The Pivot Adaptive Method for Solving Linear Programming Problems

Transcription

1 America Joural of Operatios Research ISS Olie: ISS Prit: he Pivot Adaptive Method for Solvig Liear Programmig Problems Saliha elahcee Philippe Martho 2 Mohamed Aidee 3 Mouloud Mammeri Uiversity of izi-ouzou izi-ouzou Algeria 2 Site ESEEIH de l IRI Uiversité Fédérale oulouse Midi-Pyréées oulouse Frace 3 L2CSP Laboratory Desig ad Cotrol Systems Productio izi-ouzou Algeria How to cite this paper: elahcee S Martho P ad Aidee M (28) he Pivot Adaptive Method for Solvig Liear Programmig Problems America Joural of Operatios Research Received: February 3 28 Accepted: March 2 28 Published: March Copyright 28 by authors ad Scietific Research Publishig Ic his work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC Y 4) Ope Access Abstract A ew variat of the Adaptive Method (AM) of Gabasov is preseted to miimize the computatio time Ulike the origial method ad its some variats we eed ot to compute the iverse of the basic matrix at each iteratio or to solve the liear systems with the basic matrix I fact to compute the ew support feasible solutio the simplex pivotig rule is used by itroducig a matrix that we will defie his variat is called the Pivot Adaptive Method (PAM); it allows presetig the resolutio of a give problem uder the shape of successive tables as we will see i example he proofs that are ot give by Gabasov will also be preseted here amely the proofs for the theorem of the optimality criterio ad for the theorem of existece of a optimal support ad at the ed a brief compariso betwee our method ad the Simplex Method will be give Keywords Optimizatio Liear Programmig Simplex Method Adaptive Method Itroductio As a brach of mathematics liear programmig is the domai that had the most successful i optimizatio [] [2] [3] [4] [5] Sice its formulatio from 93 to 94 ad the developmet of the Simplex method of Datzig i 949 [] [6] researchers i various fields have bee led to formulate ad solve liear problems Although the Simplex algorithm is ofte effective i practice the fact that it is ot a polyomial algorithm as show by Klee ad Mity [7] has icited researchers to propose other algorithms ad led to the birth of the iterior poit DOI: 4236/aor28828 Mar America Joural of Operatios Research

2 S elahcee et al algorithms I 979 L Khachiya proposed the first polyomial algorithm for liear programmig [8]; it is based o the ellipsoid method studied by Arkadi emirovski ad David Yudi a prelimiary versio of which havig bee itroduced by aum Z Shor Ufortuately this ellipsoid method has a poor efficiecy I 984 K Karmakar published a iterior poit algorithm which has a polyomial covergece [9] ad this caused a reewed iterest to iterior poit methods as well i liear programmig that i oliear programmig [] [] [2] [3] [4] Gabasov ad Kirillova have geeralized the Simplex method i 995 [5] [6] [7] ad developed the Adaptive Method (AM) a primal-dual method for liear programmig with bouded variables As the Simplex method the Adaptive Method is a support method but it ca start from ay support (base) ad ay feasible solutio ad ca move to the optimal solutio by iterior or boudary poits; this method was exteded later to solve geeral liear ad covex problems [8] [9] [2] ad also optimal cotrol problems [2] I liear programmig several variats of the AM have bee proposed hey geerally address the problem of iitializatio of the method ad the choice of the directio [22] [23] I this work a ew variat of the adaptive method called Pivot adaptive method (PAM) is preseted Ulike the origial method ad its variats we eed ot to compute the iverse of the basic matrix at each iteratio [6] or to solve the liear systems with the basic matrix [23] I fact to compute the ew feasible solutio ad the ew support we oly eed compute the decompositio of vectors colums of the matrix of the problem i the curret basis For this computatio we use the simplex pivotig rule [4] by itroducig a ew matrix GAMMA which allows to miimize the computatio time his ew variat of adaptive method allows us to preset the resolutio of a give problem uder the shape of successive tables (as is doe with the Simplex method) as we will see i example he proofs for the theorem of the optimality criterio ad for the theorem of existece of a optimal support that are ot give by Gabasov [6] will be preseted here ad without givig too may details a brief compariso betwee our method ad the Simplex method will be give at the ed of this article he paper is orgaized as follows I Sectio 2 after givig the statemet of the problem some importat defiitios are give I Sectio 3 we describe step by step the Pivot Adaptive Method ad i parallel the proofs of the correspodig theorems are give I Sectio 4 a example will be solved to illustrate the PAM ad more details will be give to explai how preset the resolutio uder the shape of successive tables I Sectio 5 we give a brief compariso betwee the PAM ad the Simplex Method by solvig the Klee-Mity problem Sectio 6 cocludes the paper DOI: 4236/aor America Joural of Operatios Research

