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Applied Mathematics, 2010, 1, 8-17 doi:10.4236/am.2010.11002 Published Online May 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM A Modified Limited SQP Method For Constrained Optimization* Gonglin Yuan1, Sha Lu2, Zengxin Wei1 1Department of Mathematics and Information Science, Guangxi University, Nanning, China 2School of Mathematics Science, Guangxi Teacher’s Education University, Nanning, China E-mail: glyuan@gxu.edu.cn Received December 23, 2009; revised February 24, 2010; accepted March 10, 2010 Abstract In this paper, a modified variation of the Limited SQP method is presented for constrained optimization. This method possesses not only the information of gradient but also the information of function value. Moreover, the proposed method requires no more function or derivative evaluations and hardly more storage or arith- metic operations. Under suitable conditions, the global convergence is established. Keywords: Constrained Optimization, Limited Method, SQP Method, Global Convergence 1. Introduction Consider the constrained optimization problem Ijxg Eixhts xf j i ,0)( ,0)(.. )(min (1) where RRghf n ji :,, are twice continuously diffe- rentiable, },,,2,1{mE 0},,,2,1{ llmmmI is an integer. Let the Lagrangian function be defined by )()()(),,( xhxgxfxLTT (2) where and are multipliers. Obviously, the La- grangian function L is a twice continuously differenti- able function. Let S be the feasible point set of the problem (1). We define I to be the set of all the sub- scripts of those inequality constraints which are active at x, i.e., }.0)(|{ xgandIiiI i It is well known that the SQP methods for solving twice continuously differentiable nonlinear programming problems, are essentially Newton-type methods for find- ing Kuhn-Tucher points of nonlinear programming problems. These years, the SQP methods have been in vogue [1-8]: Powell [5] gave the BFGS-Newton-SQP method for the nonlinearly constrained optimization. He gave some sufficient conditions, under which SQP me- thod would yield 2-step Q-superlinear convergence rate (assuming convergence) but did not show that his mod- ified BFGS method satisfied these conditions. Coleman and Conn [2] gave a new local convergence qua- si-Newton-SQP method for the equality constrained non- linear programming problems. The local 2-step Q-superlinear convergence was established. Sun [6] proposed quasi -Newton-SQP method for general 1 LC constrained problems. He presented the locally conver- gent sufficient conditions and superlinear convergent sufficient conditions. But he did not prove whether the modified BFGS-quasi-Newton-SQP method satisfies the sufficient conditions or not. We know that, the BFGS update exploits only the gradient information, while the information of function values of the Lagrangian func- tion (2) available is neglected. If n R x holds, then the problem (1) is called un- constrained optimization problem (UNP). There are ma- ny methods [9-13] for the UNP, where the BFGS method is one of the most effective quasi-Newton method. The normal BFGS update exploits only the gradient informa- tion, while the information of function values available is neglected for UNP too. These years, lots of modified BFGS methods (see [14-19]) have been proposed for UNP. Especially, many efficient attempts have been made to modify the usual quasi-Newton methods using both the gradient and function values information (e.g. [19,20]). Lately, in order to get