Research Article Global Sufficient Optimality Conditions for a Special Cubic Minimization Problem

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1 Mathematcal Problems n Engneerng Volume 2012, Artcle ID , 16 pages do: /2012/ Research Artcle Global Suffcent Optmalty Condtons for a Specal Cubc Mnmzaton Problem Xaome Zhang, 1 Yanjun Wang, 1 and Wemn Ma 2 1 Department of Appled Mathematcs, Shangha Unversty of Fnance and Economcs, Shangha , Chna 2 School of Economcs and Management, Tongj Unversty, Shangha , Chna Correspondence should be addressed to Wemn Ma, mawm@tongj.edu.cn Receved 9 February 2012; Accepted 11 June 2012 Academc Edtor: Janmng Sh Copyrght q 2012 Xaome Zhang et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. We present some suffcent global optmalty condtons for a specal cubc mnmzaton problem wth box constrants or bnary constrants by extendng the global subdfferental approach proposed by V. Jeyakumar et al The present condtons generalze the results developed n the work of V. Jeyakumar et al. where a quadratc mnmzaton problem wth box constrants or bnary constrants was consdered. In addton, a specal dagonal matrx s constructed, whch s used to provde a convenent method for justfyng the proposed suffcent condtons. Then, the reformulaton of the suffcent condtons follows. It s worth notng that ths reformulaton s also applcable to the quadratc mnmzaton problem wth box or bnary constrants consdered n the works of V. Jeyakumar et al and Y. Wang et al Fnally some examples demonstrate that our optmalty condtons can effectvely be used for dentfyng global mnmzers of the certan nonconvex cubc mnmzaton problem. 1. Introducton Consder the followng cubc mnmzaton problem wth box constrants: mn f x b T x xt Ax a T x, n s.t. x D u,v, CP1 where u,v R, u v, 1, 2,...,n,anda a 1,...,a n T R n,b b 1,...,b n T R n, A S n, where S n s the set of all symmetrc n n matrces. x 3 that means x 3 1,...,x3 n T.

2 2 Mathematcal Problems n Engneerng The cubc optmzaton problem has spawned a varety of applcatons, especally n cubc polynomal approxmaton optmzaton 1, convex optmzaton 2, engneerng desgn, and structural optmzaton 3. Moreover, research results about cubc optmzaton problem can be appled to quadratc programmng problems, whch have been wdely studed because of ther broad applcatons, to enrch quadratc programmng theory. Several general approaches can be used to establsh optmalty condtons for solutons to optmzaton problems. These approaches can be broadly classfed nto three groups: convex dualty theory 4, local subdfferentals by lnear functons 5 7, and global L- subdfferental and L-normal cone by quadratc functons The thrd approach, whch we extend n ths paper, s often adopted to develop optmalty condtons for specal optmzaton forms: quadratc mnmzatons wth box or bnary constrants, quadratc mnmzaton wth quadratc constrants, bvalent quadratc mnmzaton wth nequalty constrants, and so forth. In ths paper, we consder the cubc mnmzaton problem, whch generalzes the quadratc functons frequently consdered n the mentoned papers. The proof method s based on extendng the global L-subdfferentals by quadratc functons 8, 12 to cubc functons. We show how an L-subdfferental can be explctly calculated for cubc functons and then develop the global suffcent optmalty condtons for CP1. We also derve the global optmalty condtons for specal cubc mnmzaton problems wth bnary constrants. But when we use the suffcent condtons, we have to determne whether a dagonal matrx Q exsts. It s hard to dentfy whether the matrx Q exsts. So we rewrte the suffcent condtons n an other way through constructng a certan dagonal matrx. Ths method s applcable to the quadratc mnmzaton problem wth box or bnary constrants consdered n 8, 12. Ths paper s organzed as follows. Secton 2 presents the notons of L-subdfferentals and develops the suffcent global optmalty condton for CP1. The global optmalty condton for specal cubc mnmzaton wth bnary constrants s presented n Secton 3. In Secton 4, numercal examples s gven to llustrate the effectveness of the proposed global optmalty condtons. 2. L-Subdfferentals and Suffcent Condtons In ths secton, basc defntons and notatons that wll be used throughout the paper are gven. The real lne s denoted by R and the n-dmensonal Eucldean space s denoted by R n. For vectors x, y R n, x y means that x y,for 1,...,n. A B means that the matrx A B s a postve semdefnte. A dagonal matrx wth dagonal elements α 1,...,α n s denoted by dag α 1,...,α n.letl be a set of real-valued functons defned on R n. Defnton 2.1 L-subdfferentals 13. LetL be a set of real-valued functons. Let f : R n R. An element l L s called an L-subgradent of f at a pont x 0 R n f f x f x 0 l x l x 0, x R n. 2.1 x 0. The set L f x of all L-subgradents of f at x 0 s referred to as L-subdfferental of f at Throughout the rest of the paper, we use the specfc choce of L defned by L {b T x 3 12 } xt Qx β T x Q dag α 1,...,α n,α R, β R n. 2.2

