Optimizatio Volume 2013, Article ID 346131, 6 pages http://dx.doi.org/10.1155/2013/346131 Research Article Quasicovex Semidefiite Miimizatio Problem R. Ekhbat 1 ad T. Bayartugs 2 1 Natioal Uiversity of Mogolia, Mogolia 2 Mogolia Uiversity of Sciece ad Techology, Mogolia Correspodece should be addressed to R. Ekhbat; rekhbat46@yahoo.com Received 23 May 2013; Accepted 7 November 2013 Academic Editor: Jei-Sha Che Copyright 2013 R. Ekhbat ad T. Bayartugs. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We itroduce so-called semidefiite quasicovex imizatio problem. We derive ew global optimality coditios for the above problem. Based o the global optimality coditios, we costruct a algorithm which geerates a sequece of local imizers which coverge to a global solutio. 1. Itroductio Semidefiite liear programg ca be regarded as a extesio of liear programg ad solves the followig problem: A j,x b j, C, X, X 0, j=1,2,...,s, where X R is a matrix of variables ad A j R,j = 1,2,...,s. X 0 is otatio for X is positive semidefiite., deotes robeius orm ad X = A, A. Semidefiite programg fids may applicatios i egieerig ad optimizatio [1]. Most iterior-poit methods for liear programg have bee geeralized to semidefiite covex programg [1 3]. There are may works devoted to the semidefiite covex programg problembutlessattetiosoforhasbeepaidtoquasicovex programg semidefiite quasicovex imizatio problem. The aim of this paper is to develop theory ad algorithms for the semidefiite quasicovex programg. The paper is orgaized as follows. Sectio 2 is devoted to formulatio of semidefiite quasicovex programg ad its global (1) optimality coditios. I Sectio 3, we cosider a approximatio of the level set of the objective fuctio ad its properties. 2. Problem Defiitio ad Optimality Coditios Let X be matrices i R, ad defie a scalar matrix fuctio as follows: f:r R. (2) Defiitio 1. Let f(x) be a differetiable fuctio of the matrix X.The f f (X) (X) = ( ). (3) x ij Itroduce the robeius scalar product as follows: X, Y = x ij y ij, X,Y R. (4) i=1j=1 If f( ) is differetiable, the it ca be checked that f (X+H) f(x) = f (X),H +o( H ). (5) Defiitio 2. AsetD R is covex if αx + (1 α)y D for all X, Y D ad α [0,1].
2 Optimizatio Defiitio 3. The fuctio f : D R is said to be quasicovex o D if f (αx + (1 α) Y) max {f (X),f(Y)} X, Y D, α [0, 1]. The well-kow property of a covex fuctio [3]ca be easily geeralized as follows. Lemma 4. Afuctiof:R oly if the set is covex for all c R. Proof (6) R is quasicovex if ad L c (f)={x R f(x) c} (7) Necessity. Suppose that c R is a arbitrary umber ad X, Y L c (f). By the defiitio of quasicovexity, we have f (αx + (1 α) Y) max {f (X),f(Y)} c α [0, 1], which meas that the set L c (f) is covex. Sufficiecy. Let L c (f) be a covex set for all c R.or arbitrary X, Y R, defie c o = max{f(x), f(y)}. The X L c o(f) ad Y L c o(f).cosequetly,αx + (1 αy) L c o(f),forayα [0,1].Thiscompletestheproof. Lemma 5. Let f : R R be a quasicovex ad differetiable fuctio. The the iequality f(x) f(y) for X, Y R implies that (8) f (Y),X Y 0, (9) where f (X) = ( f(x)/ x ij ) ad, deotes the robeius scalar product of two matrices. Proof. Sice f is quasicovex, f (αx + (1 α) Y) max {f (X),f(Y)} =f(y) (10) for all α [0,1]ad X, Y R such that f(x) f(y).by Taylor s formula, there is a eighborhood of the poit Y o which: f (Y+α(X Y)) f(y) =α( f (Y),X