International Journal of Mathematics Trends and Technology (IJMTT) Volume 37 Number 3 September 2016

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1 Inernaional Journal of ahemaics Trends and Technology (IJTT) Volume 37 Number 3 Sepember 6 Conveiy in he s Class of he Rough ahemaical Programming Problems T... Aeya, Kamilia l-bess, Deparmen of Physics and ngineering ahemaics, Faculy of ngineering, Tana Universiy, Tana, gyp Absrac Conve opimizaion is considered o be a reliable compuaional ool in engineering as i is used o solve many engineering problems in an efficien and fas mehod. The goal of his paper is o discuss, in deph, he conveiy in he s class of rough mahemaical programming and o presen some relaed resuls. Keywords conveiy, opimizaion, rough programming. I. INTRODUCTION Opimizaion (mahemaical programming) is a subfield of operaions research and i has a widely grown in he las hree decades. The goal of any opimizaion problem is o maimize (or minimize) one or more of obecive funcions under a deermined se of condiions. Opimizaion can be applied o many fields lie business, mining and engineering. Opimizaion is used in our daily life (e.g. moving from a place o anoher). The model of a simple mahemaical programming problem is: where g : U R is he crisp obecive funcion and U is he feasible se of he problem. U is he universe of he problem. Conve opimizaion problem involves maimizing concave funcions over conve ses. They can be convered ino a minimizaion problem of conve funcions by muliplying he obecive funcion by minus one. One of he advanages of conve opimizaion is ha i covers a broad range of pracical opimizaion problems. Also, here are some nonconve problems ha can be reformulaed ino conve problems. The oher advanage is ha if he decision se of he problem is conve, any local opimum is also a global opimum. There are some boos ha discuss conve opimizaion (e.g. Rocafellar [], Soer and Wizgall [3], Holmes [4], Bazaraa and Shey [5], eland and Temam [6], Ioffe and Tihomirov [7], Barbu and Precupanu [8] and Ponsein [9]). We assume ha he reader has a previous nowledge of conve opimizaion. In many acual opimizaion problems, he decision maer is no able o define he obecive funcion and/or he se of consrains precisely bu raher can define hem in a "rough sense". Rough se heory (RST), inroduced by Pawla [], provides a fleible mahemaical ool o he decision maer o solve such problems. Recenly, Youness [] combined rough se heory wih mahemaical programming. He described a new ype of mahemaical programming problems in which he feasible region is rough and called i RPP. He defined new conceps, namely, "conve rough se", "local rough opimal soluion" and "global rough opimal soluion". Osman e al. [] eended he previous wor and demonsraed ha he roughness may eis in he obecive funcion, he feasible se or boh of hem. They classified rough programming problems ino hree classes according o he place of roughness. They discussed he conveiy in he s class of he RPPs in which he decision se is rough and he obecive funcion is crisp. In heir discussion, hey showed ha he lower and/or upper approimaions of rough feasible se could be conve. They also inroduced new conceps such as: "Upper conve" and "Lower conve". In his paper, we eend he above menioned wors, and propose and prove some heorems relaed o he conveiy in he s class of RPPs. II. ROUGH ST THORY RST has been proven o be an ecellen mahemaical ool dealing wih vague or imprecise descripions of obecs. Therefore, many researchers applied RST o many domains such as paern recogniion, daa mining, arificial inelligence, image processing, machine learning and medical applicaions []. The rough se mehodology proposed by Pawla [], in 98, assumed ha any imprecise concep is characerized by a pair of precise conceps called he lower and he upper approimaions. RST is based on equivalence relaion ha pariions he universe ino classes of indiscernible obecs. RST epresses imprecision by employing a boundary