Combinatorial Optimization and Integer Programming

Spring 2015

MATP6620 / ISYE6760

Course basics:

• Course outline.
• LMS.
• Some books are on reserve in the library. You can borrow the materials for up to an hour.

Homeworks:

• Homework 1. Here are the solutions.
• Homework 2. Here are the solutions.
• Homework 3.

Notes:

Introduction:

• Introduction: example problems, linear programming.
  • Lecture 1, Jan 27: Pages 1-7.
  • Lecture 2, Jan 30: Pages 8-15, including the typewritten handouts on the simplex algorithm and an iteration of it.
• Graph theory: introduction, max flow.
  • Lecture 1, Jan 27: Pages 1-6, via the typewritten handout on graph theory (also available as a pdf with more pictures but more page breaks).
  • Lecture 2, Jan 30: Pages 14-20, 24
• Graph theory: minimum spanning trees, matchings. Most of this material is not covered in class.
  • Lecture 3, Feb 3: Pages 1-3, 6-8 (skimmed).

• Introduction to NP-completeness. See also some typewritten notes.
  • Lecture 3, Feb 3: Pages 1-10.
  • Lecture 4, Feb 6: Pages 11-14.
• NP-completeness reductions: Hamiltonian cycle. For a nice presentation of the reduction of 3-SAT to Hamiltonian cycle, see this link from Carl Kingsford.
  • Lecture 4, Feb 6: Pages 1-10.
  • Lecture 5, Feb 10: Page 11.
• NP-completeness: node packing, knapsack, linear programming.
  • Lecture 4, Feb 6: Pages 1-3.
  • Lecture 5, Feb 10: Pages 4-14.
• Polynomial equivalence of separation and optimization.
  • Lecture 5, Feb 10: Pages 1-4.

• Introduction to polyhedral theory. There is a typewritten version of most of these notes.
  (See also the four typewritten handouts from February 13.)
  • Lecture 6, Feb 13: Pages 1-10.
• Facets.
  • Lecture 6, Feb 13: Pages 1-3.
  • Lecture 7, Feb 20: Pages 4-10. See also these typewritten notes for some of the definitions.
• Chvatal-Gomory rounding. See also these typewritten notes for some of the definitions.
  • Lecture 7, Feb 20: Pages 1-5, 13 (skip 6-12).
  • Lecture 8, Feb 24: Pages 14-19. See also this method for showing the Chvatal rank is at least 2.
• Gomory's cutting plane algorithm.
  • Lecture 8, Feb 24: Pages 1-11. See also the recent paper by Fischetti et al. (pdf).
• Mixed integer programming, including Gomory's mixed integer cut.
  • Lecture 9, Feb 27: Pages 2-3.
  • Lecture 10, Mar 3: Pages 4-6.
  • Lecture 15, Mar 20: Pages 1, 7-8.
• Maxcut problems and linear ordering problems.
  • Lecture 10, Mar 3: Pages 4-7.
  • Lecture 11, Mar 6: Pages 1-3.
  See also the handout on the Barahona-Mahjoub routine for finding violated constraints for MAXCUT problems.
• Node packing. Lifting.
  • Lecture 11, Mar 6: Pages 1-5.
  • Lecture 12, Mar 10: Pages 6-9.
• Knapsack problems and satisfiability problems.
  • Lecture 12, Mar 10: Pages 1-3.
• The traveling salesman problem. The Concorde TSP solver is available from this TSP website at Georgia Tech. The website also includes images of TSPs and information about the history of the TSP. Optima 90 contains a discussion of approximation algorithms for the TSP.
  • Lecture 13, Mar 13: Pages 1-10.
• Lift and project, and disjunctive cuts.
  • Lecture 14, Mar 17: Pages 1-9.
  • Lecture 15, Mar 20: Pages 10-11.
• Total unimodularity.
  • Lecture 16, Mar 31: Pages 1, 3, and 4.
  • Lecture 17, Apr 3: Pages 2 and 5.
• Perfect graphs.
  • Lecture 17, Apr 3: both pages.

• Part 1
  • Lecture 18, Apr 7: Pages 1-6.
  • Lecture 19, Apr 10: Page 7.
• Part 2.
  • Lecture 19, Apr 10: Pages 1-8.
• See also the handouts of an example of branch-and-bound (lecture 17), an example of branch-and-cut, and the branch-and-cut algorithm. Also handed out were some computational results from a paper by Achterberg, Koch, and Martin (see here for more tables of results).

• MaxCut, Lovasz Theta number, node packing, completely positive programs.
  • Lecture 20, Apr 14: Pages 1-6, 10, 12-14.
  • Lecture 21, Apr 17: Pages 7, 9, 17-23.

• Lagrangian relaxation.
  • Lecture 22, Apr 21: Pages 1-5
  • Lecture 24, Apr 28: Pages 6-11.
• Cutting plane and bundle methods for solving Lagrangian relaxations.
  • Lecture 24, Apr 28: Pages 2-7. (Skimmed.)
• Benders decomposition.
  • Lecture 25, May 1: all pages.
• Branch and price.
  • Lecture 25, May 1: all pages.

• Lecture 26, May 5.