3 Mathematcal Problems n Engneerng 3 Proposton 2.2. Let f x b T x 3 1/2 x T Ax a T x and x x 1,...,x n T R n.then { L f x b T x xt Qx β T x A Q 0, Q dag α 1,...,α n β AQ x a, α R }. 2.3 Proof. Suppose that there exsts a dagonal matrx Q dag α 1,...,α n, such that A Q 0. Let l x b T x xt Qx β T x, β A Q x a. 2.4 Then t suffces to prove that l x L f x.let φ x f x l x 1 2 xt A Q x ( a β ) T x. 2.5 Snce 2 φ x A Q 0, for all x R n, we know that φ x s a convex functon on R n.note that φ x A Q x a β 0, and so x s a global mnmzer of φ x,thats,φ x φ x, for all x R n. Ths means that l x L f x. Next we prove the converse. Let l x L f x, l x b T x 3 1/2 x T Qx β T x. By defnton, f x f x l x l x, x R n. 2.6 Hence φ x f x l x 1 2 xt A Q x ( a β ) T x f x l x, x R n. 2.7 Thus, x s a global mnmzer of φ x. So, φ x 0and 2 φ x 0, that s, A Q 0, A Q x ( a β ) 0, 2.8 hence β A Q x a. For x x 1,...,x n T D, defne 1 f x u, x 1 f x v, Ax a f x u,v. 2.9 X dag x 1,..., x n.

4 4 Mathematcal Problems n Engneerng For Q dag α 1,...,α n,α R, 1,...,n, defne α mn{0,α }, Q dag α 1,..., α n By Proposton 2.2, we obtan the followng suffcent global optmalty condton for CP1. Theorem 2.3. For CP1, letx x 1,...,x n T D and u u 1,...,u n T,v v 1,...,v n T. Suppose that there exsts a dagonal matrx Q dag α 1,...,α n,α R, 1,...,n, such that A Q 0, and for all x D, b x x 0, 1,...,n. If X Ax a 1 2 Q v u 0, 2.11 then x s a global mnmzer of problem CP1. Proof. Suppose that condton 2.11 holds. Let l x b T x xt Qx β T x, β A Q x a Then, by Proposton 2.2, l x L f x, thats, f x f x l x l x, x R n Obvously f l x l x 0 for each x D, then x s a global mnmzer of CP1. Note that l x l x n α 2 x x 2 Ax a T x x b T( x 3 x 3) If each term n the rght sde of the above equaton satsfes α ( ) 2 x x 2 Ax a x x b x 3 x3 0, 1,...,n, x u,v, 2.15 then, from 2.14, t holds that l x l x 0. So x s a global mnmzer of l x over box constrants. On the other hand, suppose that x s a global mnmzer of l x, x D. Then t holds that l x l x n α 2 x x 2 Ax a T x x b T( x 3 x 3) 0, x D. 2.16