a higher order accuracy in approximating the second curvature of the objective function, Wei, Yu, Yuan, and Lian [18] proposed a new BFGS-type method for UNP, and the reported numerical results show that the average performance is better than that of the standard BFGS method. The superlinear con- vergence of this modified has been established for un- iformly convex function. Its global convergence is estab- lished by Wei, Li, and Qi [20]. Motivated by their ideas, Yuan and Wei [21] presented a modified BFGS method *This work is supported by the Chinese NSF grants 10761001 and the Scientific Research Foundation of Guangxi University (Grant No. X081082), and Guangxi SF grants 0991028. G. L. Yuan ET AL. Copyright © 2010 SciRes. AM 9 which can ensure that the update matrix are positive de- finite for the general convex functions. Moreover, the global convergence is proved for the general convex functions. The limited memory BFGS (L-BFGS) method (see [22]) is an adaptation of the BFGS method for large-scale problems. The implementation is almost identical to that of the standard BFGS method, the only difference is that the inverse Hessian approximation is not formed explicitly, but defined by a small number of BFGS updates. It is often provided a fast rate of linear convergence, and requires minimal storage. Inspired by the modified method of [21], we combine this technique and the limited memory technique, and give a limited SQP method for constrained optimization. The global convergence of the proposed method will be established for generally convex function. The major contribution of this paper is an extension of, based on the basic of the method in [21], the method for the UNP to constrained optimization problems. Unlike the standard SQP method, a distinguishing feature of our proposed method is that a triple },,{ iii Ays being stored, where 1iii s xx ,,)()( 1iiixixi sAzLzLy 1i z 111 (,, ) iii x ,),,( iiii x z , i and i are the multipliers which are according to the Lagrangian objec- tive function at i x, while 1i and 1i are the mul- tipliers which are according to the Lagrangian objective function at 1i x, and i A is a scalar related to Lagran- gian function value. Moreover, a limited memory SQP method is proposed. Compared with the standard SQP method, the presented method requires no more function or derivative evaluations, and hardly more storage or arithmetic operations. This paper is organized as follows. In the next section, we briefly review some modified method and the L-BFGS method for UNP. In Section 3, we describe the modified limited memory SQP algorithm for (2). The global con- vergence will be established in Section 4. In the last sec- tion, we give a conclusion. Throughout this paper, |||| denotes the Euclidean norm of vectors or matrix. 2. Modified BFGS Update and the L-BFGS Update for UNP We will state the modified BFGS update and the L-BFGS update for UNP in the following subsections, respectively. 2.1. Modified BFGS Update Quasi-Newton methods for solving UNP often need to update the iterate matrixk B. In tradition, }{ k Bsatisfies the following quasi -Newton equation: kkk SB 1 (3) where kkkxxS 1,)()( 1kkk xfxf .The very famous updatek Bis the BFGS formula k T k T kk kk T k k T kkk kkSSBS BSSB BB 1 (4) Let k H be the inverse of k B, then the inverse up- date formula of (4) method is represented as ,)()( )( )()( )( )( 2 2 1 k T k T kk k T k T kk k k T k T kk k T k T kkkk T kkkk k T k T kkkkk T k kk S SS S S IH S S I S HSSSHS S