3 S elahcee et al 2 Statemet of the Problem ad Defiitios I this article we cosider the primal liear programmig problem with bouded variables preseted i the followig stadard form: where: J { } ( P) = max F( x x2 x) = cx = ax i = bi i I = + d x d J = = is the idex set of the variables x x2 x I = m is the idex set of the parameters b b2 bm (correspods to the costraits) put: J = J J J J = J = m ad itroduce the vectors: x x = x( J) = ( x J) = ( x x2 x) = with: x x x J x J x = x J = x J = = ( ) ( ) ( ) c c( J) ( c J) ( c c c ) c with: c = c J = c J = = = 2 = c = = ( ) ( ) ( ) c c J c J = = i = 2 m ; b b ( J) ( b i I) ( b b b ) d = d J = d J = d d d ; d < + d = d J = d J = d d d d < + ad the ( m ) () { } matrix A a A= A( I J) = ( ai i I J) = ( a a2 a ) = with: a = A m a m J ( a : the th colum of the matrix A) ad Ai = ( ai ai 2 ai ) i I ( A i : the i th row of the matrix A) A= ( A A) A = A( I J) A = A( I J) We assume that rak ( A) = m < he the problem () takes the followig form: max F( x) = c x ( P) Ax = b (2) + d x d Ax = b are called the geeral costraits of (P) + H = x R Ax = b d x d Deote the feasible regio of (P) as: { } 2 Defiitio Each vector of the set H is called a feasible solutio of (P) 22 Defiitio 2 Ay vector x R that satisfies the geeral costraits Ax b = of the prob- DOI: 4236/aor America Joural of Operatios Research

4 S elahcee et al lem (P) is called a pseudo-feasible solutio 23 Defiitio 3 A feasible solutio 24 Defiitio 4 F x = c x = c x x is called optimal if: max x H For give value the solutio x is called a -optimal (or suboptimal) if: F x F x where x is a optimal solutio for the problem (P) ( ) 25 Defiitio 5 he set of m idices J J ( J = m) is called a support of (P) if the submatrix A = A( I J) is o-sigular ( det A ) the the set J = J \ J of ( m) idices is called o-support A is the matrix support ad A = A( I J) is the matrix o-support 26 Defiitio 6 he pair { xj } formed by a feasible solutio x ad the support J is called a support feasible solutio (SF-solutio) 27 Defiitio 7 + xj is called o-degeerate if: d < x < d J Recall that ulike the Simplex Method i which the feasible solutio x ad the basic idices J are related itimately i the adaptive method there are idepedet he SF-solutio { } 3 Optimality Criterio For the smooth ruig of calculatios i the rest of the article we assume that the sets J ad J are vectors of idices (ie: the order of idices i these sets is importat ad must be respected) 3 Formula of the Obective Value Icremet Let { xj } be the iitial SF-solutio of the problem (P) ad x x = x( J) = a arbitrary pseudo-feasible solutio of (P) We set x x= x + x = x x the the icremet of the obective fuctio value is give by: F x = F x F x = c x c x= c x= c x + c x (3) ad sice: the A x = A x + A x = Ax Ax = (4) x =A A x (5) DOI: 4236/aor America Joural of Operatios Research

5 S elahcee et al Substitutig the vector x i (3) we get: F x = c c A A x (6) Here we defie the m-vector of multipliers y as follows: y = c A (7) I [6] Gabasov defies the support gradiet ad oted it as: the k th support de- = y A c but he also recogizes that for every k J rivative is equal to k = ck yak i fact k is the rate of chage of the obective fuctio whe the k th o-support compoets of the feasible solutio x is icreased ad all the other o-support compoets are fixed at the same time the support compoets are chaged i such way to satisfy the costraits Ax = b he i the pivot adaptive method we defie the -vector of reduced gais (or support gradiet) δ as follows: δ = = δ J = c y A= δ δ where : δ = δ = c y A (8) o reduce the computatio time i the adaptive method of Gabasov [6] we defie the ( m ) matrix Γ as follows: Γ= A A= Γ Γ where: Γ = I Γ = A A (9) m which is the decompositio of the colums of the matrix A o the support J We have: Γ =Γ ( I J ) = ( Γi i I J ) Γ i is the product of the i th row of the matrix A ad the th colum of the matrix A he the -vector of support gradiet give i (8) ca be computed as follows: ( J) c ca A c c ( ) δ = δ = = Γ= δ δ where: δ = δ = c c Γ We see that to compute the quatity ad ot y c A A we begi to compute () Γ= A A = c A ; therefore we eed ot to compute the iverse of the basic matrix A From (6) ad () we obtai 32 Defiitio 8 δ F x = x = δ x () J For a give support J ay vector χ χ( J ) χ = = χ of R that satisfies χ = d if δ < + χ = d if δ > + χ = d or d if δ = χ = A ( b Aχ) ( J ) (2) DOI: 4236/aor America Joural of Operatios Research

