OPTIMALITY CONDITIONS FOR NONLINEAR PROGRAMMING WITH MIXED CONSTRAINTS AND ρ -LOCALLY ARCWISE CONNECTED FUNCTIONS

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1 HE PUBLSHNG HOUSE PROCEEDNGS OF HE ROMANAN ACADEMY, Seres A, OF HE ROMANAN ACADEMY Volume 7, Number /6, pp - OPMALY CONDONS FOR NONLNEAR PROGRAMMNG WH MXED CONSRANS AND ρ -LOCALLY ARCWSE CONNECED FUNCONS oan M SANCU-MNASAN, Andreea Mădălna SANCU he Romanan Academy, nsttute of Mathematcal Statstcs and Appled Mathematcs, Calea Septembre nr, Ro-57, Bucharest 5, Romana, E-mal: stancum@csmro A nonlnear prorammn problem wth mxed constrants s consdered, where the functons nvolved are ρ -locally arcwse connected, ρ -locally Q-connected, ρ -locally P-connected and locally PQ-connected (a noton ntroduced n ths paper and dfferentable wth respect to an arc Suffcent optmalty condtons are obtaned n terms of the rht dfferentals wth respect to an arc of the functons Our results eneralze to the case of mxed constrants the results obtaned by Stancu-Mnasan [] and Stancu-Mnasan and Andreea Mădălna Stancu [] PRELMNARES n ths secton we ntroduce the notaton and defntons whch are used throuhout the paper Let R n be the n-dmensonal Eucldean space and R n ts nonneatve orthant { x R n, x j, ј =,, n} hrouhout the paper the follown conventons for vectors n R n wll be followed: x > y f and only f x > y, =,, n, x y f and only f x y, =,, n, x y f and only f x y, =,, n, but x y secton Let Also, all defntons and theorems are numbered consecutvely n a snle numeraton system n each X R n be a nonempty and compact subset of R n Defnton Let x, x X A contnuous mappn H ( = x, H ( x x, x = H, : [,] R n wth s called an arc from x to x Defnton ([] We say that the set X R n s a locally arcwse connected set at x ( x X ( X s LAC( x, for short f for any x X there exst a postve number a ( x, x, wth < a ( x, x, and a contnuous arc H, ( λ X for any λ (, a ( x, x H We say that the set X s locally arcwse connected f X s locally arcwse connected at any x X f we choose the functon H of the form H (λ = ( λ x λ x, we retreve the defnton of locally starshaped set as ven by Ewn [] Recommended by Marus OSFESCU, member of the Romanan Academy

2 oan M SANCU-MNASAN, Andreea Mădălna SANCU Defnton ([7] Let f : X R be a functon, where X R n s a locally arcwse connected set at x X wth the correspondn functon H (λ and a maxmum postve number a ( x, x satsfyn the requred condtons (from Defnton Also, let ρ R and d(, : X X R d ( x, x for x x We say that f s: ( ρ -locally arcwse connected at x ( f s ρ - LCN( x, for short f for any x X there exst a postve number d ( x, x a ( x, x and an arc H n X on [, d( x, x ] f (, ( λ λf ( x ( λ f ( x ρλ d( x, x, λ d( x, x ( H ( ρ -locally Q-connected at x ( ρ - LQCN(x f for any x X there exst a postve number d ( x, x a ( x, x and an arc H n X on [, d( x, x ] f ( x f ( x f ( H λ d( x,x, ( λ f ( x ρλd( x, x x X there exst a postve number ( ρ -locally P-connected at x ( ρ - LPCN( x f for any d ( x, x a ( x, x, an arc H n X on [, d(x, x ], and a postve number γ x, x f ( x < f ( x f ( H λ d( x,x x, x ( λ f ( x λγ x, x ρλd( x, x ( ρ -locally strctly P-connected at x ( ρ - LSPCN( x f for any x X there exst a postve number d ( x, x a ( x, x, an arc H n X on [, d(x, x ], and a postve number γ x, x x x, f ( x < f ( x λ d( x,x f ( H, ( λ < f ( x λγ, ρλd( x, x he functon f s sad to be ρ -locally strctly arcwse connected at x X ( ρ - LSCN( x f for each x X, x x, nequalty ( s strct f f s ρ - LCN( x ( ρ - LSCN( x at each x X, then f s sad to be ρ - LCN (ρ - LSCN on X f f s ρ - LQCN at each x X, then f s sad to be ρ - LQCN on X f f s ρ - LPCN at each x X, then f s sad to be ρ - LPCN on X Defnton Let f : X R be a functon, where X R n s a locally arcwse connected set at x X wth the correspondn functon H (λ and a maxmum postve number a ( x, x satsfyn the requred condtons (from Defnton We say that f s locally PQ-connected at x ( LPQCN( x f for any x X there exst a postve number d( x, x a ( x, x and an arc H n X on [, d( x, x ] f( x = f( x f( Hxx, ( λ f( x < λ < d( x,x Defnton 5 ([] Let f : X R be a functon, where X R n s a locally arcwse connected set at x X, wth the correspondn functon (λ and a maxmum postve number H