Y + o(α X Y ) ) 0, α > 0. α (11) α o rom the fact that o(α x y )/α 0,weobtai f (Y), X Y 0which completes the proof. Cosider the problem of imizig a differetiable quasicovex matrix fuctio subject to costraits f (X) (12) subject to g j (X) b j, j=1,2,...,s, (13) X 0, (14) where g j : R R,j=1,2,...,sare scalar fuctios ad X 0are positive semidefiite matrices, b j R. We call problem (12) (14) as the semidefiite quasicovex imizatio problem. Deote by D a costrait set of the problem as follows: D ={X R g(x) j b j,j=1,2,...,s;x 0}. (15) The problems (12) (14)reduceto f (X). (16) x D I geeral, the set D is ocovex. Problem (16)isocovex ad belogs to a class of global optimizatio problems i Baach space. We formulate a ew global optimality coditio for problem (16)ithefollowig.orthispurpose,weitroduce the level set E f(z) (f) of the fuctio f:r R at a poit Z R : E f(z) (f)= {Y R f(y) =f(z)}. (17) The global optimality coditios for problem (16) cabe formulated as follows. Theorem 6. Let Z be a solutio of problem (16).The f (X),X Y 0 Y E f(z) (f), X D, (18) where E c (f) = {Y R f(y)=c}.if,iadditio, lim f (X) =+, X f (X+αf (X)) =0 (19) holds for all X Dad α 0, the coditio (18) becomes sufficiet. Proof Necessity. Assume that z is a solutio of problem (16). Let X D ad Y E f(z) (f).thewehave0 f(z) f(x) = f(y) f(x),adlemma 5 implies f (X), X Y 0. Sufficiecy. Suppose, o the cotrary, that Z is ot a solutio of (16). The there exists a U Dsuch that f(u) < f(z). Costruct a ray Y α for α>0defied by Y α =U+αf (U). (20) We claim that f(y α )>f(u)holds for all positive α. By Taylor s formula, we have f(u+αf (U)) f(u) =α( f 2 o(α + f ) ) α (21)
Optimizatio 3 for small α > 0, where lim α 0+ o(α f (U) )/α = 0. Therefore, there exists α o > 0 such that f(y α ) f(u) > 0 holds for all α (0,α o ).Hece,byLemma 5, wehave f (U + α o f (U)), f (U) 0sice f (U) =0 ad f (U + α o f (U)) =0bytheassumptio.Notethatforallγ>1,we also have f(u + γα o f (U)) > f(u + α o f (U));forotherwise, we would have f(u+γα o f (U)) f(u+α o f (U)),adcosequetly, by Lemma 5, f (U + α o f (U)), α o (γ 1)f (U) 0, which would imply γ 1 which is cotradictig to the assumptio that γ>1.moreover,weca showthatf(u + γα o f (U)) is icreasig i γ > 0.Iff(U + γ α o f (U)) < f(u + γα o f (U)) holds for some γ > γ,theα o (γ γ) f (U + γα o f (U)), f (U) 0, which would cotradict the fact that γ >γ.theseproveourclaimf(y α )>f(u)for all α>0. Now it is obvious that the fuctio φ:r + Rdefied as φ (α) =f(y α ) (22) is cotiuous o [0, ). Also, with assumptio (19) implies lim α φ(α) = +, ad therefore, there exists a α such that φ( α) > f(z). Usigthecotiuityofφ(α) ad the iequalities φ( α) > f(z) > f(u), there exists a α such that f(y+αf (U)) =f(z), (23) which meas that Y α E f(z) (f). O the other had, we have f (U) = (1/α)(Y α U).Thusweget f (U),U Y α = 1 α Y α U,U Y α = 1 α Y α Y 2 <0, (24) which cotradicts (18). This meas that Z must be a solutio of (16). Example 7. Cosider the followig problem: X D (f (X) = X 2 ), (25) subject to D ={X R X=( 2 3 4 1 ) X X =( 7 5 8 9 ),X 0}. Example 8. Cosider the fractioal programg problem (26) (f (x) = f 1 (X) X D f 2 (X) ), (27) where f 1 is covex ad differetiable o R ad f 2 is cocave ad differetiable o R.Supposethatf 1 ad f 2 are defied positively o a ball B cotaiig a subset D R ; that is, f 1 (X) >0, f 2 (X) >0 X D B. (28) We will call this