5 Mathematcal Problems n Engneerng 5 When x s chosen as a specal pont x D as follows: x x 1, x 2,...,x 1,x, x,...,x n, x u,v,x / x, 1,...,n, 2.17 we stll have l x l x α 2 x x 2 Ax a T x x b T( x 3 x 3) 0, 1,...,n Ths means that f x s a global mnmzer of l x over box constrants. Then 2.15 holds. Combnng the above dscusson, we can conclude that x s a global mnmzer of l x over box constrants f and only f 2.15 holds. So next, we just need to prove 2.15 n order to show that x s a global mnmzer of l x. We frst see from 2.11, for each 1,...,n,that α 2 v u x Ax a Snce α 0, then for each x u,v, 1,...,n, α 2 x u x Ax a 0, 2.20 and α 2 x v x Ax a For each 1,...,n, we consder the followng three cases. Case 1. If x u,v, then x Ax a.by 2.20, α 2 x u Ax a So, α 0and Ax a 0, and then α ( ) 2 x x 2 Ax a x x b x 3 x3 α ( ) 2 x x 2 b x 3 x3 ( ) b x x x 2 x x x

6 6 Mathematcal Problems n Engneerng Case 2. If x u, then x 1. By 2.20, α 2 x u Ax a So we have α ( ) 2 x x 2 Ax a x x b x 3 x3 α ) 2 x u 2 Ax a x u b (x 3 u3 { α ( ) 2 x u Ax a } x u b x 3 u Case 3. If x v, then x 1.By 2.21, α 2 x v Ax a Then α ( ) 2 x x 2 Ax a x x b x 3 x3 α ) 2 x v 2 Ax a x v b (x 3 v So, f condton 2.11 holds, then 2.15 holds. And, from 2.14, we can conclude that x s a global mnmzer of CP1. Theorem 2.3 shows that the exstence of dagonal matrx Q plays a crucal role because f ths dagonal matrx Q does not exst, then we have no way to use ths theorem. If the dagonal matrx Q exsts, then the key problem s how to fnd t. These questons also exst n 8, 12. The followng corollary wll answer the questons above. Corollary 2.4. For CP1, letx D. Assume that, for all x D, t holds that b x x 0 1,...,n. Then one has the followng concluson. 1 When A s a postve semdefnte matrx, f X Ax a 0, 2.28 then x s a global mnmzer of CP1.

7 Mathematcal Problems n Engneerng 7 2 When A s not a postve semdefnte matrx, f there exsts an ndex 0, 1 0 n, such that x 0 Ax a 0 > 0, 2.29 then there s no such dagonal matrx Q that meets the requrements of the Theorem Let α 2 x Ax a, 1,...,n, v u Q dag α 1,...,α n When A s not a postve semdefnte matrx and the condton X Ax a 0 holds, f A Q 0 holds, then x s a global mnmzer of CP1. Otherwse, one can conclude that there s no such dagonal matrx Q that meets the requrements of the Theorem 2.3. Proof. 1 Suppose that A 0 and the condton X Ax a 0 holds. Choosng Q Q 0, by Theorem 2.3, we can conclude that x s a global mnmzer of CP1. 2 When A s not a postve semdefnte matrx, f there exsts an ndex 0, 1 0 n, such that x 0 Ax a 0 > 0, 2.31 then 2 x 0 Ax a 0 v 0 u 0 > Suppose there exsts a dagonal matrx Q that meets all condtons n Theorem 2.3. Condton 2.11 can be rewrtten n the followng form: x Ax a 1 2 α v u 0, 1,...,n Then t follows that α 2 x Ax a v u, 1,...,n For the ndex 0, we stll have the followng nequalty: α 0 2 x 0 Ax a 0 v 0 u 0 > Ths conflcts wth the fact that α mn{0,α } 0, 1,...,n.