6 S elahcee et al is called a primal pseudo-feasible solutio accompayig the support J 33 Propositio A primal pseudo-feasible solutio accompayig a give support J χ verifies ust the geeral costraits of the problem (P) ie Aχ = b If more the support compoets χ of the vector χ verifies: d χ d + the χ is feasible ad optimal solutio for (P) 34 heorem (he Optimality Criterio) Let { xj } be a SF-solutio of the problem (P) ad δ be the support gradiet computed by () the the relatios δ < for x = d + δ > for x = d J + δ = for d x d are sufficiet ad i the case of o-degeeracy of the SF-solutio { xj } also ecessary for the optimality of the feasible solutio x Proof ) he sufficiet coditio: for the SF-solutio { xj } suppose that the relatio (3) are verified we must proof that x is optimal it amouts to show that: x H F( x) = F( x) F( x) (4) Let x H ad x = x x the: F x = F x F x = δ x J δ ( x x) δ ( x x) δ ( x x) J J J δ δ δ = + ( d x ) δ ( d x ) = δ + J J δ δ = + + the x is optimal 2) he ecessary coditio: Suppose that { xj } a SF-solutio o-degeerate ie: (3) + J d < x < d (5) ad that x optimal ie: x H F( x) = F( x) F( x) = δ x (6) Reasoig by absurdity: Suppose that the relatios (3) are ot verified thus we have at least oe of the followig three cases: ) k J δk ad xk > dk + 2) k J δ ad x < d k k k J DOI: 4236/aor America Joural of Operatios Research

7 S elahcee et al + 3) k J δ = ad x = d or x = d k k k Suppose the we have the first case ie k J δ ad x > d (7) k k k (the proof uses the same reasoig for the secod ad the third cases) Costruct the -vector x( θ) ( x( θ) ) ( x( θ) ) ( x( θ) ) 2 ( x( θ )) = x J \ { k} ( ( θ) ) = θ θ k ( x( θ) ) = x + θγ k J where k of the matrix we have ad = as follows: x xk xk d k Γ is the product of the th row of the matrix For ( ( θ) ) A Let θ = x d > k k A ( θ) ( ( θ) ) ( ( θ) ) (8) ad the k th colum Ax = A x + A x = (9) k ( ( θ )) + J d x d (2) J x = x + θγ the for have: a value of θ is chose such that: ie + d x Put: θ2 = mi Γ k > Γ J we have: θ 2 > 3 ( ( θ )) + J d x d (2) + J d x + θγ d (22) k + d x < θ for Γ k > Γ k d x < θ for Γ k < Γ k k θ3 d x = mi Γ Γ k < J θ > because { } Let θ = mi ( θ θ θ ) the 2 3 x k ( J ) xj is o-degeerate (23) θ H (24) or ( ) F x F x θ = δ x = δ x = δθ < (25) k k k J the cotradictio with (6) ( ) F( x) F x θ > (26) DOI: 4236/aor America Joural of Operatios Research

8 S elahcee et al 35 Defiitio 9 he pair { } x J satisfyig the relatios (3) will be called a optimal SF-solutio of the problem (P) 36 he Suboptimality Criterio Let x a optimal solutio of (P) ad x a feasible solutios of (P) From () we obtaied: Let = max δ ( x x) + max δ ( x x) max F x = F x F x x H x H J x H J δ > δ < + ( d x ) δ ( d x ) δ + J J δ > δ < + ( xj ) = ( d x) + ( d x) (27) β δ δ J J δ > δ < β ( xj ) is called the suboptimality estimate of the SF-solutio { xj } sice ( F( x ) F( x) ) β ( xj ) We have the followig theorem of suboptimality (28) 37 heorem 2 (Sufficiet Coditio of Suboptimality) For a feasible solutio x to be -optimal for give positive umber it is sufficiet of the existece of a support β xj 38 Propositio 2 From (2) ad (27) we deduce: 39 he Dual Problem J such that ( xj ) ( x ) ( x) β = δ χ = δ χ (29) he dual of the primal problem (P) is give by the followig liear problem mi φ y v w b y d v d w = + + = m y R v w D A y v w c + (3) where A a = with J a is the traspose of the th colum a of a m+ 2 variables the matrix A he problem D has geeral costraits ad DOI: 4236/aor America Joural of Operatios Research

