1、Outline,Deep into SG algorithm。Solve parallel machine scheduling using L-G methodSimulationanalysis,Deep into SG method,Widely used in non-differentiable optimization;Min f(x),dual,1.寻找到下界 2.得到一个初始的解,illustration for dual problems,原问题：寻找使得目标值最小的x,对偶问题：寻找使得目标函数最大的,Steps of SG,Step1:选一个初始拉格朗日乘子如 0,t=1
2、; Step2：对于 从中任选一个次梯度 ；若 0则达到了最优解而停止计算；否则，t=t+1,重复STEP2；,Illustration of SG,upper1,上升很慢,Gama,Lemma1：whenLemma2： and for all k, thenHalve the dual gap,Illustration for gama,upper,lower,description,Basic parallel machine scheduling,job4,job3,job2,job1,M2,M1,Pro_time, weight, duedate, start_time,目 标2.能力
3、约束3.自身约束,Formulation,problem formulation,Steps of LG method,step1 .relaxation of the hard constrain step2 .solve the L-G dual problem using S-G method step3. get feasible solution using heuristic alg0rithm,Why using LG method,Good primal sequenceGood bound.,First step relaxation,Dual problem,Max min
4、 problem (LSS),Dual problem,decomposition,给定 将上述问题分解成N个子问题,Solve dual problem,给定一个,将 带回原问题 得到关于 的函数 用次梯度法 得到一个 新的,heuristic,Get feasible solutionLG-solution is a good primal solution The sequence is good but violate the capacity constrain. The one who cost much goes first,so it goes on.,Flow chart o
5、f heuristic,List 按照（LG start） K=1,i=1,Mk=0?,no,有任务在k完成吗,yes,Mk+;(k+1 N),yes,Set K= k+1,Reform the list,i+;调度下一个job,i=job_num?,re- fresh start,Refresh start,no,end,yes,Simulation results,eg1: 12 jobs and 2 identical machines ; Eg2:25 jobs and 4 identical machines ; Result: Eg1 bound=31.82 with best h
6、euristic result 34; Eg2 bound=37.74 with best heuristic result 38;,analysis,Deep into paithe LG multiplier What is the meaning of pai in physical sense? How it acts and changes during the itaration?The LG solution is a good primal solution though non-feasible.Why the LG solution is a good primal sol
7、ution?,Pai (what),It tells us How important is the time K How important is the machine of time k Conclusion The cost of using the resource of time k,Mathematical sense (pai),Pai stands for the cost of resource at time k,Illustration for pai,不同时刻的pai在整个迭代过程中的趋向,K在,Illustration for pai,1.2.3.4.5分别代表不同的迭代时间,经过迭代pai46最终的值,Illustration for pai,Pai !=0,Pai !=0;How can the solution of a completely different problem gives you a good primal solution?,原问题一定跟对偶问题很不一样,Formulation,problem formulation,Dual problem,My explanation,Good beginning promises success!,Thanks for listening!,