SSHS HH (5) which is the dual form of the DF P update formula in the sense thatkk BH , 11 kk BH , and kk ys . It has been shown that the BFGS method is the most ef- fective in quasi-Newton methods from computation point of view. The authors have studied the convergence of fand its characterizations for convex minimization [23-27]. Our pioneers made great efforts in order to find a quasi-Newton method which not only possess global convergence but also is superior than the BFGS method from the computation point of view [15-17,20,28-31]. For general functions, it is now known that the BFGS method may fail for non-convex functions with inexact line search [32], Mascarenhas [33] showed that the non- convergence of the standard BFGS method even with exact line search. In order to obtain a global convergence of BFGS method without convexity assumption on the objective function, Li and Fukushima [15,16] made a slight modification to the standard BFGS method. Now we state their work [15] simply. Li and Fukushima (see [15]) advised a new quasi-Newton equation with the fol- lowing form 1 1kkk SB , where, 1 kkkkk Sgt 0 k t is determined by }0, |||| max{1 2 k k T k kS S t . Un- der appropriate conditions, these two methods [15,16] are globally and superlinearly convergent for nonconvex minimization problems. In order to get a better approximation of the objective function Hessian matrix, Wei, Yu, Yuan, and Lian (see [18]) also proposed a new quasi-Newton equation: ,)3()2( 2 1kkkkkk SASB where 2 |||| )]()([)]()([2 )3( k k T kkkkkkkk kS Sxfdxfdxfxf A . Then the new BFGS update formula is G. L. Yuan ET AL. Copyright © 2010 SciRes. AM 10 . )2( )2()2( )2()2( 2 22 1 k T k T kk kk T k k T kkk kk SSBS BSSB BB (6) Note that this quasi-Newton formula (6) contains both gradient and function value information at the current and the previous step. This modified BFGS update for- mula differs from the standard BFGS update, and a higher order approximation of )( 2xf can be obtained (see [18,20]). It is well known that the matrix k B are very impor- tant for convergence if they are positive definite [24,25]. It is not difficult to see that the condition 0 2 k T k S can ensure that the update matrix )2( 1k B from (6) in- herits the positive definiteness of)2( k B. However this condition can be obtained only under the objective func- tion is uniformly convex. If f is a general convex function, then 2 k T k S and k T k S may equal to 0. In this case, the positive definiteness of the update matrix k B can not be sure. Then we conclude that, for the gen- eral convex functions, the positive definiteness of the update matrix k B generated by (4) and (6) can not be satisfied. In order to get the positive definiteness of )2( k B based on the definition of 2 k and k for the general convex functions, Yuan and Wei [21] give a modified BFGS update, i. e., the modified update formula is de- fined by , )3( )3()3( )3()3( 3 33 1 k T k T kk kk T k k T kkk kk S SBS BSSB BB (7) where }0),3(max{, 3 kkkkkk AASA . Then the corresponding quasi-Newton equation is 3 1)3( kkk SB (8) which can ensure that the condition 0 3 k T k S holds for the general convex function f(see [21] in detail). Therefore, the update matrix 1k B from (8) inherits the positive definiteness of k B for the general convex function. 