9 S elahcee et al Like (P) the problem (D) has a optimal solutio λ = ( y v w ) ( ) ( = φ λ ) where x is the optimal solutio of (P) F x 3 Defiitio Ay ( m+ 2) -vector ( yvw ) D is called a dual feasible solutio 3 Defiitio Let ad λ = satisfyig all the costraits of the problem J be a support of the problem (P) ad δ the correspodig support gra- m 2 λ = yvw which satisfies: diet defied i () the ( + ) -vector y = ca v = δ w = if δ J v = w = δ if δ > is called the dual feasible solutio accompayig the support 32 Propositio 3 A dual feasible solutio ( yvw ) dual feasible ie: For ay support J (3) λ = accompayig a give support J is m Ay v+ w= cv w y R J we have c χ = φ λ (32) where λ ad χ are respectively the dual feasible solutio ad a primal pseudo-feasible solutio accompayig the support 33 Defiitio 2 Let y be dual feasible solutio: λ = yv w such that: J + = he ( m+ 2) -vector Ay v w cv w v = w = c a y if c a y v = c a y w = if c a y is called a coordiated dual feasible poit to the m-vector y 34 Propositio 4 he coordiated dual feasible poit ( yv w ) A y v + w = c v w ( J) (33) λ = to y is dual feasible ie: 35 Decompositio of the Suboptimality Estimate of a xj SF-Solutio { } β be its suboptimali- Let { xj } be a SF-solutio of the problem (P) ( xj ) ty estimate calculated by (27) ( yvw ) λ = be the dual feasible solutio accompayig the support J χ the primal pseudo-feasible solutio accompay- ig the support J x be the optimal solutio for the problem (P) the optimal solutio for the dual problem (D) λ be DOI: 4236/aor28828 America Joural of Operatios Research

10 S elahcee et al the From (9) () (29) (32) ad propositio we have: ( xj ) = ( x ) = ( x) = ( c cγ)( x) β δ χ δ χ χ where: β( J ) φ( λ) φ( λ ) J ad ( x) F( x ) F( x) = c χ c x c A Aχ + c A Ax= c χ c x F( x) F( x ) F( x) = φ λ = φ λ φ λ + ( xj ) ( x) ( J ) β = β + β (34) = is the degree of o-optimality of the support β = is the degree of o-optimality of the feasible solutio x 36 Propositio 5 he feasible solutio x will be optimal if β ( x) = ad the support J will be optimal if β ( J ) = but the pair { xj } will be a optimal SF-solutio if β x = β J = so the optimality of the feasible solutio x ca be ot idetified if it is examied with a ufit support because i this case there will be β ( J ) ad β ( xj ) β J = φ λ φ λ the the support J will be optimal if its accompayig dual feasible solutio λ is optimal solutio for the dual problem D As 37 heorem 3 (Existece of a Optimal Support) For each problem (P) there is always a optimal support Proof As (P) has a optimal solutio its dual (D) give by (3) has also a optimal λ = yvw this solutio ad defie the sets: solutio let ad + { / ( ) = } I { J / ( c ) a y } I J c a y = m { ( + / ) ad ( ) } C = y R c a y i I c a y i I we have y C Let y C ad defie the followig liear problem: ( D) + ( y b y d c a y) d ( c a y) ψ = + + y C R mi + J J m yvw is the coordiated dual feasible poit to the vector y ad y is a optimal feasible solutio of the problem ( D ) O the other had there exists a vertex y of the set C that is a optimal feasible solutio for ( D ) ad this vertex is the itersectio of at least m + hyperplae a y = c ( I I ) + J = i I I / a y = c where J = m We have ψ ( y) = φ( yvw ) where ( ) Put { } (35) DOI: 4236/aor28828 America Joural of Operatios Research