3 Optmalty condtons for nonlnear prorammn a ( x, x satsfyn the requred condtons he rht dfferental of f at x wth respect to the arc ( s defned as provded the lmt exsts f f s dfferentable at any x H λ (df (x, H ( = lm [ f ( H ( λ f ( x ] ( λ λ X, then f s sad to be dfferentable on X SUFFCEN OPMALY CRERA Consder the nonlnear prorammn problem Mnmze f( x (P subject to (, ( =, x h X, where X R n s a nonempty open locally arcwse connected set; f : X R; = ( m : X R m ; = ( h j j k : X R k ; v the rht dfferentals of f,, =,!, m, and hj, j =,, k at x exst wth respect to the same arc H ( λ Let X = { x X (x, hx ( = } be the set of all feasble solutons to (P Let N ε (x = { x R n x x < ε } Defnton a x s sad to be a local mnmum soluton to problem (P f x X and there exsts ε > x N ε ( x X f ( x f (x b x s sad to be the mnmum soluton to problem (P f x X and f ( x = mn f ( x For x X we denote by = ( x = { ( x = } the set of ndces of actve constrants at x, by J = J ( x = { ( x < } the set of ndces of nonactve constrants at x, and set = ( Obvously, J = {,,, m} Let u R m be u and u ( x = Obvously, u and u J =, where u and u J denotes the subvectors of u correspondn to the ndex sets and J, respectvely Let = { : u > } and L = { : u = }; L = Let and L be the subvectors of correspondn to the ndex sets and L, respectvely n ths secton we ve suffcent optmalty theorems for problem (P Frst, we ve a suffcent optmalty theorem of the uhn-ucker type he functons f, and h are not dfferentable but are drectonal dfferentable wth respect to the same arc H x, x ( λ at λ = Let,, } be a partton of the ndex set ; thus for each =,,, = Ø { for each rs {,,}, wth r s, and " = = he next result does not requre the functon h to be drectonally dfferentable x X r s

4 oan M SANCU-MNASAN, Andreea Mădălna SANCU heorem Let x X R n, where X s a locally arcwse connected set and let u R m Assume that there exst the rht dfferentals at x wth respect to the same arc H, of f and and ( x, u satsfes the condtons below: Assume furthermore that (d f ( x, H ( u (d ( x, H (, x X, ( x,x x,x u ( x =, ( (x, h( x = ( u, u (,, s α - LQCN( x, (5 u s β - LQCN( x (6 f u s γ - LPCN( x (7 αu β γ hen x s a mnmum soluton to Problem (P he follown result s a specal case of heorem, where the condtons are specal cases of (5 throuh (8 heorem Let x X R n, where X s a locally arcwse connected set and let u R m Assume that there exst the rht dfferentals at x, wth respect to the same arc H, of f and and ( x, u satsfes condtons ( - ( Assume furthermore that any one of the hypotheses below s satsfed f u s γ - LPCN( x, where γ ; a,, s α - LQCN( x, b f s γ - LPCN( x, c α u γ ; a u s β - LQCN( x, b f s γ - LPCN( x, c β γ ; a u s β - LQCN( x, b f u s γ - (,, s a partton of, c β γ ; 5 a,, s α - LQCN( x, b f u s γ - LPCN( x, where {, } s a partton of, c α u γ ;,, s α - LQCN( x, 6 a LPCN x where { } (8

5 5 Optmalty condtons for nonlnear prorammn b u s β - LQCN( x, c f s γ - LPCN( x, d αu β γ, where {, } s a partton of hen x s a mnmum soluton to problem (P Let v R k and defne P= { v > } and Q= { v < } Let { P, P, P } and { Q, Q, Q } be parttons of the sets P and Q, respectvely Let h P and h Q (=,, be the subvectors of h correspondn to the ndex sets P and Q ( =,,, respectvely Let vp and v ( = Q,, be the subvectors of v correspondn to the ndex sets P and Q ( =,,, respectvely he next result does not requre the functon to be drectonally dfferentable heorem Let x X R n, where X s a locally arcwse connected set and let v R k Assume that there exst the rht dfferentals at x wth respect to the same arc H, of f and h and ( xv, satsfes the condton (d f ( x, Hxx, ( v (d h ( x, H xx, (, x X Assume furthermore that h P, s LPQCN( x,, h, Q, s LPQCN( x, s LPQCN ( x P P Q Q f s τ - LPCN( x, ( τ P P Q Q hen x s a mnmum soluton to Problem (P u R Let { } m Let L= u > Let {,,, }, {,,, } and {,,, } k v R and defne = { > } P v and = { <} Q v Let L L L L P P P P Q Q Q Q be parttons of the sets LP, and Q, respectvely he follown result s a combnaton of heorems and horem 5 Let x X R n, where X s a locally arcwse connected set and let u R m and k v R Assume that there exst the rht dfferentals at x, wth respect to the same arc H, of f, and h and ( xuv,, satsfes the condtons below: (d f ( x, H ( u (d ( x, H ( v (d h ( x, H (, x X, xx, Assume furthermore that u ( x =, ( x (, hx =, u, u