8 8 Mathematcal Problems n Engneerng 3 Next we wll consder the case that A s not a postve semdefnte matrx, and condton X Ax a 0 holds. We construct a dagonal matrx Q dag α 1,...,α n where α 2 x Ax a / v u, 1,...,n,andA Q 0. Then t suffces to show that condton 2.11 n Theorem 2.3 hold. Note that x Ax a 0, 1,...,n, and then α 2 x Ax a v u 0, 1,...,n Snce α 0, we have α α.so α 2 x Ax a v u 0, 1,...,n Rewrtng the above nequalty, we have x Ax a 1 2 α v u 0, 1,...,n Apparently ths means that the constructed dagonal matrx Q also satsfes condton Accordng to Theorem 2.3, we can conclude that x s a global mnmzer of CP1. If the constructed dagonal matrx Q does not meet the condton A Q 0, then we can conclude that there s no such dagonal matrx Q that can meet the requrements of Theorem 2.3. To show ths, suppose that there exsts a dagonal matrx Q dag α 1,...,α n, whch satsfes A Q 0and From 2.11, we have 0 α 2 x Ax a v u α, 1,...,n Obvously f A Q 0, then there must exst a dagonal matrx Q dag α 1,...,α n such that A Q 0. Ths conflcts the assumpton. We now consder a specal case of CP1 : mn s.t. f x n n b x 3 r n 2 x2 a x n x D u,v, CP2 where u,v,a,b,r R and u v, 1, 2,...,n.

9 Mathematcal Problems n Engneerng 9 Corollary 2.5. For CP2,letx D. If, for each 1,...,n, x r x a 1 2 r v u 0, b x x 0, 2.40 then x s a global mnmzer of CP2, where r mn{0,r }. Proof. For CP2, choose Q A dag r 1,...,r n.if 2.40 holds, then, by Theorem 2.3, x s a global mnmzer of CP2. 3. Suffcent Condtons of Bvalent Programmng In ths secton, we wll consder the followng bvalent programmng: mn s.t. f x b T x xt Ax a T x n x D B {u,v }, CP3 where A, a, b,andu, v, 1,...,nare the same as n CP1. Smlar to Theorem 2.3, we wll obtan the global suffcent optmalty condtons for CP3. Theorem 3.1. For CP3, letx x 1,...,x n T D B. Suppose that there exsts a dagonal matrx Q dag α 1,...,α n,α R, 1,...,nsuch that A Q 0, and for all x D B, b x x 0 1,...,n.If X Ax a 1 Q v u 0, then x s a global mnmzer of problem CP3. Proof. Suppose that condton 3.1 holds. Let l x b T x xt Qx β T x, β A Q x a. 3.2 Then f x f x l x l x, x R n. 3.3 Obvously f l x l x 0 for each x D B, then x s a global mnmzer of CP3.

10 10 Mathematcal Problems n Engneerng Note that l x l x n α 2 x x 2 Ax a T x x b T( x 3 x 3). 3.4 Thus, x s a global mnmzer of l x wth bnary constrants f and only f, for each 1,...,n, x {u,v }, α ( ) 2 x x 2 Ax a x x b x 3 x Frstly, we note from 3.1, for each 1,...,n,that Next we only show t from the followng two cases. Case 1. If x u, then 3.6 s equvalent to α 2 v u x Ax a α 2 v u Ax a It s obvous that, for each x {u,v }, α ) 2 x u 2 b (x 3 u3 Ax a x u So 3.5 holds. Case 2. If x v, then 3.6 s equvalent to α 2 v u Ax a It s obvous that, for each x {u,v }, α ) 2 v x 2 b (v 3 x 3 Ax a v x So 3.5 holds. From 3.5, we can conclude that x s a global mnmzer of problem CP3 Smlar to Corollary 2.4, we have the followng corollary. Corollary 3.2. For CP3,letx D B. Suppose that, for all x D B, b x x 0 1,...,n. (1) When A s a postve semdefnte matrx, f X Ax a 0, 3.11 then x s a global mnmzer of CP3.