11 S elahcee et al A I J y = c ad so: y = c A further the coordiated dual he: feasible poit to y is the dual feasible solutio accompayig the support ad its optimal feasible solutio for the problem (D) the accordig to the propositio (4) we coclude that J is a optimal support for (P) 4 he Descriptio of the Pivot Adaptive Method PAM As the Simplex Method to solve the problem (P) the adaptive method (ad J therefore also the pivot adaptive method) has two phases: the iitializatio phase ad the secod phase we will detail each oe i the followig ) he iitializatio phase: I this phase a iitial support feasible solutio SF-solutio { xj } will be determied ad thus the matrix Γ defied i (9) We assume that b i I ad d = J Remark i otice that each problem (P) ca be reduced to this case ie: bi i I ad d = J i fact: a) if i I / b < the both term of the correspodig costrait is multiplied by () i b) if k J / d k the we do the chage of variable: ( x ) = x k k d k the: ( x ) d + d (36) k k k ad the geeral costraits Ax = b of (P) will be write as: ax + ax + + a x + a x + d + a x + + ax = b (37) k k k k k k+ k+ where J a is the th colum of the matrix A from this relatio we obtai: ax + ax + + a x + a x + a x + + ax = bad (38) k k k k k+ k+ k k i (36) if ( b ad k k ) < we multiple the two terms by () he to determie a iitial SF-solutio to the problem (P) defie the artificial problem of (P) as follows: = m max F( x ) = x = ( P ) (39) Ax+ x = b + d x d x b where bi i I ad x = x ( J) = ( x i i I) = ( x x 2 x m) are the m artificial variables added to the problem (P) We resolve the artificial problem P with the pivot adaptive method startig with the obvious iitial feasible solutio: ( xx ) x= x = b ad the caoical support (determied by the idices of the artificial variables) ad we R take the matrix Γ= A If H is ot empty at the optimum of P + x = Ax = b ad d x d DOI: 4236/aor America Joural of Operatios Research

12 S elahcee et al so x is a feasible solutio of (P) ad the optimal SF-solutio of the problem P ca be take as the iitial SF-solutio for the problem P 2) he secod phase: Assume that a iitial SF-solutio { xj } of the problem (P) is foud i the iitializatio phase ad let Γ the correspodet matrix computed by (9) a arbitrary umber the problem will be solved with the pivot adaptive method that we describe i the followig At the iteratios of the pivot adaptive method the trasfer { xj } { xj } from oe SF-solutio to a ew oe is carried out i such a way that: β xj β xj (4) the trasfer will be realized as two procedures makig up the iteratio: i) he procedure of the feasible solutio chage x x : Durig this procedure we decrease the degree of o-optimality of the feasible solutio: β( x) β( x) ad the improve the primal criterio F( x) F( x) : i) compute the support gradiet δ by () ad the suboptimality estimate by (27) i2) If β( xj ) ε the stop the resolutio with: x is a -optimal feasible solutio for (P) (P) i3) If β( xj ) > ε we compute the o-support compoets χ of the primal pseudo-feasible solutio accompayig the support J by (2) ad the search directio l = ( l l) with: l = χ x (4) l= Γl i4) Fid the primal step legth θ with: + d x if l > ; l = = = if l < ; l ; if l = { } d x ( J) θ mi θ where : θ mi θ θ J Let J = which is the positio of i the vector of idices J i5) Compute the ew feasible solutio: x = x+ θ l ad F( x) = F( x) + θ β( x J ) i6) If θ = the stop the resolutio with: { xj } is the optimal SF-solutio of (P) i7) If θ < the compute: such that: ( xj ) ( ) ( xj ) (42) β = θ β (43) xj is the -optimal SF-solutio of (P) i9) if β ( x J ) > the go to ii) to chage the support J to J i8) If β ( x J ) the stop the resolutio with: { } DOI: 4236/aor America Joural of Operatios Research

13 S elahcee et al ii) he procedure of the support chage J J : Durig this procedure we decrease the degree of o-optimality of the sup- β J β J φ λ φ λ port: ad the improve the dual criterio where λ is the dual feasible solutio accompayig the support J ad λ is the dual feasible solutio accompayig the support J I this procedure there are two rules to chage support: rule of the short step ad rule of the log step the two rules will be preseted ii) to the foud i (i4) compute χ = x + l ad ii2) compute the dual directio t with: t = sig( α ) ; t = J \ { } ; t = tγ ii3) compute: δ if δ t > ; t σ = if δ = t < ξ = d ; J + if δ = t > ξ = d ; i other cases ad arrage the obtaied values i icreasig order as: α = χ x (44) (45) J k { p} (46) σ σ σ σ 2 p k k As was said before the chage of support ca be doe with two differet rules to use the short step rule go to a) ad to use the log step rule go to b) a) Chage of the support with the short step rule : a) From (46) put: = = = mi (47) σ σ σ σ J Let such that: J ( ) = which is the positio of idices J Put: J J \ { } J J \ J = a2) Compute: { } = ( ) { } { } i the vector of = the J = δ = δ σ t ( x J) ( x J) β = β + σα (48) go to ii4) b) Chage of the support with the log step rule : b) for every k { p} k of (47) compute + ( d d ) α = Γ (49) k k k k b2) as we choose q such that DOI: 4236/aor America Joural of Operatios Research

