Introduction to Nonlinear Combinatorial Optimization DingZhu Du University
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 • A combinatorial optimization problem with nonlinear objective function and/or constraints. • A nonlinear optimization problem with discrete (combinatorial) structure.
st 1 Example Topological Control in Wireless Networks
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.
Energy in Wireless Network r
Symmetric xxx Asymmetric xxx
Min Spanning Tree Minimum Spanning Tree Problem is polynomial-time solvable.
Symmetric Topological Control Problem is NP-hard!
Asymmetric Topological Control Problem is NP-hard!
nd 2 Example Job Scheduling in Cloud Computing
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.
Alternating Direction Method • Fixed energy assignment, find optimal or good approximate solution for classic scheduling problem. • Fixed job scheduling, find a good energy assignment.
Work of Bingsheng He • Convergent with two group of variables • Not convergent with 3 or more.
rd 3 Example Influence Maximization in Social Networks
Subareas • 0 -1 program: set function optimization • Integer program: lattice point optimization (including discrete DC programming) • Mixed integer program
Related Books
5 Discrete Convex Analysis Kazuo Murota
Discrete Newton Method • Inverse optimization problem • Survey by Zhao Zhang 6
Relaxation 7/8 • LP-relaxation (Linear Program) • SD-relaxation (Semi-definite Program) • Convex-relaxation (Convex Program); primaldual method with convex program (STOC, SODA) • Continuous-relation (Discrete DC Program); a prima-dual method with discrete DC program is possible.
What is discrete DC program? 8 • DC: Difference of Convex • Convex-extensible
What is discrete DC program? • Discrete DC function • Lin-vex extension
Convex closure & extension Convex closure Convex extension
Toland-Singer duality Why a primal-dual method exists?
First try on discrete DC program Dachuan Xu
9 Why NCO may develop well now? Final Remark
What more important? • In development of applied math, what is more important? • Deeper mathematics? • Solid application?
Previously • One proposed some NCO problems from theoretical point of view. • But, NCO didn’t get strong support from applications.
Today • NCO gets a lot of applications. • Its theoretical foundation starts to grow.
Thanks, End
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