DUALITY FOR MINIMUM MATRIX NORM PROBLEMS

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1 HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMNIN CDEMY, Seres, OF HE ROMNIN CDEMY Vole 6, Nber /2005, DULIY FOR MINIMUM MRI NORM PROBLEMS Vasle PRED, Crstca FULG Uverst of Bcharest, Faclt of Matheatcs ad Iforatcs, Str. cadee 4, 0004 Bcharest, Roaa cade of Ecooc Sceces, Pata Roaa 6, 0074 Bcharest, Roaa, Corresodg athor: Vasle PRED, E-al: We vestgate a atrx or roble wth ealt costrats ad ts dal. exlct rle for fdg a ral solto fro the solto of the dal roble s gve. he case wth ealt costrats s also cosdered. Kewords: atrx or soltos, dalt, relatos betwee ral ad dal soltos.. PRELIMINRIES he roble of deterg a or solto arses varos alcatos, see refereces [,2,6,7]. We cosder the roble P F( ) =, sbject to B where < <,, B are We eto that = j=, real atrces, = x j ad that = x j j B eas j a x j, =,, j =, where a s the th row of, b j s the jth col of B = ( b j ) j It s assed throghot ths aer that the feasble rego = { B} s the real atrx of ows., x j s the jth col of. R s ot et. Hece the exstece ad eess of the solto of roble (P) are esred b the rojecto theore [7]. he dal dex of s deoted b, that s + =. We wll establsh that the dal roble of (P) has the for ( D) (,..., ) = b b axg 0,..., 0,..., R Reccoeded bmars IOSIFESCU, eber of the Roaa cade

2 Vasle PRED, Crstca FULG 2 where Y =... R l, wth ( l ) =... ( l ) + + R, l =, ad = (... ).We recall that = c j. l= j= Next, we wll vestgate the relatos betwee the two robles, the ral oe, (P), ad ts dal (D). 2. DULIY RESULS I order to rove that (D) s the dal of (P), we eed soe frther otato. Let c j be the jth col Z Y = Y... Y be a real atrx, where of ad let l ( ( Y ) R, l, ad = l +... l + =, For the cooet ( l ) + j of the atrx ( l ) + j = c j sg( c j ), l =,, j =,. Z we also se the otato. he followg leas are eeded the roof of or a reslt. jl Lea 2. We have grad =... R Proof. B drect calcls we have = ( ) Y = cj == aj c sg( c ) = ( l ) + j l + j l= j= = = a = a Y j l ( l ) + j. Lea 2.2 We have = = Proof. We ca sccessvel wrte

3 3 Dalt for atrx or robles l l + j j l + j j l + = l= j= l= j= = Y = a Y = a Y = = ( Y) aj l l j ( l ) ( Y) c + + = + = l= = j= l= = = c c c = c = Y sg l= = l= = Lea 2.3. We have = Z Proof. Clearl, Z = ( l ) j c j sg( c j ) c j. + = = = l= j= Let g g jl j=, l=, l= j= = be a real Lea 2.4. We have real atrx wth g ( Z) = g jl l= j= F =. x jl Proof. Clearl, ( ) ( ) g Z Y = Y sg Y = c sg c = c g Z Y = Y jl jl jl j j j. heore 2.5 Let be the otal solto of (P); the there exsts Y R sch = Z Y ad Y s the otal solto of (D). Coversel, let Y be the otal solto of (D); that the ( Y = Z ) s the e solto of (P). We also have F( ) ealt D F, for a feasble soltos,y of (P), (D). D = ad the classcal ral-dal Proof Usg the Kh-cer otalt codtos we have that s a otal solto for (P) f ad ol f there exsts Y R sch that x x x = ˆ... ; Y 0; Y... = 0; g Y x b b Bt, b Lea 2.4, g ( Z ( Y ) = ˆ( Y ), hece Z( Y ) ( b )... ; Y 0; =. Sbstttg ths to (2.) elds... ( b ) whch are the otalt codtos for (D). We have roved that f s the otal solto of (P), the the corresodg Lagrage vector Y R s the otal solto of (D). = 0 = 0 (2.) (2.2)

4 Vasle PRED, Crstca FULG 4 Coversel, let Y R be the otal solto of (D). he the otalt codtos (2.2) hold = Z Y satsfes (2.), whch roves that s the otal solto of (P). I both cases, b Leas , we fd that ad Y satsf D Y = b b Y = ad = = == Z = = F. Moreover, for all feasble soltos, Y of (P), (D) we have = b b D = F F. D Y ortat secal case of (P) occrs whe the sste case the ral roble ca be wrtte as ( P2) lwas der the hothess F = B = B s relaced b = B. I ths F = ( P3) B B. { M, R = B}, b heore 2.5 the dal of (P3) has the for ax R v R = b,, v b = 0, =, 0, =, v ( U, V ) ˆ U, V = where v... v... s a real v =, = elds the followg reslt., 2 atrx,, v R, =,. Sbstttg Corollar 2.6 he dal of the or roble (P2) s ax R = b, =,. If s the otal solto of (P2), the there exsts Y R solto of (D2). Coversel, f. sch that Z ( Y ) Y s the otal solto of (D2) the Z( Y ) (P2). We also have F( ) DY soltos,y = ad Y s the otal = s the e solto of D = ad the classcal ral-dal ealt F( ) of (P2), (D2). for a feasble

5 5 Dalt for atrx or robles REFERENCES. CDZOW, J.., fte algorth for the l -solto to a sste of cosstet lear eatos, SIM Nercal alss, 0, , D., exteded Kacar s ethod for l - or soltos, echcal Reort, Hdrologcal Servce of Israel, D., O or soltos, Joral of Otato heor ad lcatos, 76, o., , FULG C., exteso of the or roble, Matheatcal Reorts, 7 (57), , FULG C., O or robles, alele Uverstat Bcrest Mat., 52, o. 2, , HERMN G.., LEN., fal of teratve adratc otato algorths for ars of ealtes wth alcato dagostc radolog, Matheatcal Prograg Std, 9,. 5-29, LUENBERGER D.G., Otato b Vector Sace Methods, Wle, New Yor, PRED V., Otalt codtos for a class of atheatcal rograg robles, Bll. Soc. Sc. Math. Roae, 34, (82), , PRED V., Nolear rograg ad atrx gae evalece, J. stral. Math. Soc. Ser. B, 35, , ROCKFELLR R.., Covex alss, Prceto Uverst Press, Prceto, N.J., 970. Receved Jaar 24, 2005