Optimum Solution of Quadratic Programming Problem: By Wolfe s Modified Simplex Method

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1 Volume VI, Issue III, March 7 ISSN 78-5 Optimum Solutio of Quadratic Programmig Problem: By Wolfe s Modified Simple Method Kalpaa Lokhade, P. G. Khot & N. W. Khobragade, Departmet of Mathematics, MJP Educatioal Campus, RTM Nagpur Uiversity, Nagpur, Idia. Departmet of Statistics, MJP Educatioal Campus, RTM Nagpur Uiversity, Nagpur, Idia. Abstract: -I this paper, a alterative approach to the Wolfe s method for Quadratic Programmig is suggested. Here we proposed a ew approach based o the iterative procedure for the solutio of a Quadratic Programmig Problem by Wolfe s modified simple method. The method sometimes ivolves less or at the most a equal umber of iteratio as compared to computatioal procedure for solvig NLPP. We observed that the rule of selectig pivot vector at iitial stage ad thereby for some NLPP it takes more umber of iteratio to achieve optimality. Here at the iitial step we choose the pivot vector o the basis of ew rules described below. This powerful techique is better uderstood by resolvig a cyclig problem. Key Words Ad Phrases: Optimum solutio, Wolfe s method, Quadratic Programmig Problem. Q I. INTRODUCTION uadratic Programmig Problem is cocer with the Noliear Programmig Problem (NLPP) of maimizig (or miimizig) the quadratic obective fuctio subect to a set of liear iequality costraits. I Geeral Quadratic Programmig Problem (GQPP) is writte i the form: Maimize M k Subect to costraits: i,,..., m. ad,,,......,. k k i where k k for all ad k, ad also i. Philip Wolfe (959) has give algorithm which based o fairly simple modificatio of simple method ad coverges i a fiite umber of iteratios. Terlaky proposed a algorithm which does ot require the elargemet of the basic table as Frak-Wolfe (956) method does. Terlaky s algorithm is active set method which starts from a primal i, feasible solutio costruct dual feasible solutio which is complemetary to the primal feasible solutio. But here we proposed a ew approach based o the iterative procedure for the solutio of a Quadratic Programmig Problem by Wolfe s modified simple method. Let the Quadratic form semi-defiite. k k k be egative The New approach to Wolfe modified simple Algorithm to solve the above QPP is stated below: Rule : Itroduce the slack variable P i i the correspodig ith costrait to covert the iequality costrait ito equatios, where i m. ad itroduce P m i the th o-egatively costrait,.. Rule : Costruct the Lagragia fuctio m L(, P, ) M i i i Pi m P m where,,....., P P, P,....., P m,,....., m Differetiate L (, P, ) partially with respect to the compoet, P ad equate the first order derivative equal to zero. Derive the Kuh-Tucker coditio from the resultig equatios. Rule : Itroduce the o-egative artificial variable,,,...,. i the Kuh-Tucker coditios Page

2 Volume VI, Issue III, March 7 ISSN 78-5 k m k ii k i for,,......,. m Costruct a obective fuctio M.... Rule : Obtai a iitial basic feasible solutio to LPP:- Miimize M... Subect to costraits:- k m k ii m, k i,,......,. i i, i,,......, m. ad,,......,.,,,......, m.,,,......, m.,,,......, m. Above modificatio states that to became a basic variable wheever is ot permitted is already a basic variable ad vice verse for,,......, m. This esures for each value of, whe optimal solutio to this problem is the desired optimal solutio to the origial QPP. Rule 5: Obtai a optimum solutio to the LPP i above metioed rule by usig ew techique for determie the pivot basic vector by choosig maimum value of give by i, where is i the sum of correspodig colum to the each Z C. Let it be for some k, hece y k eter ito the basis. Select the outgoig vector by mi Bi, let yik it be for some If i r.hece y rk the pivot elemet. is same for two or more vectors the the vector with positive Z C eters the basis. If all Z C, the optimum solutio is obtaied. The optimal solutio must satisfy feasibility coditio that Z*=ΣC B X B = ad it should satisfy restrictio o sigs of Lagrage s multipliers. Rule 6: The optimum solutio obtaied i above metioed rule is a optimum solutio to the give QPP. II. STATEMENT OF THE PROBLEM I what follows we shall illustrate the problem where the iteratios are less (by our method) tha the solutio obtaied by eistig method. Use Alterative Approach To Solve The Followig QPP: Eample : Maimize z Subect to the costraits:, III. SOLUTION OF THE PROBLEM Covert the iequality costraits ito equatios by itroducig slack variable P ad P respectively, also itroduce P, P i, to covert them ito equatios. Maimize: M Subect to the costraits: P P P P where P, P, P, P are slack variables. Costruct the Lagragia fuctio: L L(,, P, P, P, P,,,, ) Page

