LINEARLY CONSTRAINED MINIMIZATION BY USING NEWTON S METHOD

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1 Jural Karya Asl Loreka Ahl Matematk Vol 8 o 205 Page Jural Karya Asl Loreka Ahl Matematk LIEARLY COSTRAIED MIIMIZATIO BY USIG EWTO S METHOD Yosza B Dasrl, a Ismal B Moh 2 Faculty Electrocs a Computer Egeerg, Uverst Tekkal Malaysa Melaka, Malaysa 2 Isttut Matematk Keuruteraa, Uverst Malaysa Perls, Malaysa yosza@yahoocom Abstract : I ths paper we propose a alteratve approach to f the optmum soluto of quaratc programmg problems QPP by employg the ewto s metho It ca be oe by covertg the orgal costrae problem to a sequece of ucostrae mmzato va pealty fucto Also we escrbe how to obta the optmal soluto of the problem f locate at the bouary or at the tersecto of the two a more costrats Keywors: Optmum soluto, quaratc, programmg, feasble, eplorato Itroucto The theory of quaratc programmg problem eals wth problems of costrae mmzato, where the costrat fuctos are lear a the obectve s postve efte quaratc fucto [-3] Although t represets a atural trasto from theory of lear programmg to the theory of olear programmg problem, there are some mportat ffereces betwee ther optmal solutos If the optmum soluto of quaratc programmg problem ests the t s ether a teror pot or bouary pot whch s ot ecessarly a etreme pot of the feasble rego I ths paper we propose a alteratve metho whch s calle the costrat eplorato metho for eplorg a obtag the optmal soluto of QPP wthout usg ay atoal formato for > 3 [3,4] otato A quaratc fucto o R to be cosere ths paper s efe by T T f A b q 2 whch A a, b b,, b T m,,, T a q s a costat The set of all feasble solutos whch s calle the feasble rego s a close asset efe by where B s m matr a c a vector Ucostrae Mmzato F R, B c, 0 2 R The statoary pot of f gve by ca be eterme by A b 3 whch s calle ucostrae mmum of f f A s a postve efte matr As metoe prevous secto ca be a teror pot or a bouary pot of the feasble rego However, 205 Jural Karya Asl Loreka Ahl Matematk Publshe by Pustaka Ama Press S Bh

2 there s a possblty that ca be a eteror pot Therefore, f A s postve efte matr, the problem Suppose that F s efe by 2 a let mmum of f whch s gve by Suppose that eotes the costrae mmum of f subect to F A Sgle Equalty Costrae Optmal Yosza a Ismal s the feasble rego a becomes the optmal soluto of the quaratc programmg gve by 3 eote the ucostrae I ths secto we escrbe how to search a pot o the equalty costrat whch becomes a caate of the optmal soluto to the problem -2 Let us coser the quaratc programmg problem wth = 3 Suppose that the costrat s gve by a ts equalty costrat s Clearly, the pot a a2 2 a3 3 c a a2 2 a3 3 c 4 c c c,, a a2 a3 5 whch les o the plae a a2 2 a3 3 c s uquely eterme sce there s oe to oe correspoece betwee the pot a ther respectve a therefore, by substtutg the above pt to the obectve fucto, we ca obta the fucto wth a as the epeet varables from whch the statoary pot of f, ca be acheve through fferetatg f, wth respect to a a equatg wth zero If a eotes the statoary pot, the we obta the pot c, c, c a a2 a3 6 whch refers to the costrae mmum of f o the equalty costrat gve by 4 prove the hessa of the obectve fucto s postve efte The Itersecto of Two Equalty Costrats I ths secto we escrbe a metho for obtag the costrae mmum of f o the tersecto of two hyperplaes Suppose that a are eotes the costrae mmum subect to the th a th costrats gve by 6 respectvely Let us efe a are the ormal vectors of th a th costrats respectvely see Fg To obta the etreme pot o the tersecto of the th a th costrats, 2, we coser the followg theorem 85

3 Jural KALAM Vol 8 o, Page [] [] Fg : Itersecto of two hyperplaes [] a [] Theorem : the pots o the th costrat ca be eterme by = + + where = 2 o the tersecto of th a th costrats ca be eterme by 2 = + where = 86 Proof: Usg th costrat A pot o the th costrat ca be eterme by usg = + + Iee, ths pot wll be the th costrat f a oly f = Sce = = + + = By usg the pot, hypothess Theorem above, a eplore the bouary of th costrat, the the pot 2 must satsfy the th costrat So 2 = or = 2 = + = = Teorem 2: If the hypothess of Theorem s val

4 Yosza a Ismal 87 Ay pot o the th hyperplae ca be eterme by usg = + + where = Ay pot 2 o the tersecto le of th a th costrats ca be eterme by 2 = + where = Proof: By usg th costrat A pot, wll be o the th costrat f a oly f = such that = = + + a = Furthermore, 2 wll satsfy th costrat f a oly f = 2 = + = = = Corollary If the hypotheses of Theorem a Theorem 2 are val, the the optmal pot of the QPP problem s 2 Costrae Optmal By Usg ewto s Metho The pealty fucto to problem subect to the equato 2, we efe the ew obectve fucto ; m P K f K F 7 By choosg K p > 0 as a postve ecreasg sequece, the mmzg P;K yels K p, a t ca be show that as p, K p a K p

5 Jural KALAM Vol 8 o, Page The ewto s algorthm at the kth terato s gve as follows: Solve F k k = -g k for k Set k+ = k + k Where R m m m a g: R R I ths case F s the Jacoba matr of equato 6 at ; that s, F s the matr whose etry s g /,, =,2,, The frst step requres the soluto of system equatos Cocluso I ths paper we trouce a alteratve metho to obta the optmal soluto of the quaratc programmg problem orgal form wthout usg ay atoal formato such gve smple Wolfe s metho I ths approach we also o ot ee to apply the Lagrage maultpler metho as see most of the fovourte methos gve ltertaure The optmal soluto of quaratc programmg problem whch s occurrg ether at the corer pot, bouary a se feasble rego ca be easly obtae The alteratve metho also ca be use to eterme the costrats whch s volate by the ucostrae optmal Refereces [] I B Moh a Y Dasrl, A ote o the volate costrats approach for solvg the quaratc programmg problem, J of Ultra Scetst of Physcal Sceces 02, 998: 4-48 [2] I B Moh a Y Dasrl, Costrat eplorato metho for quaratc programmg problem, J Apple Mathematcs a Computato 2, 2000: 6-70 [3] I B Moh a Y Dasrl, Cross-Prouct recto eplorato approach for solvg the quaratc programmg problems, J of Ultra Scetst of Physcal Sceces 22, 2000: [4] H Tel, C Va epae, Quaratc programmg as a eteso of covetoal quaratc mamzato, J Ist Maagemet Sc 7,96, -20 [5] P Wolfe, The smple metho for quaratc programmg, Ecoometrca 27, 959,