2.2. Limited Memory BFGS-Type Method The limited memory BFGS (L-BFGS) method (see [22]) is an adaptation of the BFGS method for large-scale problems. In the L-BFGS method, matrix k H is ob- tained by updating the basic matrix )0 ~ ( 0mH times using BFGS formula with the previous m ~ iterations. The standard BFGS correction (5) has the following form T kkkkk T kkSSVHVH 1 (9) where k T k kS 1 , T kkkkSIV , I is the unit ma- trix. Thus, 1k H in the L-BFGS method has the fol- lowing form: . ][][ ][][ ][ 12 ~ 2 ~ 1 ~ 2 ~ 11 ~ 1 ~ 1 ~ 1 ~ 111111 1 T kkk kmk T mkmk T mk T kmk kmkmk T mk T k T kkkk T kkkkk T k T k T kkkkk T kk SS VVSSVV VVHVV SSVSSVHVV SSVHVH (10) 3. Modified SQP Method In this section, we will state the normal SQP method and the modified limited memory SQP method, respectively. 3.1. Normal SQP Method The first-order Kuhn-Tucker condition of (2) is .0)( ,,0)(,0,0)( ,0)()()( xh Ijforxgxg xhxgxf jjj TT (11) The system (11) can be represented by the following system: ,0)( zH (12) where Szz ),,( and lmnlmn RRH : is defined by . )( }),(min{ )()()( )( xh xg xhxgxf zH TT (13) Since ,, gf and h are continuously differentia- ble functions, it is obviously that )(zH is continuously differentiable function. Then, for all lmn Rd , the directional derivative ):( dzH of the function )(zH exists. Denote the index sets by )}(|{)( xgiz ii (14) and )}.(|{)( xgiz ii (15) G. L. Yuan ET AL. Copyright © 2010 SciRes. AM 11 Under the complementary condition, it is clearly that )(z is an index set of strongly active inequality con- straints, and )(z is an index set of weakly active and inactive inequality constraints. In terms of these sets, the directional derivative along the direction ),,( dddd x is given as follows , )( )}(,min{ )( ):( )( )( x T zix T i zix T i dxh dgd dg Gd dzH i (16) where Gis a matrix which elements are the partial deriv- atives of )(zL x to , x d, d, drespectively. If ii ddgd zix T i )( )}(,min{ holds, then the set . 000)( 000 000)( )()()( )( T T xh I xg xhxgxgV zW (17) By (33) in [6], we know than the system ),( kkk zHdW (18) where ),,(kkk dddd xk and )( kk zWW, define the Kuhn-Tucker condition of problem (2), which also defines the Kuhn-Tucker condition of the following qua- dratic programming :),( kk VzQP ,0)()( ,0)()( ,0)()(.. , 2 1 )(min sxhxh sxgxg sxgxgts sVssxf T kk T kk T kk k TT k (19) where ).(, 2 kxxkk zLVxxs Generally, suppose that )1( k B is an estimate of k V and )1( k B can be updated by BFGS method of qua- si-Newton formula , )1( )1()1( )1()1( 1 k T k T kk kk T k k T kkk kk sy yy sBs BssB BB (20) where kkk xxs 1, )()( 1kxkxk zLzLy , ),,,( 1111 kkkk xz ),,,( kkkk xz k and k are the multipliers which are according to the Lagrangian objective function at k x, while 1k and 1k are the multipliers which are according to the Lagrangian objec- tive function at 1k x. Particularly, when we use the up- date formula (20) to (19), the above quadratic program- ming problem can be written as :),(kk BzQP .0)()( ,0)()( ,0)()(.. ,)1( 2 1 )(min sxhxh sxgxg sxgxgts sBssxf T kk T kk T kk k TT k (21) Suppose that ),,( s is a Kuhn-Tucker triple of the sub problem),( kk BzQP, therefore, it is obviously that 0 s if ),,( kk x is a Kuhn-Tucker triple of (2). 