region of a se. If he boundary region of a se is empy, hen he se is crisp (eac) oherwise he se is rough (ineac). RST uses equivalence relaion o group obecs wih similar characerisics ino indiscernibiliy classes and any vague se is characerized by a pair of precise ses called he lower and he upper approimaions. The lower approimaion includes all obecs ha surely belong o he concep of ineres, where he upper approimaion includes all obecs which possibly belong o ha imprecise concep. The main advanage of using RST in handling imprecise conceps is ha i does no need any addiional informaion. ISSN: hp:// Page 4

2 Inernaional Journal of ahemaics Trends and Technology (IJTT) Volume 37 Number 3 Sepember 6 Le U be a non-empy finie se of obecs, called he universe, and U U be an equivalence relaion on U. The ordered pair A=(U, ) is called an approimaion space generaed by on U. generaes a pariion U / { Y, Y,..., Y } m where Y, Y,..., Y are he equivalence classes of he m approimaion space A. Based on he equivalence relaion, he mapping [ ] : U U is given by [ ] { y U y }. Shorly, he subse [ ] U is he equivalence class conainin In RST, any subse U is defined in erms of he equivalence classes of he approimaion space A by is lower and upper approimaions (i.e. ( ) and ( ), respecively) as follows: ( ) { U [ ] } U I ( ) { [ ] } Therefore, ( ) ( ) The difference beween he upper and he lower approimaions is called he boundary of and is denoed by ( ) ( ) ( ). For simpliciy, le ( ), ( ). ( ) and III. TH ST CLASS OF RPPS [] Le A ( U, ) be an approimaion space generaed by an equivalence relaion on he universe U. Therefore, U / { Y, Y,..., Y } m is he pariioned universe generaed by on U where Y, Y,..., Y are he equivalence classes of he m approimaion space A. A RPP over he universe U aes he following form: () s.. where g : R is he crisp obecive funcion. U is he se of consrains of he problem, ha is roughly defined in he universe U by and where: { U [ ] } U { ([ ] ) }, Definiion 3.: In problem (), he opimal value g of he obecive funcion is defined by is lower and upper bounds g and g g where ma{, } ma{, } ma min g ( ) Y Y g, respecively, such ha: Definiion 3.: In problem (), a poin is a surely-feasible soluion, if and only if. Definiion 3.3: In problem (), a poin is a possibly-feasible soluion, if and only if. Definiion 3.4: In problem (), a poin is a surely-no feasible soluion, if and only if. Definiion 3.5: In problem (), a poin is a g( ) surely-opimal soluion, if and only if Definiion 3.6: In problem (), a poin is a g( ) possibly-opimal soluion, if and only if Definiion 3.7: In problem (), a poin is a surely-no opimal soluion, if and only if g( ) Definiion 3.8: In problem (), here are four opimal ses covering all possible degrees of feasibiliy and opimaliy, as follows: The se of all surely-feasible, surely-opimal soluions is denoed by FO s s, and i is defined by: FO { () } s s g The se of all surely-feasible, possiblyopimal soluions is denoed by FO s p, and i FO { ( ) } s p g The se of all possibly-feasible, surelyopimal soluions is denoed by FO p s, and i FO { ( ) } p s g The se of all possibly-feasible, possiblyopimal soluions is denoed by FO p p, and i F O { ( ) } p p g IV. CONVXITY IN ST CLASS OF RPPS Conve ses and concave funcions have many aracive properies in mahemaical programming. For eample, any local maimum poin of a concave funcion over a conve se is also a global maimum poin. In his secion, we presen some significan ISSN: hp:// Page 5

