An Algebraic Elimination Method for the Linear Complementarity Problem

Transcription

1 Volume-3, Issue-5, October-2013 ISSN No: Iteratioal Joural of Egieerig ad Maagemet Research Available at: wwwijemret Page Number: A Algebraic Elimiatio Method for the Liear Complemetarity Problem P Rajedra 1, P Padia 2 1,2 Departmet of Mathematics, School of Advaced Scieces, VIT Uiversity, Vellore-14, Tamiladu, INDIA ABSTRACT A ew method amely, algebraic elimiatio method is proposed for fidig a complemetarity feasible solutio to the liear complemetarity problem which have applicatios i o-liear programmig, ecoomics, game theory ad egieerig The, the algebraic elimiatio method is exteded to quadratic programmig problems The solutio procedure is illustrated by meas of umerical examples Keywords: Liear complemetarity problem, Algebraic elimiatio method, Quadratic programmig problem I INTRODUCTION The liear complemetarity (LC) problem is oe of the most widely studied problems of mathematical programmig sice it arise i a variety of applicatios [3, 4, 6] i egieerig, ecoomics ad applied scieces Several methods [11, 10, 14, 16, 4 ] have bee proposed for solvig LC problems Iterative methods for the solutio of LC problem were cosidered i [2, 12, 7, 9] Morales etal [13] proposed a improvemet of a method described by Kocvara ad Zowe [9] for the solutio of mixed symmetric liear complemetarity problems Yassie [18] discussed a comparative study betwee Lemke s method ad the Iterior poit method for the mootoe liear complemetarity problem Hadjidimos etal[8] proposed the scaled extrapolated block modulus algorithm for solvig the LC problem whose real matrix is a H+-matrix Elfoutayei etal [5] have proposed a mimax algorithm for solvig LC problems liear iequalities ad complemetarity slackess Also, QP problem ca be trasformed ito LC problem usig the of KKT coditios ad the, it ca be solved by Lemke s algorithm Recetly, Quazzafi Rabbai etal [15] applied Fourier s variables elimiatio method to solve QP problem I this paper, we propose a ew method amely, algebraic elimiatio method for fidig a complemetarity feasible solutio to the LC problem I the algebraic elimiatio method, we first reduce the LC problem ito a iequality system by usig the relatio W 0 ad the, the resultig reduced iequality system is solved by matrix / algebraic method The proposed method for solvig LC problem is very simple ad easy to uderstad ad also, to apply Further, we exted the algebraic elimiatio method for solvig quadratic programmig problems with liear costraits after covertig to LC problem Numerical examples are give for better uderstadig the solutio procedures of the proposed method II LINEAR COMPLENTARITY PROBLEM Cosider the followig problem kow as a LC problem to fid colum vectors W R ad Z R satisfyig the system A quadratic programmig (QP) problem is a special case of o- liear programmig problem The two importat methods for solvig QP problem are Wolfe s method [17] ad Beale s method [1] Wolfe [17] used Kuh Tucker coditios to trasform the QP problem ito 51

2 A colum vector ( W, Z) R R satisfyig (21) ad (22) is called a feasible solutio to the LC problem A colum vector ( W, Z) R R satisfyig (21) to (23) is called a complemetarity feasible solutio to the LC problem III ALGEBRAIC ELIMINATION METHOD Now, we propose a ew method amely, algebraic elimiatio method for fidig a complemetarity feasible solutio to the LC problem The proposed method proceeds as follows: Step 1 If M 0 ad q 0, the reduce the give system MZ = q, Z 0 by takig W = 0 ad solve the reduced system by matrix / algebraic method Say Z = Z The, ( W = 0, Z = Z ) is a complemetarity feasible solutio to LC problem Step 2 If there is o egative term i q, the ( W = q, Z = 0) is a complemetarity feasible solutio to LC problem Step 3 If q < 0 for atleast oe term, move o to Step 4 Step 4 Costruct the iequality system MZ q ; Z 0 from the give system usig W 0 Step 5 Elimiate the Z-variables oe by oe upto oly oe of the Z-variable obtaied Say z k Step 6 Fid the greatest lower boud of all maximum possible values of z k Say, z k The values of all possible z j ' s are computed usig backward substitutio method Step 7 If m =, go to Step 9 Otherwise, move o to Step 8 Step 8 (a) Take w j = 0 if z j 0, j = 1,2,, (b) Substitutig the values of w j, j = 1,2,, ad the values of z j, j = 1,2,, i W MZ = q obtaied i 9(a), compute the rest of W ad Z-values Thus, W-value ad Z-value are obtaied Say, W ad Z (c) ( W, Z ) is a complemetarity feasible solutio to the LC problem Step 9 (a) Take w j = 0 if z j 0 (b) Substitutig the values of w j, j = 1,2,, ad the values of z j, j = 1,2,, i W MZ = q obtaied i 9(a), compute the rest of W-values Thus, W-value is obtaied Say, W (c) ( W, Z ) is a complemetarity feasible solutio to the LC problem Now, the proposed method is illustrated with the help of the followig examples Example 31: Cosider the followig LC problem Now, by the Step 1, the complemetarity feasible solutio to the give LC problem is Example 32: Cosider the followig LC problem ad basic algebraic method by makig the ivolved iequality ito equality Say z j = z j, j = 1,2,, 52