11 Mathematcal Problems n Engneerng 11 (2) Let α 2 x Ax a, 1,...,n, v u Q dag α 1,...,α n When A s not a postve semdefnte matrx, f A Q 0 holds, then x s a global mnmzer of CP3. Otherwse, one can conclude that there s no such dagonal matrx Q that meets the requrements of Theorem 3.1. We just show the proof of 2. Proof. We construct the dagonal matrx Q dag α 1,...,α n, where α 2 x Ax a / v u, 1,...,n.IfA Q 0, we just need to test condton 3.1. Because then rewrtng the above equatons, we have α 2 x Ax a v u, 1,...,n, 3.13 x Ax a 1 2 α v u 0, 1,...,n It obvously means that the dagonal matrx Q also satsfes condton 3.1. Accordng to Theorem 3.1, x s a global mnmzer of CP3. Note that there s dfference between formula 3.1 and formula In formula 3.1, the dagonal elements α of a dagonal matrx Q are allowed to be postve or nonpostve. But n formula 2.11, the dagonal elements α of a dagonal matrx Q must meet the condtons α 0. So we have to dscuss the sgn of the terms x Ax a 1,...,n n Corollary 3.2. We now consder a specal case of CP3 : mn s.t. f x n n b x 3 r n 2 x2 a x n x D B {u,v }, CP4 where b,r,a,u,v R and u v, 1,...,n. Corollary 3.3. For CP4,letx D B. If, for each 1,...,n, x r x a r 2 v u 0, b x x 0, 3.15 then x s a global mnmzer of CP4.

12 12 Mathematcal Problems n Engneerng Proof. For CP4, choose Q A dag r 1,...,r n. If condtons 3.15 hold, then, by Theorem 3.1, x s a global mnmzer of CP4. 4. Numercal Examples In ths secton, sx examples are gven to test the proposed global suffcent optmalty condton. Example 4.1. Consder the followng problem: mn s.t. 7 3 x3 1 5x3 2 2x x2 1 x x2 3 2x 1x 2 x 1 x 3 x 2 x x 1 5x 2 3x 3 x D 3 1, Let A and a 3/2, 5, 3 T,b 7/3, 5, 2 T. Obvously A s a postve semdefnte matrx. Consderng x 1, 1, 1 T, obvously we have, for each x 1, 2, b x x 0 1, 2, 3. NotethatAx a 15/2, 10, 6 T and X dag 1, 1, 1, andso 15 2 X Ax a 10 < Accordng to Corollary 2.4 1, x 1, 1, 1 s a global mnmzer. Example 4.2. Consder the followng problem: mn x 3 1 3x x3 4 x2 1 x x x2 4 2x 1x 2 x 1 x 4 2x 2 x 4 x 1 4x 2 5x 4 4 s.t. x D 1,

13 Mathematcal Problems n Engneerng 13 Let A and a 1, 4, 0, 5 T,b 1, 3, 0, 1/2 T. Obvously A not s a postve semdefnte matrx. Consderng x 1, 1, 0, 1 T, obvously we have, for each x 1, 1, b x x 0 1, 2, 3, 4. Note that Ax a 6, 10, 0, 9 T. Then lettng X dag 1, 1, 0, 1, we have X Ax a 6, 10, 0, 9 T 0. Letα 1 2 x 1 Ax a 1 / v 1 u 1 6, α 2 10,α 3 0andα 4 9. Then Q dag 6, 10, 0, 9, whch satsfes A Q 0. Accordng to Corollary 2.4 3, x 1, 1, 0, 1 s a global mnmzer. Example 4.3. Consder the followng problem: mn x x3 2 3x3 3 2x x2 2 4x x2 4 2x 1 5x 2 3x 3 4 s.t. x D 1, Let A dag 4, 6/5, 8, 1 and a 2, 5, 3, 0 T, b 1, 2/3, 3, 0 T,r 4, 6/5, 8, 1 T. Consder x 1, 1, 1, 0 T. Let X dag 1, 1, 1, 0, r 4, 6/5, 8, 0 T. Then x 1 r 1 x 1 a r 1 v 1 u 1 2 < 0, x 2 r 2 x 2 a r 2 v 2 u 2 5 < 0, x 3 r 3 x 3 a r 3 v 3 u 3 3 < 0, 4.7 x 4 r 4 x 4 a r 4 v 4 u 4 0, x 1, 1, b x x 0, 1, 2, 3, 4. Accordng to Corollary 2.5, x 1, 1, 1, 0 s a global mnmzer.