problem as the mixed fractioal imizatio problem. By Lemma 4, wecaeasilyshowthatf(x) is quasicovex. Hece, the optimality coditio (13) at a solutio Z of (27)isasfollows: i=1j=1 ( f 1 (X) f x 2 (X) f 2 (X) f ij x 1 (X)) (x ij y ij ) ij f2 2 (X) 0 Y E f(z) (f), 3. A Algorithm for the Covex Miimizatio Problem X D. (29) We cosider the quasicovex imizatio problem as a special case of problem (16): X D f (X), (30) where f:r R is strogly covex ad cotiuously differetiable ad D is a arbitrary compact set i R.I this case, the we ca weake coditio (19)asshowithe ext theorem. Theorem 9. Let Z beasolutioofproblem(30).the f (X),X Y 0 Y E f(z) (f), X D. (31) If, i additio, X D f (X) >0 (32) holds, the coditio (31) is also sufficiet. Proof Necessity. Assume that z is a solutio of problem (30). Cosider X Dad Y E f(z) (f). The by the covexity of f,wehave 0 f(z) f(x) =f(y) f(x) f (X),Y X. (33) Sufficiecy. Let us prove the assertio by cotradictio. Assume that (31) holds ad there exists a poit U Dsuch that f (U) <f(z). (34) Clearly, f (U) =0 by assumptio (32). Now defie U α as follows for α>0: The, by the covexity of f,wehave U α =U+αf (U). (35) f(u α ) f(u) f (U),U α U =α f 2, (36)
4 Optimizatio which implies f(u α ) f(u) +α f 2 >f(u). (37) The fid α=αsuch that f (U) + α f 2 =f(z) ; (38) that is, f (Z) f(u) α= f (U) 2 >0. (39) Thus we get f(u α ) f(u) + α f 2 =f(z) >f(u). (40) Defie a fuctio h:r + Ras h (α) =f(u+αf (U)) f(z), (41) where R + ={α R α 0}.Itisclearthath is cotiuous o [0, + ).Notethath(α) 0 ad h(0) < 0. There are two cases with respect to the values of h(α) which we should cosider. Case a. h(α) = 0 (or f(u + αf (U)) = f(z)), the f (U),U U α = f (U), αf (U) (42) = α f 2 <0, cotradictig coditio (31). Case b. h(α) > 0 ad h(0) < 0. Siceh is cotiuous, there exists a poit α o (0,α) such that h(α o ) = 0 (or f(u + α o f (U)) = f(z)).thewehave f (U),U U αo = α o f 2 <0, (43) agai cotradictig (31). Thus, i both cases, we fid cotradictios, provig the theorem. Now usig the fuctio P(Y) = X D f (X), X Y, Y R, we reformulate Theorem 9 i terms of fuctio ψ(z) defied as follows: ψ (Z) = Y E f(z)(f) P (Y), Z D. (44) Theorem 10. Assume that f:r R is strogly covex ad cotiuously differetiable ad D is a compact set i R. Let X D f (X) > 0. Ifψ(Z) = 0, the the poit Z is a solutio to problem (30). Proof. This is a obvious cosequece of the followig relatios: 0=ψ(Z) P(Y) f (X),X Y, (45) which are fulfilled for all Y E f(z) (f) ad X D. Now we are ready to preset a algorithm for solvig problem (30). We also suppose that oe ca efficietly solve the problem of computig X D f (X), X Y for ay give Y R. Algorithm MIN Iput. A strogly quasicovex fuctio f ad a compact set D. Output.AsolutioX to the imizatio problem (30). Step 1. Choose a feasible poit X 0 D.Setk:=0. Step 2. Solve the followig problem: Y E f(xk )(f) P (Y). (46) Let Y k be a solutio of this problem (i.e., P(Y k ) = X D f (X), X Y k = Y Ef(Xk )(f)p(y)), ad let X k+1 realizes P(Y k ) (i.e., ψ(x k ) = P(Y k )= f (X k+1 ), X k+1 Y k ). Step 3. If ψ(x k ) = 0 the output X = X k ad terate. Otherwise, let k:=k+1ad retur Step 2. The covergece of this algorithm is based o the followig theorem. Theorem 11. Assume that f:r R is strogly covex ad cotiuously differetiable ad D is a compact set i R.Let X D f (X) >0. The the sequece {X k, k = 0, 1,...