MthSc810 Mathematical Programming
Fall 2011, Solutions to Homework #6
Problem 1. Implement, in AMPL, a solution to the following problem (it might be useful in the next midterm): A classroom can be thought of as a set of n two-dimensional vectors, indicat
Dual feasibility MthSc 810: Mathematical Programming Lecture 13
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
Consider a primal LP and its dual: max c x s.t. Ax b x 0 min u b s.t. u A c u 0
Add slack variables to the primal, surplus1 va
Dual of a problem in standard form MthSc 810: Mathematical Programming Lecture 12
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
If the primal is min c x s.t. Ax = b x0 then, by applying the magic table, the dual is max u b s.t. A uc Not
MthSc 810: Mathematical Programming Lecture 12
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
October 11, 2011 Reading for today: Sections 4.3-4.4 Reading for Oct. 13: Sections 4.5-4.6 Midterm 2: November 1, 2011, 2pm
Dual of a problem i
How fast is the simplex method? MthSc 810: Mathematical Programming Lecture 11
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
Consider an LP in standard form identified by (n, m, c, A, b). How much time does it take to solve this problem
MthSc 810: Mathematical Programming Lecture 11
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
October 6, 2011 Reading for today: Sections 3.7 and 4.1-4.2, textbook Reading for Oct. 11: Sections 4.3-4.4, textbook Homework #7 is out! Due O
Recap: the simplex method MthSc 810: Mathematical Programming Lecture 10
Reduce problem to standard form Pietro Belotti
Dept. of Mathematical Sciences Clemson University
Add slack variables Get initial dictionary If infeasible, do Phase I
If no feasible s
MthSc 810: Mathematical Programming Lecture 10
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
September 29, 2011 Reading for today: Sections 3.3, textbook Reading for Oct. 4: Sections 3.7 and 4.1-4.2, textbook
Recap: the simplex method
R
MthSc 810: Mathematical Programming Lecture 9
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
September 22, 2011 Reading for today: Sections 3.4-3.5, textbook Seminar at 4:45pm @M-103 by Banu Soylu Midterm next Tuesday. Open or Closed boo
MthSc 810: Mathematical Programming Lecture 8
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
September 20, 2011 Reading for today: Chapters 3.3 and 3.4, textbook Reading for Sep. 22: Chapters 3.5, textbook Midterm exam: Next Tuesday, 2:0
Recap MthSc 810: Mathematical Programming Lecture 7
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
Consider the Linear Optimization problem LP : mincfw_c x : Ax = b, x 0. If P = cfw_x Rn : Ax = b contains 1 extreme points, and + LP admit
MthSc 810: Mathematical Programming Lecture 7
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
September 15, 2011 Reading for today: textbook, Sections 3.1 and 3.2 Reading for Sep. 20: Sections 3.3 and 3.4 Homework #4 out today due Septemb
AMPL MthSc 810: Mathematical Programming Lecture 6
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
a modeling language for optimization problems an interface between problems and solvers easy, intuitive syntax it's interpreted1 that means
MthSc 810: Mathematical Programming Lecture 6
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
September 13, 2011 Sep. 13: AMPL. Reading: Bob Fourer's notes, Chapters 1-2 Reading for Sep. 15: textbook, Sections 3.1 and 3.2
AMPL
a modeling
MthSc 810: Mathematical Programming Lecture 14
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
October 25, 2011 Reading for today: Section 4.6 and 4.7 Reading for Oct. 27: Section 4.8-4.10 Midterm 2: Next Tuesday, 2pm3:15pm
Farkas' lemma
Farkas' lemma MthSc 810: Mathematical Programming Lecture 14
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
Lemma Consider A Rmn and b Rm . Then (1) x 0 such that Ax = b, or (2) p such that p A 0 and p b < 0, Therefore, either a (non-neg
MthSc810 Mathematical Programming
Pietro Belotti
January 23-27, 2012
An example
You work at a company that sells food in tin cans, and are charged with designing the next generation can, which is a cylinder made of tin The can must contain V = 20 cu.in. (
The Traveling Salesperson Problem (TSP) MthSc 810: Mathematical Programming Lecture 21
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
A salesperson has to visit n cities and then return home. She/he would like to spend as little as possi
MthSc 810: Mathematical Programming Lecture 21
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
December 8, 2011 Reading for today: Section 6.3 Homework #13 due today, 6pm EST.
The Traveling Salesperson Problem (TSP)
A salesperson has to v
Recap: Lagrangian relaxation MthSc 810: Mathematical Programming Lecture 20
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
Consider a problem zOPT = min c x s.t. Ax = b d x=f x 0. Lagrangian relaxation applied to the last constraint yiel
MthSc 810: Mathematical Programming Lecture 20
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
December 6, 2011 Reading for today: Sections 6.1, 6.2 Reading for Thursday: Sections 6.3, 6.4.
Recap: Lagrangian relaxation
Consider a problem
Cuts MthSc 810: Mathematical Programming Lecture 19
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
An s - t cut of a (flow) network G = (V, A) is a partition of V into S and T = V \ S such that s S and t T For flow f , net flow across a
MthSc 810: Mathematical Programming Lecture 19
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
December 1, 2011 Reading for today: Sections 7.5 Reading for Tuesday: Sections 6.1, 6.2.
Cuts
An s - t cut of a (flow) network G = (V, A) is a
(Undirected) graphs MthSc 810: Mathematical Programming Lecture 18
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
A graph1 G is a tuple (V, E) where V is the set of nodes and E is the set of edges. Each edge is a subset of V of cardinali
MthSc 810: Mathematical Programming Lecture 18
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
November 29, 2011 Reading for today: Sections 7.1, 7.2 Reading for Thursday: Sections 7.5. Final: take home, out Tue Dec. 6, turn in Tue Dec. 1
Value functions in Optimization MthSc 810: Mathematical Programming Lecture 17
Pietro Belotti
Dept. of Mathematical Sciences Clemson University
Consider the function F(b) = mincfw_c x : Ax = b, x 0. Note: F(b) = F(b) for any 0. Also, F(b) is a function of