3.2. Modified Limited Memory SQP Method The normal limited memory BFGS formula of qua- si-Newton-SQP method with k H for constrained opti- mization (2) is defined by ][][ ][][ ][ 12 ~ 2 ~ 1 ~ 2 ~ 11 ~ 1 ~ 1 ~ 1 ~ 111111 1 kmk T mkmk T mk T kmk kmkmk T mk T k T kkkk T kkkkk T k T k T kkkkk T kk VVssVV VVHVV ssVssVHVV ssVHVH (22) where , 1 k T k kys , T kkkk syIV I is the unit matrix. To maintain the positive definiteness of the li- mited memory BFGS matrix, some researchers suggested to discard correction },{ kk ys if 0 k T kys does not hold (e.g. [34]). Another technique was proposed by Powell [35] in which k y is defined by ,,)1( ,2.0, otherwisesBy sBsysify y kkkkk kk T kk T kk where , 8.0 k T kkk T k kk T k kyssBs sBs kk HB 1 of (22). How- ever, if the Lagrangian objective function ),,( xL is a general convex function, then k T kys may equal to 0. In this case, the positive definiteness of the update matrix k H of (22) can not be sure. Whether there exists a limited memory SQP method which can ensure that the update matrix are positive de- finite for general convex Lagrangian objective func- tion ),,( xL . This paper gives a positive answer. Let 2 11 |||| )]()([)]()([2 ~ k k T kxkxkk ks szLzLzLzL A . Con- sidering the discussion of the above section, we discuss k A ~ for general convex Lagrangian objective function G. L. Yuan ET AL. Copyright © 2010 SciRes. AM 12 ),,( xL in the following cases to state our motivation. case i: If ,0 ~ k A we have .0|||| ~ ) ~ (2 k T kkkk T kkkk T kyssAyssAys (23) case ii: If 0 ~ k A, we get , |||| |||| )]()([)(2 |||| )]()([)]()([2 ~ 0 2 2 11 2 11 k k T k k k T kxkxkkx k k T kxkxkk k s ys s szLzLszL s szLzLzLzL A (24) which means that 0 k T kys holds. Then we present our modified limited memory SQP formula 1 111 111 111 1121121 [] [][] [][] kkk TT kkkk TTT T kkkkkkk kkkk TT kkmkmkmk TT T km kkmkmkm kmk T kkk HVHV ss VVHVss Vss VVH VV VV ssVV ss (25) where , 1 k T k kys , T kkkk syIV and kkkk sAyy }0, ~ max{ . It is not difficult to see that the modified limited memory SQP formula (25) contains both the gradient and function value information of La- grangian function at the current and the previous step if 0 ~ k A holds. Let k B be the inverse of k H. More generally, sup- pose that k B is an estimate of k V. Then the above quadratic programming problem (19) can be written as :),( kk BzQP .0)()( ,0)()( ,0)()(.. , 2 1 )(min sxhxh sxgxg sxgxgts sBssxf T kk T kk T kk k TT k (26) Suppose that ),,( s is a Kuhn-Tucker triple of the subproblem ),( kk BzQP , therefore, it is obviously that 0s if ),,( kk x is a Kuhn-Tucker triple of (2). Now we state our algorithm as follows. Modified limited memory SQP algorithm 1 for (2) (M-L-SQP-A1) Step 0: Star with an initial point ),,( 0000 xz and an estimate 0 H of )( 0 2 0zLV xx , 0 H is a symmetric and positive definite matrix, positive con- stants 10 ,0 0m is a positive constant. Set 0 k; Step 1: For given k z and k H, solve the subproblem ,0)()( ,0)()( ,0)()(.. , 2 1 )(min 1 sxhxh sxgxg sxgxgts sHssxf T kk T kk T kk k TT k (27) and obtain the unique optimal solution k d; Step 2: k is chosen by the modified weak Wolfe-Powell (MWWP) step-size rule ,)()()(k T kxkkkkk dzLzLdzL (28) and ,)()(k T kxk T kkkx dzLddzL (29) then let . 1kkkkdxx Step 3: If 1k z satisfies a prescribed termination cri- terion (18), stop. Otherwise, go to step 4; Step 4: Let },1min{ ~ 0 mkm . Update 0 H for m ~ times to get 1k H by formula (25). Step 5: Set 1 kk and go to step 1. Clearly, we note that the above algorithm is as simple as the limited memory SQP method, form storage and cost point of a view at each iteration. In the following, we assume that the algorithm updates k B-the inverse of k H. The M-L-SQP-A1 with Hessian approximation k B can be stated as follows. Modified limited memory SQP algorithm 2 for (2) (M-L-SQP-A2) Step 0: Star with an initial point ),,( 0000 xz and an estimate 0 B of )( 0 2 0zLV xx , 0 B is a sym- metric and positive definite matrix, positive constants 10 , 0 0m is