14 S elahcee et al k= q k= q q k q k k= k= (5) α = α + α < α = α + α Let such that: J ( ) = which is the positio of i the vector of idices J b3) put: σ = σ ad { } { J = J \ } J = ( J \ { }) { } the J ( ) = J ( ) = b4) Compute: δ = δ σ t k= q β( x J) = β( x J) + α ( σ σ ) where: k = k k k Go to ii4) ii4) compute Γ as follows: Γ Γ = J; Γ Γ i =Γ i Γ \ { } i I J i Γ α = α σ = Γ is the pivot Go to ii5) ii5) Set: x = x J = J β( xj ) = β( xj ) δ = δ ad Γ=Γ Go to i2) (5) 5 he Pivot Adaptive Method Let { xj } be a iitial SF-solutio for the problem (P) Γ the matrix computed by (9) ad are give he Pivot Adaptive Method (rule of the short step ad rule of the log step are preseted) is summarized i the followig steps: Algorithm (he Pivot Adaptive Method) ) compute the support gradiet δ by () ad β ( xj ) by (27) 2) if β ( xj ) the SOP with: x is a -optimal feasible solutio for (P) 3) if β ( xj ) > the compute the o-support compoets χ of the primal pseudo-feasible solutio accompayig the support J by (2) ad the search directio l with (4) 4) fid the primal step legth θ with (42) Let such that: J ( ) = which is the positio of i the vector of idices J 5) Compute the ew feasible solutio: x = x+ θ l ad F( x) = F( x) + θ β( x J ) 6) if θ = the SOP with: { xj } is the optimal SF-solutio for (P) 7) if θ < the compute β( xj ) = ( θ ) β( xj ) 8) if β ( x J ) the SOP with: { xj } is the -optimal SF-solutio for (P) 9) if β ( x J ) > the chage the support J to J ) compute χ = x + l ad: α = χ x ) compute the dual directio t with (44) 2) compute σ for J with (45) ad arrage the obtaied values i icreasig order DOI: 4236/aor America Joural of Operatios Research

15 S elahcee et al 3) do the chage of support by oe of the rules: a) Chage of the support with the short step rule as follows: a) compute σ with (47) ad set: J ( { }) { = J \ } J = ( J \ { }) { } a2) compute β( x J ) = β( x J ) + σα Go to (4) b) Chage of the support with the log step rule as follows: b) for every k { p} b2) as k we choose + of (46) compute α ( d d ) q such that (5) is verified = Γ k k k k { } = { } { } b3) set: σ = σ ad { } J \ J J = J \ k= q b4) compute β( x J) = β( x J) + α ( σ ) k σ k where: α k = α k= σ = Go to (4) (2) 4) compute Γ with (5) ad 5) set: x = x J J 6 Example δ = δ σ t = β( xj ) β( xj ) = δ = δ ad Γ=Γ Go to I this sectio a liear problem will be resolved with the Pivot Adaptive Method usig the short step rule ad i parallel we will explai how to realize these calculatios uder the shape of successive tables as is give i Figure Cosider the followig liear problem with bouded variables: max F x = c x P Ax = b + d x d where: c = ( 655) x= ( x x x x x ) A = d + = ( ) Put J = { 23 45} = 3 First phase (Iitializatio): ( ) b = d = 5 Let { } x = ( ) J = { 3 45} so: J = { 2} ( ) A = ( a a ) the he matrix Γ is give by: Γ= ( Γ Γ ) where: R (52) x J be the iitial SF-solutio of the problem (P) where; Γ = A = a a a = I Γ = I 3 ote that for the iitial support we have chose the caoical support ad for the iitial feasible solutio we took a iterior poit of the feasible regio of the problem (P) I the preamble part of the tables (Figure ) which has four rows we put sequetially: c ( d ) ( d + ) x he tables have 8 colums DOI: 4236/aor America Joural of Operatios Research

16 S elahcee et al Secod phase: Startig with the SF-solutio { x J } ad the matrix Γ to resolve the problem (P) with the PAM usig the short step rule I the first table of PAM (Figure ) (represets the first iteratio) there are 8 rows i the first 3 rows we have the matrix Γ (idicatig the vector formed A i the first colum of the tables) ad i the rest rows we have sequetially: δ l θ x 2 σ he support gradiet ( ) δ = δ δ where: ( δ ) 3 = R ( δ ) = ( 655) ad β ( x J ) 23 = > ) Iteratio : Chage solutio: we have: ( χ ) = ( 34 34) ad ( l ) = 237 ( l ) = ( ) θ = θ 4 = 5 ( J )( 2) = 4 (4 is the secod compoet of the vector of idex J ) he case of θ is marked by yellow i tables (Figure ) it correspod at the secod vector of J which is a 4 (which will come out of the basis 2 he θ < the ew feasible solutio x = ( ) ad β ( x 2 J ) = > Chage support (with the short step rule): χ 4 = 72 ad: α = 72 the dual directio t = ( 8 8) σ = σ = 52 the ( J ) = ( is the first compoet of the vector of idex J ) he case of σ is marked by gree i tables (Figure ) his row correspod to vector a which will go ito the basis 2 2 J J \ 4 = 35 J = J \ 4 = 4 2 ( ) { } { } So = { } { } ( ) { } { } Figure ables of the pivot adaptive method DOI: 4236/aor America Joural of Operatios Research

