 Applied Mathematics, 2011, 2, 1387-1392 doi:10.4236/am.2011.211196 Published Online November 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Higher-Order Duality for Minimax Fractional Type Programming Involving Symmetric Matrices Caiyun Jin, Cao-Zong Cheng College of Applied Sciences, Beijing University of Technol ogy, Beijing, China E-mail: {jincaiyun, czcheng}@bjut.edu.cn Received September 1, 2011; revised October 16, 2011; accepted Octo ber 23, 2011 Abstract Convexity and generalized convexity play important roles in optimization theory. With the development of programming problem, there has been a growing interest in the higher-order dual problem and a lot of related generalized convexities are given. In this paper, we give the convexity of (,,,,,)Fdb vector-pseudo- quasi-Type I and formulate a higher-order duality for minimax fractional type programming involving sym-metric matrices, and give the weak, strong and strict converse duality theorems under the condition of higher-order (,,,,,)Fdb vector-pseudoquasi-Type I. Keywords: Higher-Order (,,,,,)Fdb Vector-Pseudoquasi-Type I, Higher-Order Duality, Minimax Fractional Type Programming, Positive Semidefinite Symmetric Matrix 1. Introduction In this paper, we focus on the following nondifferen- tiable minimax fractional programming problem: 1212(, )() supmin(, )()TnyYxR Tfxy xBxhxy xCx ()Psubjectto () 0,,ngxxR where is a compact subset of ,YlR,:nlfhR RR and :nmgRRnlRR are continuously differentiable func- tions on and , respectively, and nR12(, )()0,Tfxy xBx 12(, )()>0,(, )ThxyxCxxyRRnl, B and C are two positive semidefinite symmetric matrices. nnWhen , (P) is a differentiable minimax fra- ctional programming problem. ==0BCThe duality of programming problem involving sym- metric matrix has been investigated widely. Schmiten- dorf  established necessary and sufficient optimality conditions for a particular case of the following problem under convexity conditions. ()P12min( ,)()supTyYfxyxBx ()P subject to () 0.gxUnder the optimality conditions of , Tanimoto  defined a first-order dual problem of , which gene- ralized the duality theorems for convex minimax pro- gramming problems considered by Weir  and relaxed the convexity assumptions in the sufficient optimality of . Mishra and Rueda  introduced generalized second-order type I functions and considered the mini- max programming problem involving those fun- ctions and established second-order duality theorems for problem ()P()P()P. Husian, Anurag Jaysural and Ahmad  established two types of second-order dual models for problem ()P, which extends some previously known results on minimax programming. With the development of programming problem, there has been a growing interest in the higher-order dual pro- blem. Mangasarian  first formulated a class of second- and higher-order du al problems for a nonlinear program- ming problem involving twice differentiable functions. In , Zhang considered the following nondifferentiable mathematical programming problem: 12Minimize( )()TfxxBx ()P subject to () 0,gxunder higher-order invexity assumptions. Mishra and Rueda  generalized the results of Zhang  to higher- C. Y. JIN ET AL. 1388 order type I functions. In , Ahmad, Husain and Sharma considered the nondifferentiable minimax programming problem ()P, and formulated a unified higher -order dual of (P), and established weak, strong and strict converse duality theo - rems under higher-order (,,,)Fd-Type I assump- tions. In , Jayswal and Stancu-Minasian formulated the weak, strong and strict converse duality of (P) un- der generalized convexity of higher-order (,,,)Fd- Type I. For problem (P), H. C. Lai and K. Tanaka gave the necessary and sufficient conditions under the conditions of pseudo-conv ex, strictly pseudo-convex and qu asi-con- vex . In this paper, we will establish a higher-order dual of (P) and give the weak, strong and strict converse duality theorems under (,,,,,)Fdb  vector-pseudoquasi- Type I assumptions. The convexity conditions in this paper generalized the convexity in , and hence, pre- sents an answer o f a question raised in . 