a positive constant. Set 0 k; Step 1: For given k z and k B, solve the subproblem ),( kk BzQP and obtain the unique optimal solution k d; Step 2: Let },1min{ ~ 0 mkm . Update k B with the triples k mkiiii Ays 1 ~ },,{ , i.e., for kmkl ,,1 ~ , compute G. L. Yuan ET AL. Copyright © 2010 SciRes. AM 13 , 1 l T l T ll k l k T l l k T ll l k l k l ksy yy sBs BssB BB (30) where lll xxs 1, llll sAyy and 1 ~ mk k B for all k. Note that M-L-SQP-A1 and M-L-SQP-A2 are mathe- matically equivalent. In the next section, we will estab- lish the global convergence of M-L-SQP-A2. 4. Convergence analysis of M-L-SQP-A2 Let xbe a local optimal solution and ),,( xz be the corresponding Kuhn-Tucker triple of problem (1). In order to get the global convergence of M-L-SQP-A2, the following assumptions are needed. Assumption A. 1) i hf , and i g are twice conti- nuously differentiable functions for all Sx and S is bounded. 2) }),({}),({ IjxgEixh iiare positive li- near independence. 3) (Strict complementarity) For0, j Ij . (iv) 0Vss Tfor all0swith sxh T i)( Ei ,0 and Ijsxg T i,0)(, where )( 2 zLV xx . (v) }{k zconverges to z where 0)( zL x. (vi) The Lagrangian function )(zL is convex for all Sz . Assumption A(vi) implies that there exists a constant 0H such that .,|||| SzHV (31) Due to the strict complementary Assumption A(3), at a neighborhood of z, the method (26) is equivalent to the following equality constrained quadratic program- ming: .0)()( ,0)()(.. , 2 1 )(min sxhxh sxgxgts sBssxf T kk T kk k TT k (32) Without loss of generality for the locally convergent analysis, we may discuss that there are only active con- straints in (2). Then (18) becomes the following system with k B instead of k V: )( )( )( )( 00)( 00)( )()( k k k kxx T TzH xh xg zL d d d xh xg xhxgB k k k (33) In the case of only considering active constraints, we can suppose that 00)( 00)( )()( T T k k xh xg xhxgV W (34) And , 00)( 00)( )()( , T T k KH xh xg xhxgB D (35) when k B is close to k V, KH D, is close to k W. Lemma 4.1 Let Assumption A hold. Then there exists a positive number 1 M such that .,2,1,0, |||| 1 2 kM ys y k T k k Proof. By Assumption A, then there exists a positive number 0 M such that (see [36]) .0, |||| 0 2 kM ys y k T k k (36) Since the function )(xL is convex, then we have k T kxkk szLzLzL )()()( 1 and ,)()()( 11 k T kxkk szLzLzL the above two in- equalities together with the definition of k A ~ imply that 2 |||| || | ~ | k k T k ks ys A. (37) Using the definition of k y, we get k T kkk T kk T kysAysys }0, ~ max{ (38) and ||,||2||||||||||}0, ~ max{|||||||||| kkkkkkk yyysAyy (39) where the second inequality of (39) follows (37). Com- bining (38), (39), and (36), we obtain: .4 ||||4|||| 0 22 M ys y ys y k T k k k T k k Let 01 4MM , we get the conclusion of this lemma. The proof is complete. Lemma 4.2 Let k B is generated by (30). Then we have ,)det()det( 1 ~ 1 ~ 1 k mkl ll T l l T l mk kk sBs ys BB (40) where )det( k B denotes the determinant of k B. Proof. To begin with, we take the determinant in both sides of (20) G. L. Yuan ET AL. Copyright © 2010 SciRes. AM 14 , )1( ))1(( )])1( )1( ))1(( )(( )))1((1)( )1( )1( 1))[(1(( ) )1( )1( )1( ())1(( )) )1( )1( )1( )(1(())1(( 1 1 1 1 1 kk T k k T k k kk kk T k T kk k T k k T k k T k k T kk kk T k kk T kk k T k T kkk kk T k k T kk k k T k T kkk kk T k k T kk kk sBs sy BDet yB sBs sB sy y s sy y yB sBs sB sBDet ys yyB sBs Bss IDetBDet ys yyB sBs Bss IBDetBDet where the third equality follows from the formula (see, e.g., [37] Lemma 7.6) ).)