1. CS541 Advanced Networking 1 Introduction to Optimization Neil Tang 2/23/2009

2. CS541 Advanced Networking 2 Outline  Optimization Problems  Mathematical Programming  LP  ILP  CP

3. CS541 Advanced Networking 3 Optimization Problems  An optimization problem is composed of an objective function and a set of constraints.  Optimization problems in a graph can be solved by algorithms in graph theory.  If an optimization problem can be formulated as a Linear Programming (LP) problem, an Integer Linear Programming (ILP) problem, a Mixed Integer Linear Programming (MILP) problem or a Convex Programming (CP) problem, then the existing algorithms can be applied to solve it.

4. CS541 Advanced Networking 4 Mathematical Programming ObjectiveConstraintsVariablesSolution LPLinear RealSimplex Method and Interior Point Method ILPLinear IntegerBranch-and-Bound Algorithm (exponential) MILPLinear Real and integer Branch-and-Bound Algorithm (exponential) CPConvex RealInterior Point Method

5. CS541 Advanced Networking 5 LP LP in the standard form

6. CS541 Advanced Networking 6 Maximum Flow Problem - LP

7. CS541 Advanced Networking 7 Multi-Commodity Flow Problem - LP

8. CS541 Advanced Networking 8 Shortest Path Problem - ILP

9. CS541 Advanced Networking 9 Constrained Shortest Path Problem - ILP

10. CS541 Advanced Networking 10 CP CP in the standard form Property: if a local minimum exists, then it is a global minimum.

11. CS541 Advanced Networking 11 Rate Allocation Problem - CP