3 Volume VI, Issue III, March 7 ISSN 78-5 P P P P Derive the Kuh-Tucker coditio from the resultig equatios. Differetiate L(, P, ) partially with respect to the compoet, P ad equate the first order derivative equal to zero. Derive the Kuh-Tucker coditio from the resultig equatios. Thus we have L L L P, P L P L P P L P, P P L P, L P L P, L P After simplificatio ad ecessary maipulatios these yield: P P P P,, P, P, i I order to determie the solutio to the above simultaeous equatios, we itroduce the artificial variables ad (both o-egative) ad costruct the dummy obective fuctio M. The the problem becomes Miimize M ) P by ),,, ( here we replaced,, P by, ( here we replaced,, i, i,,,. The optimum solutio to the above LPP shall ow be obtaied by the alterate procedure described above i differet rules. Iitial step: C B Y B X B Ratio - / z c The above table idicate that ma correspods to ad so either or eters the basis, we ca eter ito the basis ad mi ratio correspods to therefore will leave the basis. Page

4 Volume VI, Issue III, March 7 ISSN 78-5 Step (): Itroduce ad drop C B Y B X B Ratio / / / / -/ -/ 9/8 / -/ -/ ¾ z c Sice the value 6 is most positive, we make as the eterig vector i the basis ad drop. Step (): Itroduce ad drop C B Y B X B - / / / 9/8 / -/6 -/6 -/ -/8 -/8 z c -/ 6/ 5/ Sice the value is 6/ (maimum), we make as the eterig vector i the basis ad drop. Step (): Itroduce ad drop Ratio C B Y B X B 5/6 /6 / / 59/6 / -/6 5/ / / z c Sice all z c, a optimum solutio has bee reached i three iteratios. Therefore optimum solutio is 5/6, 59/ 6 ad maimum M.9 Eample : Maimize z 6 Sub to :,, s ; & s s 6 L L s s s ; L 6 Page

5 Volume VI, Issue III, March 7 ISSN 78-5 L L s s ; A A 6 Iitial step c B y B B A A A - A Sice the value is 8 (maimum), we make as the eterig vector i the basis ad drop. st Iteratio c B B A A - - y B A A - - / / 7/ -5/ - - Sice =7/ maimum eters the basis ad A leaves the basis d Iteratio c B y B B A A / -/ / -/ / A - - / / -/6 /6 -/6 -ve /6 -/6 - / Sice =/6 maimum eters the basis ad A leaves the basis rd Iteratio c B y B B A A / -/ /6 / -/6 - -/ / 5/6 / /6 -/ -/6 7/ Z* Ma 5 5 z 6 6 Eample : Maimize z 8 Page 5