17 S elahcee et al For have the 2 d table (Figure ) we applied the pivotig rule where the pivot δ = δ σ t = δ δ is marked by red he ew support gradiet 2 ( 2 ) ( 2 ) where: ( δ 2 ) = R 3 ( δ ) 2 = 52 5 ad β( x J) = β( x J) + σα= As β ( x J ) > we go to aother iteratio we compute the: Γ = (( Γ 2 2) ( Γ 2) ) where: ( Γ ) = I Γ = 8 2) Iteratio 2: Chage solutio: we have: χ = ( 34) ad ( 2 ) ( 98 5) 2 ( l ) = ( ) θ = θ 3 = compoet of the vector of idice 2 J ) l = J = (3 is the first he x 3 = 2285 ad β ( x 2 J ) = 3 > Chage support (with the short step rule): χ 3 = 3 ad: α = 3 the 2 t = 5 2 σ σ 2 J 2 = 2 (2 is θ < the ew feasible solutio dual directio = = the 2 the secod compoet of the vector of idex J ) J = J \ 3 2 = 25 J = J 2 \ 2 3 = 43 So ( { }) { } { } ( { }) { } { } he ew support gradiet δ ( ) 3 δ2 σ t δ3 δ ( δ ) = 3 ( δ 3 ) = ( 32) ad ( x J) ( x J) 3 R 3 3 he { } = = where: β = β + σα = x J is the optimal SF-solutio of the problem (P) ad 3 F x = 4 7 rief umerical Compariso betwee the PAM ad the Simplex Method o realize a umerical compariso betwee the Simplex algorithm implemeted i the fuctio liprog uder the MALA programmig laguage versio (R22a) ad our algorithm (PAM) a implemetatio for the later with the short step rule has bee developed he compariso is carried o the resolutio of the Klee-Mity problem which has the followig form [7]: 2 max 2 x+ 2 x x + x x 5 4x+ x2 25 P3 8x+ 4x2 + x x+ 2 x x + x 5 x where x ( x x x ) = 2 (P 3 ) has variables costraits ad 2 vertices (53) DOI: 4236/aor America Joural of Operatios Research

18 S elahcee et al For a fixed to write (P 3 ) i the form of the problem (P) give i () we added to each of the costraits of (P 3 ) a spread variable ad we foud for each of the 2 variables of the result problem a lower bor ad a upper bor he the problem (P 3 ) ca be doe as a liear problem with bouded variables as follows: ( P ) 4 2 max 2 x+ 2 x x + x x + x+ = 5 4x+ x2 + x+ 2 = 25 8x+ 4x2 + x3 + x+ 3 = 25 2 x+ 2 x x + x + x2 = 5 x 5; x2 25; ; x 5 ; x+ 5; x+ 2 25; ; x2 5 he Simplex algorithm chose here takes as a startig solutio the origial the i our implemetatio of the (PAM) we impose the same iitial feasible solutio For the iitial support we had take the caoical oe J = { } the Γ= A We have cosidered problems with matrix A of size where { } for each size we obtaied the umber of iteratios ad the time of resolutio where the time is give ito millisecods his results are reported i Figure 2 ad i Figure 3 where Simplex(it) PAM(it) Simplex(tm) PAM(tm) represet respectively the umber of iteratios of the Simplex algorithm the umber of iteratios of the Pivot Adaptive Method the time of the Simplex algorithm the time of the Pivot Adaptive Method he time is give ito millisecods (54) Size Simplex (it) PAM (it) Simplex (tm) PAM (tm) Size Simplex (it) PAM (it) Simplex (tm) PAM (tm) Figure 2 umber of iteratios ad time for Liprog-Simplex ad PAM for Klee-Mity problem DOI: 4236/aor America Joural of Operatios Research

