1 / 30

Introduction to Nonlinear Combinatorial Optimization

Introduction to Nonlinear Combinatorial Optimization. Ding-Zhu Du University of Texas at Dallas. What is Nonlinear Combinatorial Optimization?. Ding-Zhu Du University of Texas at Dallas. Nonlinear & Discrete.

achristiansen
Download Presentation

Introduction to Nonlinear Combinatorial Optimization

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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. 1st Example Topological Control in Wireless Networks

  5. 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. Symmetric Topological Control

  6. Energy in Wireless Network r

  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. 2nd Example Job Scheduling in Cloud Computing

  12. Scheduling with energy consideration 2 • 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. 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. 5 Discrete Convex Analysis Kazuo Murota

  19. Discrete Newton Method 6 • Inverse optimization problem • Survey by Zhao Zhang

  20. Relaxation 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 closure 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

More Related