2. Preliminaries Let be the n-dimensional Euclidean space, nRnR be its nonegative orthant and X be an open subset of . Let be the set of all feasible solutions of (P). Denote nRS=1,2, ,Mm. For each (,)xySY, we define ()=: ()=0jJxjM gx 12121212(, )()()= :(, )()(,) ()=sup(,)()TTTzY Tfxy xBxYxy Yhxy xCxfxz xBxhxz xCx 1212=1()=(,,):1 1,= (,,,)with=1,=(,,,)and (),=1,2,,slssssisiiKxstyN RRsnttttRtyyyy yYxis  Definition 1: A function :nFXXRR  is said to be sublinear in its third argument, if ,,xxX 1) (subadditivity) 12,,naa R121 2(,;)(,;)(,;);Fxxa aFxxaFxxa  2) (positive homogeneous) ,,nRaR (,;)=(,;).Fxx aFxxa Let :nlfRR R,: \{XX R, , , 212=( ,)R0}nR12 , , and ,:bd XXR:neR R:RR. Let , :nl nwRRRR:nnlR R and be three diffe- rentiable functions. We assume that :kRnnRmRF is a sublinear functional throughout this paper. Definition 2 : (,)fe is said to be higher-order (,,,,Fd,)b  vector-pseudoquasi-Type I at xX with respect to , if for all npRxS and ()yYx, 1(, )(,)(, )(,,)(,,)<0(, ,(, )((,,)<(,)() (,)(,0( ,,( ,)((,)))( ,).TppTppbxxfxyxfxywxyppwxypFxx xxwxypdxxexlxpplxpFxx xxlxpxx    21222)))TTxd Definition 3: (,)fe) is said to be strictly higher-order (,,,,,Fdb  vector-pseudoquasi-Type I at xX with respect to , if for all npRxxS and ()yYx, 121222)))TTxd(, )(,)(, )(,,)(,,) 0(, ,(, )((,,)<(,)() (,)(,0( ,,( ,)((,)))( ,).TppTppbxxfxyxfxywxyppwxypFxx xxwxypdxxexlxpplxpFxx xxlxpxx     Obviously, when  is subadditive function and sa- tisfies 0(a)a0,,,,,), higher-order (BuFdb v ector-pseudo quasi-Type I is th e con- vexity condition of Theorem 3.1 in . In the following section, we will use Lemma 1 and Lemma 2 which were given in . Lemma 1: (Necessary Condition) If x, is an optimal solution of problem (P) satisfying and >0, >0TTxBx xCx () ()jgxjJx are li- near independent, then there exist (,,)()sty Kx R , and ,nuv R,mR such that =1=1(, )((, ))()=0sii iBimjjjtfxyhxyu Cvgx  (2.1) 1122(, )()(,)() 0,=1,2, ,TTiifxyxBx hxyxCxis   (2.2) =1()=mjjigx0 (2.3) Copyright © 2011 SciRes. AM C. Y. JIN ET AL.1389 s=1 =1,0, =1,2,,siijtti (2.4) 12121, 1,=() ,=() .TTTTTTuBu vCvxBu xBxxCv xBx    (2.5) Lemma 2: Let be a positive semidefinite symme- tric matrix of order . Then for all Bn,nxuR, 1122()()TT T.xBux Bxu Bu The equality holds when =Bx Bu for some 0. Evidently, if 12()TuBu 1, we have 12()TT.xBux Bx 3. Duality Model We consider the following dual model . (WD)(,,) ()(,,,,,)(,,),supmaxstyK zzuvpH sty where (,, )Hsty,,,)v p denote the set of all satisfying (,, nnn mzu RRRRRRn=1 =1(, ,)((,))0smipip jjijtwzypBuCv kzp  (3.11) 1, 1,TTuBu vCv (3.12) =1=1(, )(, ),(,,)(,, )0,sTTii iisTii piitfzyhzyzBu zCvtwzyppwzyp (3.13) =1 ()(, )((, ))0.mTjjjjp jjjgzkzp pkzp  (3.14) If for a triplet (,, )()sty Kz, the set (,, )=Hsty(,, ), then we define the supremum over Hsty to be . Next, we establish the duality of type (WD). Theorem 3.1 (Weak Duality) Let x and (,,,,,,,,)zuv styp be feasible solutions of (P) and (WD), respectively. Assume that 1) =1 =1(,)(,) ,()smiii jjijtf yhyg is higher-order (,,,,,)BuCvFdb () 0aa vect or -pseu d oquasi Type I at , z2) 0, (,)>0,bxz12120(,) (,)xz xz. Then 1212(, )().sup(, )()TyY Tfxy xBxhxy xCx Proof: From (3.14), we know that =1 ()(,)((,))0, mTjjjjp jjjgzkzp pkzp  then follows form 1) and 2(,)>0xz, we have 22=1 2,;((,))(,).(,)mpjjjFxzk zpdxzxz Since F is sublinear in its third argument, by (3.11) we can get =1=1=1=1=12220=,;( ,,)((,))=,; (,,)(,;((,)),;(,, )(,).(,)sip iimpjjjsip iimpjjjsip iiFxztwzyp BuCvkz p)Fxztwzy pBuCvFxzk zpFxztwzy pBuCvdxzxz   Furthermore, by 12120(,) (,)xz xz and 1(,)>0,xz we have 1=121,;(,)(,, )(,),sip iiFxzxztwzy pBuCvdxz which implies that =1=1=1(,)(,)(, )(, )(, )()(, ,)(, ,) 0.  