19 S elahcee et al Figure 3 ime compariso betwee Liprog-Simplex ad PAM for the Klee-Mity problem We remark that the (PAM) is better tha the Simplex algorithm i computatio time ad i ecessary umber of iteratios to solve the problems Recall that the optimal solutio of the problem (P 3 ) is give by x = ( 5 ) 8 Coclusios he mai cotributio of this article is a ew variat of the Adaptive Method that we have called Pivot Adaptive Method (PAM) I this variat we use the simplex pivotig rule i order to avoid computig the iverse of the basic matrix i each iteratio As was recogized i this work the algorithm saves our time compared to the origial algorithm (AM) ad allows us to preset the resolutio of a give problem uder the shape of successive tables as see i a example We have implemeted our algorithm (PAM) ((PAM) with the short step rule ad (PAM) with the log step rule) usig MALA ad we have doe a brief compariso with the primal simplex algorithm (usig liprog-simplex i MALA) for solvig the Klee-Mity problem Ideed the (PAM) is more efficiet i umber of iteratios ad i computatio time I a subsequet work we shall do a more thorough compariso betwee the Simplex algorithm of Datzig ad the (PAM) ad we shall exted the (PAM) to the problems with a large scale usig the costrait selectio techique Refereces [] Datzig G (95) Miimizatio of a Liear Fuctio of Variables Subect to Liear Iequalities Joh Wiley ew York [2] Mioux M (983) Programmatio mathématique héorie et al gorithmes ome Duod [3] Culioli JC (994) Itroductio à l optimisatio Elipse [4] Datzig G (963) Liear Programmig ad Extesios Priceto Uiversity Press Priceto DOI: 4236/aor28828 America Joural of Operatios Research

20 S elahcee et al [5] Gabasov R ad Kirillova FM ( ) Method of Liear Programmig Vol 2 ad 3 GU Press Misk (I Russia) [6] lad RG (977) ew Fiite Pivotig Rules for the Simplex Method Mathematics of Operatios Research [7] Klee V ad Mity GJ (972) How Good Is the Simplex Algorithm? I: Shisha III O Ed Iequalities Academic Press ew York [8] Khachiya LG (979) A Polyomial Algorithm i Liear Programmig Soviet Mathematics Doklady [9] Karmarkar K (984) A ew Polyomial-ime Algorithm for Liear Programmig Combiatorica [] Koima M Megiddo oma ad Yoshise A (989) A Uified Approach to Iterior Poit Algorithms for Liear Programmig I: Megiddo Ed Progress i Mathematical Programmig: Iterior Poit ad Related Methods Spriger Verlag ew York [] Ye Y (997) Iterior Poit Algorithms heory ad Aalysis Joh Wiley ad Sos Chichester [2] Vaderbei RJ ad Shao DF (999) A Iterior-Poit Algorithm for ocovex oliear Programmig Computatioal Optimizatio ad Applicatios [3] Peg J Roos C ad erlaky (2) A ew ad Efficiet Large-Update Iterior-Poit Method for Liear Optimizatio Joural of Computatioal echologies 6 6Â-8 [4] Cafieri S Dapuzzo M Mario M Mucherio A ad oraldo G (26) Iterior Poit Solver for Large-Scale Quadratic Programmig Problems with oud Costraits Joural of Optimizatio heory ad Applicatios [5] Gabasov RF (994) Adaptive Method for Solvig Liear Programmig Problems Uiversity of ruxelles ruxelles [6] Gabasov RF (994) Adaptive Method for Solvig Liear Programmig Problems Uiversity of Karlsruhe Istitute of Statistics ad Mathematics Karlsruhe [7] Gabasov R Kirillova FM ad Prischepova SV (995) Optimal Feedback Cotrol Spriger-Verlag Lodo [8] Gabasov R Kirillova FM ad Kostyukova OI (979) A Method of Solvig Geeral Liear Programmig Problems Doklady A SSR (I Russia) [9] Kostia E (22) he Log Step Rule i the ouded-variable Dual Simplex Method: umerical Experimets Mathematical Methods of Operatio Research [2] Radef S ad ibi MO (2) A Effective Geeralizatio of the Direct Support Method Mathematical Problems i Egieerig 2 Article ID: [2] alashevich V Gabasov R ad Kirillova FM (2) umerical Methods of Program ad Positioal Optimizatio of the Liear Cotrol Systems Zhural Vychislitel oi Matematiki i Matematicheskoi Fiziki [22] etobache M ad ibi MO (22) A wo-phase Support Method for Solvig Liear Programs: umerical Experimets Mathematical Problems i Egieerig 22 Article ID: [23] ibi MO ad etobache M (25) A Hybrid Directio Algorithm for Solvig Liear Programs Iteratioal Joural of Computer Mathematics DOI: 4236/aor28828 America Joural of Operatios Research