• GRASP.
  • Lecture 27, May 8: Pages 1-3.
• Simulated annealing.
• Tabu search.
  • Lecture 27, May 8.
• Feasibility pump.
  • Lecture 27, May 8.

Constraint programming. See also a handout on the job shop scheduling OPL example from an Interfaces paper by Lustig and Puget, and Michael Trick's powerpoint presentation on constraint programming.

• Lecture 27, May 8.

• Approximation algorithms.
  • Lecture 28, May 12: Pages 13-15.
• The primal-dual method.
  • Lecture 28, May 12: Pages 1-4.
• Matroids.
  • Lecture 28, May 12.

Handouts:

• Graph theory. (Handed out on January 27.)
• The simplex algorithm and an iteration of it. (Handed out on January 30.)
• Linear algebra. (Handed out on February 13.)
• Affine spaces. (Handed out on February 13.)
• Dimension, polyhedra, and faces. (Handed out on February 13.)
• Extreme points and extreme rays of polyhedra. (Handed out on February 13.)
• The Barahona-Mahjoub routine for finding violated constraints for MAXCUT problems. (Handed out on Mar 3.)
• A brief note on solving continuous knapsack problems. (Handed out on Apr 3.)
• An example of branch-and-bound. (Handed out on Apr 3.)
• The branch-and-cut example and algorithm were taken from this paper. (Handed out on April 7.)
• Computational results with various branching rules, from a paper by Achterberg, Koch, and Martin. (Handed out on April 7.)

Papers and resources:

• Two libraries of MINLP problems: MINLPlib and MacMINLP.
• MINLP references:
  • J.-P. Goux and S. Leyffer, Solving large MINLPs on computational grids, Optimization and Engineering 3(3), 2002, pages 327-346.
  • I. E. Grossmann, Review of Nonlinear Mixed Integer and Disjunctive Programming Techniques, Optimization and Engineering 3(3), 2002, pages 227-252.
  • S. Leyffer, Integrating SQP and branch-and-bound for mixed integer nonlinear programming, Computational Optimization and Applications 18, 2001, pages 295-309.
  • M. Tawarmalani and N. V. Sahinidis, Global optimization of mixed integer nonlinear programs: a theoretical and computational study, Mathematical Programming 99, 2004, pages 563-591.
• Tutorial on Computational Complexity, by Craig Tovey. Appeared in Interfaces 32(3), pages 30-61, 2002. Can be downloaded via the library website.
• The P=NP conjecture is one of the Millennium Prize Problems. A problem based on Minesweeper is NP-complete.
• PRIMES is in P; see also here.
• NP-Completeness columns by David S. Johnson.
• Survey papers on cutting plane algorithms, branch-and-bound, and branch-and-cut.
• An amusing interview with Vasek Chvatal regarding cutting plane methods for the TSP.
• Here is a page on the history of the TSP, with pictures (including one involving Car 54 and one of the optimal tour for a graph with 13,509 cities). This is part of a larger site at Georgia Tech on the TSP. The website also includes the downloadable software Concorde. Two further references for this problem are TSPBIB and Vasek Chvatal's page on the TSP. Instances of TSP can be obtained from the TSPLIB. Hamilton called the problem of finding a route through the vertices of a icosahedron the Icosian game. A similar problem was posed by Euler: Is it possible for a knight to visit every square of a chessboard without visiting any square twice?
• A list of selected textbooks and articles in combinatorial optimization, compiled by Brian Borchers. (Postscript file.) Updated Feb 9, 1999.
• A. Zanette, M. Fischetti, E. Balas. Lexicography and degeneracy: can a pure cutting plane algorithm work?, Mathematical Programming A, 2009, online first. (pdf).
• PORTA, a polyhedral representation algorithm. If you provide the algorithm with an integer programming problem, it will return a list of all the extreme points and information about the facets. Also available from the same site is SMAPO, a library of linear descriptions of polytopes of small instances of various integer programming problems.
• AMPL is a mathematical programming and optimization modeling language. You can input your model into AMPL in a reasonably intuitive way and it will use a solver (such as MINOS or CPLEX) for solving the problem. It is capable of solving linear, nonlinear, and integer programs. It is available from the LMS page for the course. Here is local information about AMPL, including information about using it on RCS. You can download the the whole of the book, chapter by chapter.
• A survey paper on GRASP.
• A survey paper on metaheuristics for the TSP (scroll down to the last couple of papers on the TSP, since the papers are in chronological order).
• A survey paper on genetic algorithms (volume 9, number 3 of INFORMS Journal on Computing, by Colin Reeves).
• A survey paper on tabu search.
• The semidefinite programming homepage maintained by Christof Helmberg.
• A survey paper by Mike Todd on semidefinite programming, with an emphasis on algorithms. (Acta Numerica 10 (2001), pp. 515-560.)
• A survey paper by Michel Goemans on semidefinite programming in combinatorial optimization. (Mathematical Programming, 79 (1997), pages 143-161.)
• Myths and counterexamples in optimization. This site shows that you have to be careful about your assumptions when you state some things that are "obvious" in optimization.
• A list of operations research sites.
• A compendium of approximability results for NP optimization problems.
• Some papers on the hardness of approximation by Sanjeev Arora.