3 Inernaional Journal of ahemaics Trends and Technology (IJTT) Volume 37 Number 3 Sepember 6 properies of RPPs ha have conve rough se and concave crisp funcion. Definiion 4.: A rough se is U - conve, if is upper approimaion is conve []. Definiion 4.: A rough se is L - conve, if is lower approimaion is conve []. Definiion 4.3: A rough se is conve, if is upper and lower approimaions (i.e. and ) are conve []. Definiion 4.4: A funcion g ( ) is concave on a conve se S, if g( ( ) ) g( ) ( ) g( ) for each, S and for each (,) []. Definiion 4.5: A funcion g ( ) is sricly concave on a nonempy conve se S, if g( ( ) ) g( ) ( ) g( ) for each, S and for each (,) []. Definiion 4.6: A funcion g ( ) is quasiconcave on a conve se S, if g( ( ) ) min{ g( ), g( )} for each, S and for each (,) []. Definiion 4.7: A funcion g ( ) is sricly quasiconcave on a nonempy conve se S, if for each, S wih g( ) g( ), we have g( ( ) ) min{ g( ), g( )} for each (,) []. Definiion 4.8: A funcion g ( ) is srongly quasiconcave on a nonempy conve se S, if for each, S wih, we have g( ( ) ) min{ g( ), g( )} for each (,) []. Theorem 4.: In problem (), if is a nonempy conve se and is a local opimal soluion hen: ) If g ( ) is a sricly quasiconcave funcion, hen is a surely-global opimal soluion. ) If g ( ) is a srongly quasiconcave funcion, hen is he unique surely-global opimal soluion. 3) If g ( ) is a srongly quasiconcave funcion Proof: and, hen is he unique global opimal soluion (i.e. F O F O F O F O {} ). ) Suppose, on he conrary, ha here is an ˆ wih g( ˆ ) g( ). By conveiy of, ˆ ( ), (,). Since is a local maimum by assumpion, hen g( ) g( ˆ ( ) ), (, ) for some (,). Bu since g ( ) is sricly quasiconcave and g( ˆ ) g( ), hen g( ˆ ( ) ) g( ), (,). This conradicion shows ha ˆ does no eis. ) Since is a local opimal soluion, hen here is an - neighborhood N ) around where g( ) g( ) for N ( ). Assume by conradicion o he conclusion of he heory ha here is a poin ˆ such ha ˆ and g( ˆ ) g( ). By srong quasiconcaviy, i follows ha g( ˆ ( ) ) min{ g( ˆ), g( )} g( ), (,). Bu for small enough, ˆ ( ) N ( ) and hence local opimaliy of is violaed. 3) If g ( ) is a srongly quasiconcave funcion g g Thus, and, hen is a unique surely and possibly opimal soluion. Hence, F O F O F O F O {}. ( ) Theorem 4.: In problem (), if is a nonempy conve se and is a local opimal soluion hen: ) If g ( ) is a concave funcion, hen is a surely-global opimal soluion. ) If g ( ) is a sricly concave funcion, hen is he unique surely-global opimal soluion. 3) If g ( ) is a sricly concave funcion and, hen is he unique global opimal soluion (i.e. F O F O F O F O {} ). Proof: I is similar o he above proof. Theorem 4.3: In problem (), if is a nonempy U - conve se and g ( ) is a concave funcion on, hen he poin is a surely-opimal soluion o his problem if and only if g ( ) has a subgradien a such ha ( ) for all. ( ISSN: hp:// Page 6

4 Inernaional Journal of ahemaics Trends and Technology (IJTT) Volume 37 Number 3 Sepember 6 Proof: Assume ha ( ), where. is a subgradien of g a. By concaviy of g, g( ) g( ) ( ) g( ), and herefore is a surely-opimal soluion o he problem. To show he converse, assume ha is a surelyopimal soluion o he problem, and form he following wo ses in U : S {(, y) U, y g( ) g( )} S y y {(, ), } I is easy o prove ha boh S and S are conve ses. Also S S because oherwise here would be a poin ( y, ) where, y g( ) g( ) conradicing he assumpion ha is an opimal soluion of he problem. Since S S, hen here is a hyperplane ha separaes S and S. Thus, here is a nonzero vecor (, ) and a scalar such ha: ( ) y, U, y g( ) g( ) () ( ) y,, y () If we le and y in (), hen. Ne, leing and y in () maes. Since his is rue for, hen and. Briefly, we conclude ha and. If, hen from () ( ), U. If we le, hen ( ) and hus. Since (, ) (,), hen. Dividing () and () by and denoing / by, we obain he following inequaliies: ( ) y, U, y g( ) g( ) (3) ( ) y,, y (4) By leing y in (4), we obain ( ),. From (3), i is clear ha g( ) g( ) ( ), U. Thus, ( ), is a subgradien of g a such ha. Theorem 4.4: In problem (), if is a nonempy U - conve se and g ( ) is a concave funcion on where is open, hen he poin surely-opimal soluion o his problem if and only if here is a zero subgradien of g ( ) a. is a Proof: By he previous heorem, is a surely- opimal soluion if and only if ( ), where is a subgradien of g a. Since posiive. is open, hen. Hence, for some. This means ha Theorem 4.5: In problem (), if is a nonempy U - conve se and g ( ) is