sTTii iisTTii iisTii piibxztf xyhxyxBuxCvtfzyhzyzBu zCvtwzyppwzyp Copyright © 2011 SciRes. AM C. Y. JIN ET AL. 1390 From 2) and (3.13), we can get =1=1=1(, )(, )(, )(, )()(, ,)(, ,)0. sTTii iisTTii iisTii piitfxyhxyxBu xCvtfzyhzy zBu zCvtwzyp pwzyp Therefore, following from (3.12) and Lemma 2, 1122=1=1(, )()(, )()(,)(,) 0.sTTii iisTTiiiit fxyxBxhxyxCxtfxyhxyxBu xCv  Since , , 12=( ,,,)0sttt t0, =1,2,,itisiyY and 12(, )()>0,(, )TnlhxyxCxxyRRs, at least ex- ists one , such that {1,2,, }q1212(,) (),(,) ()TqTqfxy xBxhxy xCx which implies that 1212(, )().sup(, )()TyY Tfxy xBxhxy xCx Theorem 3.2 (Strong duality) Let x>0,x be an optimal solution of (P) satisfying and let >0TTxBx Cx(), ()jgxjJx=1,2, ,is be linear independent. Assume that for any (, ,0)=0,iwx y (, ,0)=(, )(, ),pi iiwx yfx yhx y  and for any ()jJx(,0)=0, (,0)=().jpjkxkx gxj Then there exist (,, )()sty Kx )(,, ) and (,,, ,,xuv(,,, ,,xuv(,,, ,,xuvp Hsty ,,,=0)st y p,,,=0)st y p such that is a feasible solution of (WD) and the two objectives have the same values. Furthermore, if the assumptions of weak duality hold for all feasible solutions of (P) and (WD), then is an optimal solu- tions of (WD). Proof: Since x,x is an optimal solution of (P) satisfy- ing and >0 >0TTxBx Cx (), ()jgxjJx is linearly independent, by Lemma 1, there exist (,, )(),sty Kx and ,nuv R,mRR such that BuCv=1=1(, ))()=sii iimjjjtfxyygx(,0hx   (2.1) 1122() =0,TTxCx (, )()(, )=1,2, ,iifxyx Bxhxyis s (2.2) =1 ()=0mjjigx (2.3) =1=1,0, =1,2,,siijtti (2.4) 12121, 1,=() ,=() .TTTTTTuBu vCvxBu xBxxCv xBx    (2.5) By (2.1) (2.2) (2.3) (2.5) and the conditions of theorem 3.2, we know that (,,, ,,=0)(,, )xuvp Hsty  , ,,,,,=0)st yp  ,,,=0)st y p , that is is an feasible solutions of (WD). Furthermore, (3.2) implies that is an optimal solu- tions of (WD). (,,xuv,, ,,v (,xuTheorem 3.3 (Strict Converse Duality) Let x be a feasible solution of (P) and be a feasible solution of (WD). Suppose that (,,, ,zuv, )p1)' 1212(, )()=;sup(, )()TyY Tfxyx Bxhxyx Cx   2)' =1 =1(, )(, ),()smiii jjijtfy hyg  is strictly higher-order (,,,,,)BuCvFdb >0 >0,aavector-pseudoquasi-Type I at and; z)3)' ((,)>bx z 0,12120(,)(,)xz xz  Then =;zx that is z is an optimal solution of (WD). Proof: Suppose that the contradiction is not true, that is zx. Similar to the proof of Theorem 3.1 we Copyright © 2011 SciRes. AM C. Y. JIN ET AL.1391 obtain 1=121,;(,)(,, )(,),sip iiFxzxztwzypBuCvdxz  which implies that =1=1=1(,)(,)(,)(, )(, )()(, , )(,,)>0.siiiiTTsii iiTTsTii piibxztfxyhxyxBu xCvtfzy hzyzBuzCvtwzyppwzyp     From 3)' and (3.13), we can get =1=1=1(, )(, )>(,)(,)()(, ,)(,,)0.sTTii iisTiiiisTTii piitfxyhxyxBuxCvtfzyhzy zBuzCvtwzyppwzyp    Therefore, following from (3.12) and Lemma 2, 112)2=1=1(, )()(, )((, )(,)>0. sTTii iisTTiiiitfxy xBxhxy xCxtfxyhxy xBuxCv s Since , , 12=( ,,,)0sttt t 0,=1,2,,itiiyY and 12(, )()>0,(, )Tnlhxyx CxxyRR s, at least exists one , such that {1,2,, }q1212(, )()>,(, )()TqTqfxyx Bxhxyx Cx   which implies that 1212(,)( )>sup(,)( )TyY TfxyxBxhxyxCx which contradicts with 1)'. Remark: If we take place condition 2)' of this theo- rem by 2)'' =1 =1(, )(, ),()smiii jjijtfy hyg  is higher- order (,,,,,)BuCvFdbvector-pseudoquasi-Type I at , and take place condition 3)' by z3)'' () 0>0,aa (,)>0,bx z12120,(,)(,)xz xz  the strict converse duality olds too. h 4. Acknowledgements This work is supported by Youth Foundation of Beijing University of Technology (X1006011201002). 5. References  W. E. Schmitendorf, “Necessary Conditions and Suffi-cient Conditions for Static Minimax Problems,” Journal of Mathematical Analysis and Applications, Vol. 57, No. 3-4, 1977, pp. 683-693. doi:10.1016/0022-247X(77)90255-4  S. Tanimoto, “Duality for a Class of Nondifferentiable Ma thematical Programming Problems,” Journal of Ma- thematical Analysis and Applications, Vol. 79, No. 2, 1981, pp. 286-294. doi:10.1016/0022-247X(81)90025-1  T. Weir, “Pseudoconvex Minimax Programming,” Utili-tas Mathematica, Vol. 42, 1992, pp. 234-240.  S. K. 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