6 oan M SANCU-MNASAN, Andreea Mădălna SANCU 6, L, h, P h, Q, s α - LQCN( x,, s LPQCN( x,, s LPQCN( x, 5 u s β - LQCN( x, L L s LPQCN ( x P P Q Q 6 u L L s δ - LQCN ( x P P Q Q 7 f u L L s τ - LPCN ( x P P Q Q 8 αu β δ τ L hen x s a mnmum soluton to Problem (P n what follows we consder suffcent optmalty condtons of the Frtz John type Let ( xv,, vw, be a Frtz John pont, where x X (a locally arcwse connected set, v R, v R m and w R k Assume that ( xv,, vw, satsfes the condtons v (d f ( x,h ( v (d ( x,h ( w (d h ( x,h (, x X, (9 x,,,x f v = then condtons (9 ( become v ( x =, ( ( v, v, ( v, v, w ( v(d ( xh, ( w(d h ( xh, (, x X, ( xx, v ( x =, ( v, ( vw, ( Let and J be the sets defned at the bennn of ths secton By ( we have v and v J = Let L= { : v > } Let L be the subvector of correspondn to the ndex set L Also, let v L be the subvector of v correspondn to the ndex set L Let k U = w > and V = w < Let h and h w R Defne the ndex sets U and V by { } { } be the subvectors of h correspondn to the ndex sets U and V, respectvely Also, let w U and w V be the vector of w correspondn to the ndex sets U and V, respectvely heorem 6 Let x X R n, where X s a locally arcwse connected set Assume that there exst the rht dfferentals at x wth respect to the same arc H of f, and h Let ( xv,, vw, be a Frtz John pont whch satsfy condtons (9-( f v >, let the assumptons of heorem 5 hold wth u = v v, v = v w U V

7 7 Optmalty condtons for nonlnear prorammn f v =, let ( x,, v, w satsfy (-( and assume that the condtons below hold: a, L,, s α - LQCN(x, b h, U, s LPQCN( x, c h, V, s LPQCN( x, d vl L s β - LQCN( x, e vu h U w V h V s LPQCN ( x f vl L w U h U w V h V s δ - LQCN ( x, αv β δ L hen x s a lobal mnmum soluton to Problem (P he proofs wll appear n [] REFERENCES AVREL, M, ZANG,, Generalzed arcwse-connected functons and characterzatons of local-lobal mnmum propertes J Optm heory Appl,,, pp 7-5, 98 EWNG, G M, Suffcent condtons for lobal mnma of sutably convex functons from varatonal and control theory SAM Rev, 9,, pp, -, 977 AUL, R N, LYALL, V, Locally connected functons and optmalty ndan J Pure Appl Math,, pp 99-8, 99 AUL, R N, LYALL, VNOD, AUR, SURJEE, Locally connected set and functons J Math Anal Appl, pp -5, LYALL, V, SUNEJA, S, AGGARWAL, S, Frtz John optmalty and dualty for non-convex prorams J MathAnalAppl,, pp 8-5, OREGA, J M, RHENBOLD, W C, teratve Soluton of Nonlnear Equatons n Several Varables Academc Press, New York, NY 97 7 PREDA, V, NCULESCU, CRSAN, On dualty mnmax problems nvolvn ρ -locally arcwse connected and related functons Analele Unverstăţ Bucureşt, Matematca, 9,, pp85-95, 8 SANCU-MNASAN, M, Fractonal Prorammn heory, Methods and Applcatons luwer Academc Publshers, Dordrecht, SANCU-MNASAN, M, Dualty for nonlnear fractonal prorammn nvolvn eneralzed locally arcwse connected functons Rev Roumane MathPures Appl, 9,, pp87-98, SANCU-MNASAN, M, Optmalty condtons for nonlnear prorammn wth eneralzed locally arcwse connected functons Proc Ro Acad, Seres A, 5, pp-7, SANCU-MNASAN, M, SANCU, Andreea-Mădălna, Suffcent optmalty condtons for nonlnear prorammn wth ρ -locally arcwse connected and related functons Submtted SANCU-MNASAN, M, SANCU, Andreea-Mădălna, Suffcent optmalty condtons for nonlnear prorammn wth mxed constrants and eneralzed ρ -locally arcwse connected functons Submtted Receved January 6, 6