列生成算法介绍PPT推荐.ppt

1、set of items:number of times item i is requested:length of item i:length of a standard roll:set of cutting patterns:number of times item i is cut in pattern j:number of times pattern j is used,COLUMN GENERATION 4,The Cutting Stock Problem.,Set can be huge.Solution of the linear relaxation of by colu

2、mn generation.,Minimize the number of standard rolls used,COLUMN GENERATION 5,The Cutting Stock Problem.,Given a subsetand the dual multipliersthe reduced cost of any new patterns must satisfy:otherwise,is optimal.,COLUMN GENERATION 6,The Cutting Stock Problem.,Reduced costs for are non negative,hen

3、ce:,is a decision variable:the number of times item i is selected in a new pattern.The Column Generatoris a Knapsack Problem.,COLUMN GENERATION 7,Basic Observations,Keep the coupling constraints at a superior level,in a Master Problem;this with the goal of obtaining a Column Generator which is rathe

4、r easy to solve.,At an inferior level,solve the Column Generator,which is often separable in several independent sub-problems;use a specialized algorithm that exploits its particular structure.,COLUMN GENERATION 8,LP Column Generation,Optimality Conditions:primal feasibility complementary slackness

5、dual feasibility,MASTER PROBLEMColumns Dual Multipliers COLUMN GENERATOR（Sub-problems）,COLUMN GENERATION 9,Historical Perspective,G.B.Dantzig&P.WolfeDecomposition Principle for Linear Programs.Oper.Res.8,101-111.（1960）,Authors give credit to:L.R.Ford&D.R.FulkersonA Suggested Computation for Multi-co

6、mmodity flows.Man.Sc.5,97-101.（1958）,COLUMN GENERATION 10,Historical Perspective:a Dual Approach,J.E.Kelly The Cutting Plane Method for Solving Convex Programs.SIAM 8,703-712.（1960）,DUAL MASTER PROBLEMRows Dual Multipliers ROW GENERATOR（Sub-problems）,COLUMN GENERATION 11,Dantzig-Wolfe Decomposition:

7、the Principle,COLUMN GENERATION 12,Dantzig-Wolfe Decomposition:Substitution,COLUMN GENERATION 13,Dantzig-Wolfe Decomposition:The Master Problem,The Master Problem,COLUMN GENERATION 14,Dantzig-Wolfe Decomposition:The Column Generator,Given the current dual multipliers for a subset of columns:coupling

8、 constraintsconvexity constraintgenerate（if possible）new columns with negative reduced cost:,COLUMN GENERATION 15,Remark,COLUMN GENERATION 16,Dantzig-Wolfe Decomposition:Block Angular Structure,Exploits the structure of many sub-problems.Similar developments&results.,COLUMN GENERATION 17,Dantzig-Wol

9、fe Decomposition:Algorithm,Optimality Conditions:primal feasibility complementary slackness dual feasibility,MASTER PROBLEMColumns Dual Multipliers COLUMN GENERATOR（Sub-problems）,COLUMN GENERATION 18,Given the current dual multipliers（coupling constraints）（convexity constraint）,a lower bound can be

10、computed at each iteration,as follows:,Dantzig-Wolfe Decomposition:a Lower Bound,Current solution value+minimum reduced cost column,COLUMN GENERATION 19,Lagrangian Relaxation Computes the Same Lower Bound,COLUMN GENERATION 20,Dantzig-Wolfe vs Lagrangian Decomposition Relaxation,Essentially utilizedf

11、or Linear ProgramsRelatively difficult to implement Slow convergenceRarely implemented,Essentially utilizedfor Integer ProgramsEasy to implement with subgradient adjustment for multipliers No stopping rule!6%of OR papers,COLUMN GENERATION 21,Equivalencies,Dantzig-Wolfe Decomposition&Lagrangian Relax

12、ationif both have the same sub-problems,In both methods,coupling or complicating constraints go into a DUAL MULTIPLIERS ADJUSTMENT PROBLEM:in DW:a LP Master Problemin Lagrangian Relaxation:,COLUMN GENERATION 22,Equivalencies.,Column Generation corresponds to the solution process used in Dantzig-Wolf

13、e decomposition.This approach can also be used directly by formulating a Master Problem and sub-problems rather than obtaining them by decomposing a Global formulation of the problem.However.,COLUMN GENERATION 23,Equivalencies.,for any Column Generation scheme,there exits a Global Formulationthat ca

14、n be decomposed by using a generalized Dantzig-Wolfe decomposition which results in the same Master and sub-problems.,The definition of the Global Formulationis not unique.A nice example:The Cutting Stock Problem,COLUMN GENERATION 24,The Cutting Stock Problem:Kantorovich（1960/1939）,:set of available rolls:binary variable,1 if roll k is cut,0 otherwise:number of times item i is cut on roll k,COLUMN GENERATION 25,The Cutting Stock Problem:Kantorovich.,Kantorovichs LP lower bound is weak:However,Dantzig-Wolfe decomposition provides the same bound as