1. Introduction to Nonlinear Combinatorial Optimization
Ding-Zhu Du
University of Texas at Dallas

2. What is Nonlinear Combinatorial Optimization?
Ding-Zhu Du
University of Texas at Dallas

3. Nonlinear & Discrete
A combinatorial optimization problem with nonlinear objective function and/or constraints.
A nonlinear optimization problem with discrete (combinatorial) structure.

4. Topological Control in Wireless Networks
1st Example
Topological Control in Wireless Networks

5. Symmetric Topological Control
Given a wireless network, find a power assignment to every nodes to minimize the total energy consumption such that the connectivity of the network is preserved.

6. Energy in Wireless Network

7. Symmetric xxx
Asymmetric xxx

8. Min Spanning Tree Minimum Spanning Tree Problem is
polynomial-time solvable.

9. Symmetric Topological Control
Symmetric Topological Control Problem
is NP-hard!

10. Asymmetric Topological Control
Asymmetric Topological Control Problem
is NP-hard!

11. Job Scheduling in Cloud Computing
2nd Example
Job Scheduling in Cloud Computing

12. Scheduling with energy consideration2
Scheduling with energy consideration
In cloud computing, jobs’ processing time is determined nonlinearly by energy assignment.
Suppose total energy is given. How to assign energy to jobs in order to minimize the makespan.

13. Alternating Direction Method
Fixed energy assignment, find optimal or good approximate solution for classic scheduling problem.
Fixed job scheduling, find a good energy assignment.

14. Work of Bingsheng He Convergent with two group of variables
Not convergent with 3 or more.

15. Influence Maximization in Social Networks
3rd Example
Influence Maximization in Social Networks

16. Subareas 0-1 program: set function optimization
Integer program: lattice point optimization (including discrete DC programming)
Mixed integer program

17. Related Books

18. Discrete Convex Analysis5
Discrete Convex Analysis
Kazuo Murota

19. Discrete Newton Method6
Inverse optimization problem
Survey by Zhao Zhang

20. Relaxation LP-relaxation (Linear Program)
7/8
LP-relaxation (Linear Program)
SD-relaxation (Semi-definite Program)
Convex-relaxation (Convex Program); primal-dual method with convex program (STOC,SODA)
Continuous-relation (Discrete DC Program); a prima-dual method with discrete DC program is possible.

21. What is discrete DC program?8
DC: Difference of Convex
Convex-extensible

22. What is discrete DC program?
Discrete DC function
Lin-vex extension

23. Convex closure & extension
Convex extension

24. Toland-Singer duality
Why a primal-dual method exists?

25. First try on discrete DC program
Dachuan Xu

26. 9
Why NCO may develop well now?
Final Remark

27. What more important?
In development of applied math, what is more important?
Deeper mathematics?
Solid application?

28. Previously
One proposed some NCO problems from theoretical point of view.
But, NCO didn’t get strong support from applications.

29. Today NCO gets a lot of applications.
Its theoretical foundation starts to grow.

30. Thanks, End