9/16/2016)
Definition 3.1, Weak duality, Theorem 3.2 (The dual of the dual is the primal)
Definitions:
Primal, Dual
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(12 - Monday 9/19/2016)
Strong duality (Theorem 3.1)
- (Tuesday 9/20/2016) - Test 1
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(13 - Wednesday 9/21/2016)
Figure 3-1, Example of Primal and Dual problems that are both infeasible,
Example 2.1 and 3.1 - Diet Planner/Pill Maker, Theorem 3.4 Complementary Slackness
Definitions:
Complementary slackness conditions
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(14 - Monday 9/26/2016)
Theorem 3.4 Complementary Slackness proof. Two examples - determining if
a vector is optimal for the primal using complementary slackness and
solving a LP in standard form with two constraints by solving the dual graphically
and using the complementary slackness conditions. Intro to Farkas' Lemma.
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(15 - Wednesday 9/28/2016)
Passed back and discussed exam 1.
Farkas' Lemma and proof.
Definitions:
hyperplane, halfspaces generated by a hyperplane, convex polyhedron, cone generated by a set of vectors
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(16 - Friday 9/30/2016)
Dual simplex method presentation, basic sensitivity analysis (marginal/shadow prices)
Definitions:
dual feasible basis, marginal/shadow price
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(17 - Monday 10/03/2016)
zero-sum games section 8.1 in GM - Lemma - β(x) = min_j x^T M e_j and α(y) = max_i e_i^T M y
Definitions:
zero-sum game, pure strategy, mixed strategy, expected payout, β(y), α(x) and worst-case optimal.
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(18 - Wednesday 10/05/2016)
continue with zero-sum games section 8.1 (mixed) Nash equilibrium, Lemma 8.1.1, using linear programming to compute the worst case optimal solution
for Alice and Bob
Definitions:
(mixed) Nash equilibrium
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(19 - Friday 10/07/2016)
minimax theorem 8.1.3, started integer linear programming
Definitions:
value of a game, fair game, ILP (integer linear program)
(20 - Monday 10/10/2016)
max matching and min vertex cover as ILPs, dual ILPs,
satisfiability
Definitions:
relaxation of an ILP, maximum matching, minimum vertex cover, ZOLP,
satisfiability as an ILP (example 13.4 in PS)
boolean variable, boolean formula, satisfiable, clause
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(21 - Wednesday 10/12/2016)
traveling salesman as an ILP (example 13.1 in PS),
totally unimodular matrices,
Theorem 13.1 (The basic feasible solutions of a LP in standard
form are integer if the corresponding matrix is TUM and b is integer)
Definitions:
subtours, MILP, unimodular, totally unimodular
(22 - Friday 10/14/2016)
If A TUM, -A, A^T, (A|A_j), (A|-e_j) all TUM,
If P is an LP in general form and b is integer that has an optimal solution, then P
has an integer optimal solution and the same holds for the dual,
statement of 13.3 and examples
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(23 - Monday 10/17/2016)
Koenig's Theorem,
Proof of 13.3,
Intro to branch and bound
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(Tuesday 10/18/2016) - Test 2
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(25 - Wednesday 10/19/2016)
Finished branch and bound,
Discussed directed graphs and beginning of shortest path problem
Definitions:
kill a node, directed graph, node/arc incidence matrix of a directed graph,
directed path, directed cycle, shortest path problem
(26 - Friday 10/21/2016)
Section 3.4,
Solving the shortest path problem with simplex
and the dual of the shortest path problem,
implications of complementary slackness,
introduction to circulations
Definitions: Circulation
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(27 - Monday 10/24/2016)
Decomposing a circulation into at most m-1 circulations on cycles,
(s,t)-flow, network, max flow problem,
example, decomposing an (s,t)-flow into
at most m circulations on cycles and (s,t)-flows on (s,t)-paths,
there exists a max flow f that can be decomposed into (s,t)-flows on (s,t)-paths
Definitions: (s,t)-flow, network
(28 - Wednesday 10/26/2016)
Reviewed test 2, Revised simplex introduction
Definitions: CARRY matrix
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(29 - Monday 10/31/2016)
Finished Revised simplex example, started revised simplex and max flow (section 4.3 of PS)
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(30 - Wednesday 11/02/2016)
Finished revised simplex and max flow (section 4.3 of PS),
brief intro to primal dual
Definitions: arc/chain incidence matrix
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(31 - Friday 11/04/2016)
continued discussion of the primal-dual method
Definitions: restricted primal (RP),
dual of the restricted primal (DRP)
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(32 - Monday 11/07/2016)
Finished intro to the primal dual method (section 5.1 and 5.2)
5.1, 5.2 an 5.3 and example.
- (Test 3 - Tuesday 11/08/2016)
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(33 - Wednesday 11/09/2016)
Primal dual applied to shortest path (5.3)
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(34 - Friday 11/11/2016)
Discussion of the algorithm generated by the primal dual
method for shortest path and it's relation to
Dijkstra's Algorithm.
Brief intro to primal dual for max flow (5.4)
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(35 - Monday 11/14/2016)
Discussion of test 3, primal dual for max flow, Ford-Fulkerson algorithm
Definitions:
f-augmenting (s,t)-path
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(36 - Wednesday 11/16/2016)
Labeling algorithm for finding an f-augmenting (s,t)-path (from Figure 6-6 in PS),
Two issues with Ford-Fulkerson algorithm,
Section 6.1 - The Max Flow LP and its dual
Definitions:
f-augmenting (s,t)-path
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(37 - Monday 11/28/2016>
Section 6.1,
feasible solution of dual of max flow LP corresponding to (s,t)-cut (Theorem 6.1),
Complementary slackness and the max flow LP and its dual,
Max flow - min cut theorem and example
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(38 - Wednesday 11/30/2016>
Floyd-Warshall (section 6.5)
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(39 - Friday 12/02/2016>
Floyd-Warshall example, Started min-cost flow (section 7.1)
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We will finish section 7.1 and 7.2 on Monday.