} geerated by Algorithm MIN is a imizig sequece for problem (30);thatis, lim f(x k)= f (X), (47) k X D ad every accumulatio poit of the sequece {X k } is a global imizer of (30). Proof. rom the costructio of {X k },wehavex k Dad f(x k ) f for all k, wheref =f(x )= X D f(x). Clearly, f (X ) =0by assumptio. Also, ote that for all Y E f(xk )(f) ad X D,wehave ψ(x k )= Y E f(xk )(f) X D f (X),X Y f (X),X Y 0. (48) If there exists a k such that ψ(x k )=0the, by Theorem 11, X k is a solutio to problem (30) ad i this case the proof is complete. Therefore, without loss of geerality, we ca assume that igored ψ(x k ) < 0 for all k ad prove the theorem by cotradictio. If the assertio is false; that is, X k is ot a imizig sequece for problem (30), the followig iequality holds: lim k if f(x k)>f. (49)
Optimizatio 5 By the defiitio of ψ(x k ) ad Algorithm MIN, we have Takig ito accout V αk =X + α k f (X ),wehave p(y k )=ψ(x k ) ψ (X k ) f (X ),V αk X = Y E f(xk )(f) X D f (X),X Y (50) = f (X ) V α k X (61) = f (X k+1 ),X k+1 Y k ad f(y k )=f(x k ). The covexity of f implies that f(x k ) f(x k+1 )=f(y k ) f(x k+1 ) f (X k+1 ),Y k X k+1 = ψ(x k )>0. (51) Hece, we obtai f(x k+1 )<f(x k ) for all k, ad the sequece {f(x k )} is strictly decreasig. Sice the sequece is bouded from below by f, it has a limit ad satisfies lim (f (X k+1) f(x k )) =0. (52) k The, from (49) ad(50), we obtai lim ψ(x k)=0. (53) k rom (51)wehavef(X k )>f(x ) for all k. Now defie V α as follows: V α =X +αf (X ), α > 0. (54) The, by the covexity of f,wehave f (V α ) f(x ) f (X ),V α X =α f (X ) 2, (55) which implies f (V α ) f(x ) +α f (X ) 2 >f(x ), α > 0. (56) Choose α=α k such that that is, f (X ) +α k f (X ) 2 >f(xk ) ; (57) α k > f(x k) f(x ) f (X ) 2 >0. (58) Defie a fuctio h k :R + Ras h k (α) =f(x +αf (X )) f(x k ), (59) where R + ={α R α 0}.Itisclearthath k is cotiuous o [0, + ). Notethath k (α k ) > 0 ad h k (0) < 0. Sice h k is cotiuous, there exists a poit α k (0,α k ) such that h k (α k )=0;thatis,f(V αk )=f(x k ) ad V αk =x + α k f (x ). Also, ote that ψ(x k )= Y E f(xk )(f) X D f (X),X Y f (X ),X V αk. (60) X D f (X) V α k X >0. Sice lim k ψ(x k )=0,thisimplies lim V α k k =X. (62) The cotiuity of f o R yields lim f(x k)= lim f(v α k k k )=f(x ), (63) which is a cotradictio to (49). Cosequetly, {X k } is a imizig sequece for problem (30). Sice D is compact, we ca always select the coverget subsequeces {X kl } from {X k } such that lim X k l l = X D. (64) The together with (63), we obtai lim f(x k l l )=f(x) = f, (65) which completes the proof. 4. Numerical Experimets The proposed algorithm has bee tested o the followig umerical examples. Problem 12. X D AX BX 2, A=( 2 3 (66) 3 ), B=(1 3 4 2 4 ), where A 1 X, X + A 2,X 0 A { 2 X, X + A 1,X 0 D = A 1 =( 1 2 3 4 ), A 2 =( 2 3 1 4 ) (67) { { X 0. The global solutio is X =( 4 4 ). (68) 2 5 Problem 13. X D X 2 (69) subject to D ={X R X=( 2 3 4 1 ) X X. =( 7 5 8 9 ),X 0} (70)
6 Optimizatio The global solutio is X =( 4 3 ). (71) 8 9 Refereces [1] A. Bouhamidi, R. Ekhbat, ad K. Jbilou, Semidefiite cocave programg, Mogolia Mathematical Joural, pp.37 47, 2012. [2] R. T. Rockafellar, Covex Aalysis, Priceto Uiversity Press, Priceto, NJ, USA, 1970. [3] R. Ekhbat ad T. Ibaraki, O the maximizatio ad imizatio of a quasicovex fuctio, Noliear ad Covex Aalysis, pp. 43 76, 2011.
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