Combinatorial Optimization

K. Subramani, LCSEE, West Virginia University January 23, 2013

1. Given a convex set S ⊆ R^{n}, convex function, concave function. f(λ · x_{1} + (1 − λ) · x_{2}) ≤ λ · f(x_{1}) + (1 − λ) · f(x_{2}), x_{1}, x_{2} ∈ S and 0 ≤ λ ≤ 1. Linear function is both convex and concave. Advantages of concave minimization and convex minimization.

2. Size of a number, vector and matrix.

3. The classes P and NP.

4. Polytope - A polytope is conv(X) for some set X ⊂ ℜ^{n}.

5. Determinant of a square matrix. Singularity.

6. Matrix-vector multiplication - new form.

7. Rank of a matrix. r(A) ≤ min(m, n).

8. Inverse of a square matrix.

9. System of simultaneous linear equations: A · x = b. Meaning.

10. No solutions, unique solution, infinite number of solutions.

11. Gaussian elimination. Elementary row operations. Elementary matrices.

12. Infeasibility in linear systems of equalities (certificates).