NONDIFFERENTIABLE MATHEMATICAL PROGRAMS. OPTIMALITY AND HIGHER-ORDER DUALITY RESULTS

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1 HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY, See A, OF HE ROMANIAN ACADEMY Volue 9, Nube 3/8,. NONDIFFERENIABLE MAHEMAICAL PROGRAMS. OPIMALIY AND HIGHER-ORDER DUALIY RESULS Vale PREDA Uvety of Buchaet, Faculty of Matheatc ad Coute Scece E-al: Fo a geeal cla of odffeetable atheatcal oga, we gve eceay otalty codto ad dualty eult fo a hghe-ode dual of Mod-We tye. Key wod: ultobectve ogag, otzato.. INRODUCION I th ae we code a geeal cla of odffeetable atheatcal ogag oble, aely, + f ( x ( x B x = (P ubect to x X whee X = { x R g( x }, f R : R ad g R R ae twce dffeetable fucto, ad B, : g = ( g,..., g = =,, ae otve e-defte (yetc atce. Let. Fo ee, fo exale Mod[4], Peda [6], Peda ad Kolle [7]. he tudy of hghe-ode dualty otat due to the coutatoal advatage ove ft-ode dualty a t ovde bette boud fo the value of the obectve fucto whe aoxato ae ued (Magaaa [], Yag [8]. I Secto we toduce a geeal Mod-We tye [5] hghe-ode dual to oble (P ad gve oe defto of hghe-ode ρ-vexty ad geealzed hghe-ode ρ-vexty. I Secto 3, oe eceay otalty codto ae gve. I Secto 4, fo the geeal hghe-ode dual of Mod-We tye defed Secto, weak dualty, tog dualty, tct covee dualty, ad covee dualty eult ae eeted.. PRELIMINARIES AND SOME DEFINIIONS Let h, k,, k : R R R be dffeetable fucto wth eect to each aguet. I the followg, the oeato take elatve to the ft aguet whle the oeato take elatve to the ecod oe. We ut k u, = ( k,..., k whee the ybol deote taoe. Alo, let I ( {,,...,}, =,, wth I = {,,..., } ad U = I I =, fo β. β We toduce the followg geeal Mod-We tye [5] hghe-ode dual (HGD wth eect to (P:

2 Vale PREDA + ax f ( u h( u, + u B (, ( (, + w h u y g u yk u yk = ubect to hu (, + Bw = ( y ku (,, (. = yg ( u + yk yk,,, = (. wbw, {,,..., }, (.3 y, (.4 whee u, w,, w, R ad y R. ' ' ' Let ρ R, ρ = ( ρ,..., ρ R ad d : R R R. + Defto.. he obectve fucto f ad cotat fucto g, hghe-ode tye I at u wth eect to a fucto η f the equalte =,, ae ad to be (ρ,ρ' - ad = = f( x + x B w f( u u B w η xu hu Bw hu hu ρd x u (, (, + + (, ( (, + (, = g u η x u k u + k u k u ρ d x u = ' ( (, (, (, ( (, (, ;, hold fo all x. Defto.. he obectve fucto f ad cotat fucto g, =,, ae ad to be (ρ,ρ' - hghe-ode eudo-qua tye I at u wth eect to a fucto η f the lcato: η(, xu hu (, + Bw ρd (, x u = ad hold fo all x. = = f( x + x B w f( u h u B w + ( h g ( u k ( k ' ( x, u k d ( x, u,, η ρ =

3 3 Otalty ad hge ode dualty Let 3. NECESSARY OPIMALIY CONDIIONS x be a feable oluto fo (P. We defe the et S = {,,..., }, ( B x = { S x B x > }, B( x = { S x B x = }, ( Z x = { z R z g ( x, M ( x ad z f ( x + z B x B ( x ( x B x + ( z B ( x B z < }, whee M ( x = {, g ( x = }. Now eceay otalty codto fo x to be a otal oluto fo (P ae a followg. w R heoe 3. If x a otal oluto fo (P ad Z( x =, the thee ext y R, y ad, S uch that y ( g x =, y g( x = f ( x + B w ; w B w ad (x B x = x B fo S. w = 4. DUALIY RESULS FOR (P AND (HGD I th ecto, fo (P ad (HGD, we code weak dualty, tog dualty, tct covee dualty, ad covee dualty eult. heoe 4. (Weak dualty. Let y, w,..., w, fo (HGD we have η R R R uch that fo all : x X ad a feable oluto η( xu, hu (, + Bw yk ρd ( x, u = f( x + xbw f( u + ubw yg( u = = (, (, h u + (, ( (,, yk u h u yk u (4. ad yg ( u yk + k η( xu, yk ρd ( x, u, =, (4. he f (P u (HGD. + ρ. (4.3 = ρ

