一个基于妥协解的多目标线性规划分类模型

∗1 21 1000492 100190(MCLP)MCLPMCLPA compromise-based MCLP classification modelBo Wang1Yong Shi2(1Graduate University of Chinese Academy of Sciences,Beijing100049)(2CAS Research Center On Ficitious Economy and Data Science,Beijing100190)Abstract Although multiple criteria linear programming can deal with classification problemsuccessfully,an original MCLP model has some difficulties in choosing parameters.To overcomethe problems,compromise-based MCLP model is proposed to offer a good promotion of theoriginal one.In the latter model,there are also two deviations for every single point;that is,interior deviation and exterior deviation.Similar to the original MCLP model,for each point,wewant at least one of the deviations to be zero.In addition to modeling work,this paper also givesa proof of the existence of the parameter selection condition.Keywords MCLP Interior deviation Exterior deviation Compromise solution∗ ( 70921061( ),90718042( )) BHP Billiton Co.,Australia1 (MCLP)1.1Ned Freed Fred Glover (Goal Programming,GP) (Multiple Criteria Linear Programming,MCLP)MCLP1.1.1: MCLPαi βi∑minαii∑βimaxis.t.A i X=b+αi−βi,A i∈G1,A i X=b+αi−βi,A i∈G2,αi,βi 0,i=1,2,...,l1.2(Multi-objective Programming,MOP) MOPmax g(x)=(g1(x),g2(x),...,g p(x))over x∈X.Pareto Paretog i(x) gi,i=1,2,...,p.min P(α,β)s.t.g i(x)−gi=αi−βi,i=1,2,...,p,αi,βi 0,i=1,2,...,p,x∈X.Sawaragi1.1. α,β R p P(α,β) α β αi βi α∗ β∗α∗i β∗i=0,i=1,2,...,p.1.minp∑i=1a iαi−p∑i=1b iβis.t.g i(x)−gi=αi−βi,i=1,2,...,p,αi,βi 0,i=1,2,...,p,x∈X.a i=b iminp∑i=1a i(g i(x)−gi) s.t.x∈X.2. a i>b i,∀ip∑i=1a iαi−p∑i=1b iβi=p∑i=1b i(αi−βi)+p∑i=1(a i−b i)αi.α β a i>b i,∀i P(α,β) αα∗i β∗i=0,i=1,2,...,p.α∗iβ∗i=0,i=1,2,...,p1.31n X∗ A i n A i X∗ X∗ l b αi βi α=(α1,α2,...,αl) β=(β1,β2,...,βl) X∗ {A1,A2,...,A l} l α·β=0.22.1MCLP Shi Yu(1989)−∑li=1αi α∗ α∗ 0 β∗ 0∑li=1βiα∗+∑li=1αi d−α d+αd−α−d+α=α∗+l∑i=1αi,d−α+d+α=|α∗+l∑i=1αi|.β∗−∑li=1βi d−β d+βd−β−d+β=β∗−l∑i=1βi,d−β+d+β=|β∗−l∑i=1βi|.α∗,β∗ −∑li=1αi∑li=1βi|α∗+∑li=1αi| |β∗−∑li=1βi|d−α+d+α+d−β+d+β.MCLPmin d−α+d+α+d−β+d+βs.t.d−α−d+α=α∗+l∑i=1αi,d −β−d +β=β∗−l ∑i =1βi ,A i X =b +αi −βi ,A i ∈G 1,A i X =b +αi −βi ,A i ∈G 2,αi ,βi 0,i =1,2,...,lα∗ 0,β∗ 0 αi ,βi MCLP2.2αi ,βiα∗i β∗i =0,i =1,2,...,l (∗)MCLP (∗)2.1. (X ∗,b ∗) α∗i ,β∗i ,i =1,2,...,lα∗i β∗i =0,i =1,2,...,l.α∗i β∗i. α∗+∑l i =1αi d −α d +α β∗−∑l i =1βi d −β d +β (I)α∗+∑l i =1αi =−d +α<0,β∗−∑l i =1βi =d −β>0.∃ d +α− >0,d −β− >0i αi =αi + ,βi =βi +αi −βi =αi + −βi − =αi −βi .α∗+∑j =i αj +αi =−d +α+ <0,β∗−∑j =iβj −βi =d −β− >0. d +α− +d −β− =d +α+d −β−2 <d +α+d −β.(II)α∗+∑l i =1αi =d −α>0,β∗−∑l i =1βi =−d +β<0.∃i αi >0,βi >0 ∃ >0αi − >0,βi − >0,d −α− >0,d +β− >0.αi =αi − >0,βi =βi − >0.αi −βi =αi −βi ,α∗+∑j =i αj +αi − =d −α− d −α>0,β∗−∑j =i βj −βi + =−d +β+ −d +β<0.d −α+d +β=d −α+d +β−2 <d −α+d +β. (III)α∗+∑l i =1αi =d −α 0,β∗−∑l i =1βi =d −β 0.∃i αi βi >0 βi αi αi =0 βi =βi −αiαi −βi =αi −βiα∗+∑j =i αj +αi =d −α−αi ,(1)β∗−∑j =iβj −βi =d −β+αi >0. (1) 0d −α=d −α−αi ,d −β=d −α+αi .d −α+d −β=d −α+d −α.(α,β) (α,β) α·β=0.(1)<0 (I) (IV)α∗+∑l i =1αi =−d +α 0,β∗−∑l i =1βi =−d +β 0.(III)α∗+∑j =i αj +αi =−d +α−αi <0,β∗−∑j =iβj −βi =−d +β+αi .(2) (2) 0 (III) (α,β) (2)>0 (I)3αi βiαiβi=0,∀i.2.1[1]G.Kou,X.Liu,Y.Peng,Y.Shi,M.Wise and W.Xu.Multiple criteria linear programming approach to Data Mining:models,algorithm designs and software development[J].Optimization Methods and Software,V ol. 18,No.4,August2003,pp.453 473.[2]Hirotaka Nakayama and Yeboon Yun.Generating Support Vector Machines using Multiobjective Opti-mization and Goal Programming[J].Studies in Computational Intelligence,2006,V olume16/2006,173-198.[3]J.He,W.Yue and Y.Shi,Identification Mining of Unusual Patterns for Multimedia Communication Net-works[C].Abstract Proc.of Autumn Conference2005of Operations Research Society of Japan,262-263,2005.[4]N.Freed and F.Glover.Simple but powerful goal programming models for discriminant problems[J]. European Journal of Operational Research,V ol.7,Issue1,May1981,Pages44-60.[5]Y.Shi.Multiple Criteria Multiple Constraint-levels Linear Programming:Concepts,Techniques and Ap-plications.World Scientific Publishing,River Edge,New Jersey,2001.[6]Y.Shi and P.L.Yu.Goal setting and compromise solutions.Multiple Criteria Decision Making and Risk Analysis and Applications,B.Karpak and S.Zionts(eds.),Springer-Verlag,Berlin,1989.165-203.。