14 14 Mathematcal Problems n Engneerng Example 4.4. Consder the followng problem: mn 4x x3 2 3x x x x2 2 3x2 3 x2 4 2x 1x 2 x 1 x 4 x 2 x 3 x 3 x 4 5x x 2 2x 3 2x s.t. x D B 4 { 1, 1}. Let A and a 5, 9/2, 2, 2 T, b 4, 5/2, 3, 8/3 T. Obvously A not s a postve semdefnte matrx. Consderng x 1, 1, 1, 1 T, t follows that, for each x D B, b x x 0 1, 2, 3, 4. NotethatAx a 7, 13/2, 2, 6 T and X dag 1, 1, 1, 1. Letα 1 2 x 1 Ax a 1 / v 1 u 1 7. Smlarly we have, α 2 13/2, α 3 2andα 4 6. Then Q dag 7, 13/2, 2, 6, whch satsfes A Q 0. Accordng to Theorem 3.1 2, x 1, 1, 1, 1 s a global mnmzer. Example 4.5. Consder the followng problem: mn 3x 3 1 8x3 2 5x x3 4 x2 1 4x x2 3 3x2 4 x 1 2x 2 3x 3 4x 4 4 s.t. x D B { 1, 0} Let a 1, 2, 3, 4 T,b 3, 8, 5, 1/2 T,andr 2, 8, 10/3, 6 T. Consder x 1, 0, 1, 1 T. Let X dag 1, 1, 1, 1. Then x 1 r 1 x 1 a r 1 v 1 u 1 2 < 0, x 2 r 2 x 2 a r 2 v 2 u 2 6 < 0, x 3 r 3 x 3 a r 3 v 3 u < 0,

15 Mathematcal Problems n Engneerng 15 x 4 r 4 x 4 a r 4 v 4 u 4 7 < 0, x D B, b x x 0, 1, 2, 3, Accordng to Corollary 3.3, x 1, 0, 1, 1 s a global mnmzer. Example 4.6. Consder the followng problem: mn 3x 3 1 8x3 2 5x x3 4 x2 1 4x x2 3 3x2 4 x 1 2x 2 3x 3 4x 4 4 s.t. x D B { 1, 0} Let a 1, 2, 3, 4 T,b 3, 8, 5, 1/2 T and r 2, 8, 10/3, 6 T. Consder x 1, 0, 1, 1 T. Let X dag 1, 1, 1, 1. Then x 1 r 1 x 1 a r 1 v 1 u 1 0, x 2 r 2 x 2 a r 2 v 2 u 2 6 > 0, x 3 r 3 x 3 a r 3 v 3 u < 0, 4.13 x 4 r 4 x 4 a r 4 v 4 u 4 7 < 0. We can see that the condtons are not true n x 1, 0, 1, 1 T n Corollary 3.3. But x 1, 0, 1, 1 s a global mnmzer. Ths fact exactly shows that the condtons are just suffcent. Acknowledgment Ths research was supported by Innovaton Program of Shangha Muncpal Educaton Commsson 12ZZ071, Shangha Pujang Program 11PJC059, and the Natonal Natural Scence Foundaton of Chna References 1 R. A. Canfed, Multpont cubc surrogate functon for sequental approxmate optmzaton, Structural and Multdscplnary Optmzaton, vol. 27, pp , Y. Nesterov, Acceleratng the cubc regularzaton of Newton s method on convex problems, Mathematcal Programmng, vol. 112, no. 1, pp , 2008.

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17 Advances n Operatons Research Advances n Decson Scences Journal of Appled Mathematcs Algebra Journal of Probablty and Statstcs The Scentfc World Journal Internatonal Journal of Dfferental Equatons Submt your manuscrpts at Internatonal Journal of Advances n Combnatorcs Mathematcal Physcs Journal of Complex Analyss Internatonal Journal of Mathematcs and Mathematcal Scences Mathematcal Problems n Engneerng Journal of Mathematcs Dscrete Mathematcs Journal of Dscrete Dynamcs n Nature and Socety Journal of Functon Spaces Abstract and Appled Analyss Internatonal Journal of Journal of Stochastc Analyss Optmzaton