4 Vale PREDA 4 Fo the ext dualty eult we uoe that h ad k atfy oe "tal" codto (defed by (4.4 below codeed Zhag [9] ad Mha ad Rueda [3]. heoe 4. (Stog dualty. Let x be a local o global otal oluto of (P wth Z( x = ad aue that h ( x, =, k ( x, =, (, ( x h x = f, k( x, = g( x (4.4 he thee ext y R ad w,..., w R uch that ( x, y, w,..., w, = a feable oluto fo (HGD ad the coeodg value of (P ad (HGD ae equal. If the weak dualty heoe 4. alo hold, the ( x, y, w,..., w, = a otal oluto fo (HGD. heoe 4.3 (Stct covee dualty. Let x be a otal oluto of (P wth Z( x = ad aue (4.4 hold. Aue alo that the hyothee of the weak dualty heoe 4. ae atfed. If ( x, y, w,..., w, a otal oluto of (HGD ad f η xx hx Bw yk x ρd x x (, (, + (, (, = f( x + xbw f( x + xbw yg( x = = hx (, yk ( x, + hx (, ( yk ( x, >, fo ay x x, the x = x,.e., x a otal oluto of (P ad the otal value of the obectve fucto of (P ad (HGD ae equal. he oof alog the uual le of thoe of la theoe (ee, fo exale, Peda [6]. Suoe ow that h ad k, k,..., k ae twce dffeetable wth eect to the ecod aguet ad dffeetable wth eect to the ft oe. heoe 4.4 (Covee dualty. Let ( x, y, w,..., w, be a otal oluto of (HGD uch that (4.4 hold ad aue that ( the et of vecto [ h( x, ( y k ( x, ],[ ( y k ( x, ], =,, =, lea deedet, whee [ ( k y ] [ ( ( y k the th ow of ; h y k ] the th ow of h ( y k ad ( the atx aa otve o egatve defte, whee a the vecto f ( x + h( x, ( y g( x ( y k( x, ( y k ( x, + h( x,. he x a feable oluto to (P ad the coeodg value of the obectve fucto of (P ad (HGD ae equal. Futhe, f the hyothee of the weak dualty heoe 4. hold, the x a otal oluto to (P.

5 5 Otalty ad hge ode dualty Reak: Recetly oe otalty eult fo odffeetable ogag oble have bee gve []. It teetg to coae thee eult ad thoe eeted above. ACKNOWLEDGEMENS h wok wa atally uoted fo Gat PN II IDEI, code ID, o /..7. REFERENCES. BELDIMAN, M., PANAIESCU, E., DOGARU, L., Aoxate qua effcet oluto ultobectve otzato, Bull. Math. Soc. Sc. Math. Rouae 5(99, o, 9-, 8.. MANGASARIAN, O.L., Secod ad hghe ode dualty olea ogag, J. Math. Aal. Al. 5, 67-6, MISHRA, S.K., RUEDA, N.G., Hghe ode geealzed vexty ad dualty odffeetable atheatcal ogag, J. Math. Aal. Al. 7, ,. 4. MOND, B., A cla of odffeetable atheatcal ogag oble, J. Math. Aal. Al. 46, 69-74, MOND, B., WEIR, J., Geealzed covexty ad hghe ode dualty, J. Math. Sc. 6-8, 74-94, PREDA, V., O geealzed covexty ad dualty wth a quae oot te, Zetetf. Oe. Re. 36, , PREDA, V., KOLLER, E., O dualty fo a odffeetable ogag oble wth a quae oot te, Rev. Roua Math. Pue Al.. 8. YANG, X. Q., Secod ode global otalty codto fo covex coote otzato, Math Pogag 8, , ZHANG, J. Geealzed covexty ad hghe ode dualty fo atheatcal ogag oble, Ph.D. he, La obe Uvety, Autala, 998. Receved Setebe 3, 8