1. Part II General Integer Programming II.1 The Theory of Valid Inequalities 1

2. Integer Programming 2015 2

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5. Integer Rounding Integer Programming 2015 5

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7. Chvatal-Gomory (C-G) Rounding Method Integer Programming 2015 7

8. Optimizing over the First Chvátal closure Integer Programming 2015 8

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10.  If we find good but not necessarily optimal solutions to the MIP, we find very effective valid inequalities. Also heuristic methods to find good feasible solutions to the MIP are helpful.  MIP model may not be intended as computational tools to solve real problems. But we can examine the strength of rank-1 C-G inequalities to describe the convex hull of S for various problems.  For some structured problems, e.g. knapsack problem, the separation problem for the first Chvatal closure may have some structure which enables us to handle the problem more effectively. Integer Programming 2015 10

11. Modular Arithmetic Integer Programming 2015 11

12. Disjunctive Constraints Integer Programming 2015 12

13. Integer Programming 2015 13

14. Integer Programming 2015 14

15. Boolean Implications Integer Programming 2015 15

16. Geometric or Combinatorial Implication Integer Programming 2015 16 1 3 2 4 5 76