3 Now, by the Step 2, the complemetarity feasible solutio to the give LC problem is W = (1,2) ad Z = (0,0) Example 33: Cosider the followig LC problem Now, we proceed to the Step 6 to the Step 9 From the Table-3, we ca coclude that z 2 3 Now, sice z 2 3 correspods to (38) ad sice (3,8) correspods to (34) ad (33), we have z 1 = 0 Now, sice z 2 3, we have w 2 = 0 Now, substitutig the values of z 2 3, w 2 = 0 ad z 1 = 0 i the give system, we have Now, by the Step 3 ad the Step 4, we have the followig iequality system MZ q ; Z 0 The table form of the above problem ca be give as follows: Thus, the complemetarity feasible solutio to the LC problem is W = (15,0,0) ad Z = (0,3,8) IV QUADRATIC PROGRAMMING PROBLEM Now, we proceed to the Step 5 Cosider the followig quadratic programmig (QP) problem: Now, elimiatig followig table z 1 from the Table-1, we obtai the Now, usig KKT coditios, the problem (Q) ca be writte as a liear complemetarity problem as follows: Now, elimiatig z 3 from the Table-2, the followig table has bee obtaied: Result 41: The complemetarity feasible solutio to the (L) problem is the optimal solutio to the (Q) problem Now, the solutio procedure for solvig the quadratic programmig problem by the algebraic proposed method after covertig to LC problem is illustrated by the followig example 53

4 Example 41: Cosider the followig QP problem Now, sice z 3 6 correspods to (41) Now, sice z 3 6, we have w 3 = 0 Now, substitutig the values of z 3 6 ad w 3 = 0 i the give system, we have Now, cosider the followig LC problem which correspods to (Q) problem: Now, the above system ca be writte as the followig table Now, elimiatig z 1 from Table -4, we have the followig table Now, elimiatig z 2 from Table-5, we have From the Table - 6, we ca coclude that z 3 6 Thus, the optimal solutio to the give QP problem is x 1 = 4, x 2 = 0 ad Mi K = 24 V CONCLUSION I this paper, we propose a ew method amely, algebraic elimiatio method for fidig a complemetarity feasible solutio to the liear complemetarity problem The proposed method is easy to uderstad ad apply The, we exted algebraic elimiatio method to quadratic programmig problems With the help of umerical examples, the proposed method is illustrated for better uderstadig of the solutio procedures Sice complemetarity is cetral to all costraied optimizatio problems, the algebraic elimiatio method ca serve decisio makers by providig a appropriate best solutio to a variety of liear programmig complemetarity models i a simple ad effective maer I ear future, we exted the proposed method to fuzzy liear complemetarity problems ad fuzzy quadratic programmig problems REFERENCES [1] Beale, EML, O quadratic programmig, Naval Research Logistic Quart, 6, , 1959 [2] Cryer, CW, The solutio of a quadratic programmig problem usig systematic over-relaxatio, SIAM Joural o Cotrol ad Optimizatio, 9, , 1971 [3] Cottle, RW ad Datzig, GB, Complemetarity pivot theory of mathematical programmig, Liear Algebra ad Its Applicatios, 1, , 1968 [4] Cottle, RW, Jog-Shi Pag ad Richard E Stoe, The liear complemetarity problem, SIAM, Philadelphia, 2009 [5] Elfoutayei, Y ad Khaladi, M, A Mi-Max algorithm for solvig the liear complemetarity problem, Joural of Mathematical Scieces ad Applicatios, 1, 6-11, 2013 [6] Ferris, MC ad Pag, JS, Egieerig ad ecoomic applicatios of complemetarity problems, SIAM REV,39, ,

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