Combinatorial Optimization

K. Subramani, LCSEE, West Virginia University January 16, 2013

1. Vector - A vector of dimension n is an ordered collection of n elements, which are called components.

2. Vector Space - A set V that is closed under finite vector addition and scalar multiplication. A Vector is an element of a vector space.

3. Matrix, dimension, compatibility of A · x ≤ b and c · x.

4. Linear equation (cx=d), linear inequality, linear disequation.

5. System of linear inequalities, equations, constraint matrix, Subsystem.

6. Norm of a vector - positive scalability, triangle inequality, separates points.

7. L_{p} norm ((∑_{i=1}^{n} |x_{i}|^{p})^{\frac{1}{p}}, L_{1}, L_{2}, L_{∞}.

8. distance between vectors x and y.

9. Row and column sub-matrices.

10. Diagonal, upper and lower triangular.

11. Polyhedron is a set determined by linear inequalities {x : Ax ≤ b}.

12. Lattice point.

13. For any subset T ⊆ S, characteristic function, x_{T}.

14. Support of a vector.

15. Convex set.

16. Linear combination of vectors, affine combination, convex combination.

17. Linear hull, affine hull, convex hull.

18. A cone is a set C, such that if x, y ∈ C, then λ · x + μ · y ∈ C, for all λ, μ, ≥ 0. Are cones convex?

19. The cone generated by a set of vectors X, is the smallest convex cone containing X and is denoted by cone X.

20. Linear independence, affine independence.

21. Spanning set of vectors, basis.

22. Uniqueness of basis representation.