6 Volume VI, Issue III, March 7 ISSN 78-5 Sub to : 6,, s 6 s, s L 8 L s s s 6 8 L L A s 8 6 A, 6 Iitial Table c B y B B A A A 8 - A Sice =5 maimum eters the basis ad A leaves the basis st Iteratio c B y B A B A A -/ -/ - A - / / / -/ 5 Sice =5 maimum eters the basis ad A leaves the basis d Iteratio c B y B B A A / -/9 -/9 -/ / A / 6/9 /9 / - -/ / / 8/9 7/9 / - -/ Sice =8/9 maimum, so eters the basis ad A leaves the basis rd Iteratio c B y B B A A / -/ -/ -/ / / / 9/6 Page 6

7 Volume VI, Issue III, March 7 ISSN 78-5 / / -/ / 77 Ma z Z* Solutio satisfies the optimality coditio ad restrictio o Lagragia multipliers. Also by usig our modified techique oe iteratio is reduced ad solutio remais itact. Eample. Maimize, z 6 Sub to :,, L 6 s s s s Iitial Table L L 6 6 L L s s A 6 A, c B y B B A A A 6 - A Sice = maimum, so eters the basis ad A leaves the basis st Iteratio c B y B B A A A / / - / / / /6 / -/6 / / -/6 -/ /6 5/ -/ -/ / 7/ -/6 -/ - 7/6 Sice =7/ maimum, so eters the basis ad leaves the basis d Iteratio c B y B B A A A / / / / -/ / / -/ -/ -/ / -/ 5/ -/ -/ / -/ - -/ -5/ / -/ 6 Page 7

8 Volume VI, Issue III, March 7 ISSN 78-5 Sice =/ maimum, so eters the basis ad leaves the basis rd Iteratio c B y B B A A A Here there is a tie for ma ad therefore to decide eterig vector we refer value of z c. Most egative z c correspods to ad therefore eters the basis ad A leaves the basis. th Iteratio c B y B B A A - / -/ / - - -/ -/ / - - Z*. z Opt Eample 5: Maimize z Sub to : 6,, s 6, s s s L s 6 s s s L L L L s 6 s A A 6 Iitial Table c B y B B A A A - A - Page 8

9 Volume VI, Issue III, March 7 ISSN Sice =6 maimum, so eters the basis ad A leaves the basis. st Iteratio c B y B B A A -/ / A / - - -/ - / - -/ Sice = maimum, so eters the basis ad leaves the basis d Iteratio c B y B B A A -/ / A - / / -/ -/ / -/ / -/ -/ -/ / -/6 / / / Sice =/ maimum, so eters the basis ad A leaves the basis rd Iteratio c B y B B A A / / -/ / / -/ / / -/ / /9 / -/9 / -/9 -/6 / /9 -/ -8/9 / -/9 -/ / Z* - - Opt 9. z IV. CONCLUSION 9 It is see that the eistig method is more icoveiet i hadlig the degeeracy ad cyclig problems because here the choice of the vectors, eterig ad outgoig, play a importat role. Here we observed that the optimum solutio obtaied i three iteratios by our modified techique, where as Wolfe s simple method took five iteratios. Hece our techique gives efficiecy i result as compared to other method i less iteratio. Hece the umber of iteratios required is reduced by our methodology. Also we require less time to simplify Numerical Problems.. REFERENCES []. Wolfe P: The Simple method for Quadratic Programmig, Ecoometrical, 7, 8-9., 959. []. Takayama T ad Judge J. J : Spatial Equilibrium ad Quadratic Programmig J. Farm Ecom., 67-9., 96. []. Terlaky T: A New Algorithm for Quadratic Programmig EJOOR,, 9-, North-Hollad, 98. []. Ritter K : A dual Quadratic Programmig Algorithm, Uiversity Of Wiscosi- Madiso, Mathematics Research Ceter. Techical Summary Report No.7, 95. [5]. Frak M ad Wolfe P: A Algorithm for Quadratic Programmig, Naval Research Logistic Quarterly,, 95-., 956. [6]. Khobragade N. W: Alterative Approach to Wolfe s Modified Simple Method for Quadratic Programmig Problems, It. J. Latest Tred Math, Vol. No. March. P. No Page 9