a differeniable concave funcion on, hen he poin is a surelyopimal soluion o his problem if and only if g ( )( ),. Furhermore, if is open hen he poin is a surely-opimal soluion o his problem if and only if g ( ). Proof: I is sraighforward. Theorem 4.6: Consider he problem: min g ( ) subec o, where (he upper approimaion of he rough se ) is a nonempy conve se and g ( ) is a concave funcion on. If is a local opimal soluion hen ( ) for all subgradien of g a. where is a Proof: Assume ha is a local opimal soluion. Then here is an - neighborhood N ( ) where g( ) g( ), N ( ). Le, and noice ha here is ( ) N ( ) for, o. Thus, g( ( )) g( ). Le be a subgradien of g a and by concaviy of g, we have g( ) g( ( )) ( ). The above wo inequaliies imply ha ( ), and dividing by, we ge he required resul. Theorem 4.7: Consider he problem: min g ( ) subec o, where (he upper approimaion of he rough se ) is a conve se and g ( ) is a differeniable concave funcion on. If is a local opimal soluion hen g ( ), where is a subgradien of g a. Proof: I is sraighforward. Theorem 4.8: Consider he problem: min g ( ) subec o, where (he upper ISSN: hp:// Page 7

5 Inernaional Journal of ahemaics Trends and Technology (IJTT) Volume 37 Number 3 Sepember 6 approimaion of he rough se ) is a nonempy compac polyhedral se and g ( ) is a concave funcion on. Then, here is an opimal soluion o he problem, where is an ereme poin of. Proof: Since minimum a is compac, g assumes a. If is an ereme poin of, hen he resul is acquired. Oherwise, where,, and is an ereme poin of for,,...,. By he concaviy of g, we have g ( ) g ( ) g ( ) Bu since g( ) g( ), for,,...,, he above inequaliy implies ha g( ) g( ) for,,...,. Hence, he ereme poins,,..., are opimal soluions o he problem and he proof is complee. Theorem 4.9: Consider he problem: min g ( ) subec o, where (he upper approimaion of he rough se ) is a nonempy compac polyhedral se and g ( ) is a quasiconcave funcion on. Then, here is an opimal soluion o he problem, where is an ereme poin of. Proof: Since g is a funcion on and hence ges a minimum a. If here is an ereme poin whose obecive is equal o g ( ), hen he resul is acquired. Oherwise, le,,..., be ereme poins of, and suppose ha g( ) g( ) for,,...,. can be represened as,,,,...,. where Since g( ) g( ) for each, hen g( ) min g( ) () Now consider he se Noice ha conve se. Hence, for,,..., { g( ) }. and belongs o is a. By quasiconcaviy of g, g ( ), which conradics (). This conradicion shows ha g( ) g( ) for some ereme poin and he resul is obained. V. CONCLUSIONS In his paper, we provided some essenials of conve ses, conve funcions, and conve opimizaion problems in a rough environmen. RFRNCS [] Z. Pawla, Rough ses, Inernaional Journal of Compuer and Informaion Sciences,, , 98. [] R. T. Rocafellar, Conve Analysis, Princeon Universiy Press, 97. [3] J. Soer and C. Wizgall, Conveiy and Opimizaion in Finie Dimensions I, Springer-Verlag, 97. [4] Holmes, R. B., A Course on Opimizaion and Bes Approimaion, Springer-Verlag, Berlin and New Yor, 97. [5] Bazaraa,. S., and Shey, C.., Foundaions of Opimizaion, Springer-Verlag, Berlin and New Yor, 976. [6] eland, I., and Temam, R., Conve Analysis and Variaional Problems, Norh Holland Publisher, Amserdam, 976. [7] Ioffe, A. D., and Tihomirov, Y.., Theory of ernal Problems, Norh Holland Publisher, Amserdam, 979. [8] Barbu, V., and Precupanu, T., Conveiy and Opimizaion in Banach Spaces, Sihoff and Noordhoff, Alphen aan de Rin, 978. [9] Ponsein, J., "Approaches o he Theory of Opimizaion," Cambridge Universiy, Press, London and New Yor, 98. [].A. Youness, Characerizing soluions of rough programming problems, uropean Journal of Operaional Research, vol.68, no.3, pp.9-9, 6. [].S. Osman,.F. Lashein,.A. Youness, T... Aeya, ahemaical programming in rough environmen, Opimizaion, vol.6, no.5, pp.63-6,. [] ohar S. Bazaraaa, Hanif D. Sherali, C.. Shey, Nonlinear Programming: Theory and Algorihms, John Wiley and Sons Inc., 3 rd ed.; 6. ISSN: hp:// Page 8