(()1)(1()det( 324143214321 uuuuuuuuuuuuI TTTTTT Therefore, there is also a simple expression for the de- terminant of (30) .)det()det( 1 ~ 1 ~ 1 k mkl ll T l l T l mk kk sBs ys BB Then we complete the proof. Lemma 4.3 Let Assumption A hold. Then there exists a positive constant 1 such that ,||||1kk s where |||| )( k k T kx kd dzL . Proof. By Assumption A, we have ).1(||||)( ))()(( 2 1 0 1 HddtddtzVd dzLzL kkkkkk T kk k T kxkx On the other hand, using (29), we get .)()1())()(( 1k T kxk T kxkx dzLdzLzL Therefore, , 1 1 |||| kk H s let 1 1 1 H . The proof is complete. Using Assumption A, it is not difficult to get the fol- lowing lemma. Lemma 4.4 Let Assumption A hold. Then the sequence )}({ k zL monotonically decreases, and Szk for all 0k. Moreover, .))(( 0 k k T kxk dzL Similar to Lemma 2.6 in [38], it is not difficult to get the following lemma. Here we also give the proof process. Lemma 4.5 If the sequence of nonnegative numbers ),1,0(kmk satisfy k j k jkccm 0 11 ,,2,1,0, (41) then 0suplim kk m. Proof. We will get this result by contradiction. As- sume that 0suplim kk m, then, for 11 0c , there exists 0 1k, such that 1 k m for all 1 kk . Hence, for all 1 kk, 1 0 11 1 1 k j k kj j kmc 1 11 1 1 0 1 1 suplim k k j j k km c , which is a contradiction, thus, 0suplim kk m. Lemma 4.6 Let }{ k x be generated by M-L-SQP-A2 and Assumption A hold. If 0||)(||inflim kx kzL , then, there exists a constant 0 0 such that .0, 1 0 0 kallfor k k j j Proof. Assume that 0||)(||inflim kx kzL , i.e., there exists a constant 0 2csuch that ,2,1,0,||)(|| 2 kczL kx . (42) Now we prove that the update matrix 1k B will al- ways be generated by the update formula (30), i.e., 1k B inherits the positive definiteness of k B or0 k T kys always holds. For 0 k, this conclusion holds at hand. For all 1k, assume that k B is positive definite. We will deduce that 0 k T kys always holds from the fol- lowing three cases. Case 1. If 0 ~ k A. By the definition of k y and As- sumption A, we have 0}0, ~ max{ k T kkk T kk T kysAysys . Case 2. If 0 ~ k A. By the definition of k y, (24), and Assumption A, we get 0 k T kk T kysys . Case 3. If 0 ~ k A. By the definition of k y, (29), As- sumption A, )( 1 kxkk zLBd , and the positive defi- niteness of k B, we obtain 0)1()()1( kk T kkkx T kkk T kk T kdBdzLdysys , So, we have 0 k T kys , and 1k B will be generated by the update formula (30). Thus, the update matrix 1k B will always be generated by the update formula (30). Taking the trace operation in both sides of (30), we get , |||||||| )( )( 2 1 ~ 2 1 ~ 1 ~ 1 l T l l k mkl ll T l ll k mkl mk k k ys y sBs sB BTr BTr (43) G. L. Yuan ET AL. Copyright © 2010 SciRes. AM 15 where )( k BTr denotes the trace of k B. Repeating this trace operation, we have . |||||||| )( |||||||| )()( 0 2 0 2 0 2 1 ~ 2 1 ~ 1 ~ 1 k ll T l l k lll T l ll l T l l k mkl ll T l ll k mkl mk kk ys y sBs sB BTr ys y sBs sB BTrBTr (44) Combining (42), (44), )( 1 kxkk zLBd , and Lemma 4.1, we obtain .)1( )()( )()( 1 0 2 2 01 Mk zLHzL c BTrBTr k ljxj T jx k (45) Using 1k Bis positive definite, we have 0)( 1 k BTr . By (45), we obtain 2 2 10 0 2 2)1()( )()( c MkBTr zLHzL c k ljxj T jx (46) and .)1()()( 101 MkBTrBTr k (47) By the geometric-arithmetic mean value formula we get . )1()( )1( )()( 1 10 2 2 0 k k j jxj T jx MkBTr ck zLHzL (48) Using Lemma 4.2, (30), and (38), we have . 1 )det( 1 )det( )det( )det()det( 0 0 1 ~ 1 ~ 1 ~ 1 ~ 1 ~ 1 ~ 1 k jj k mkl l mk k k mkl ll T l l T l mk k k mkl ll T l l T l mk kk B B sBs ys B sBs ys BB This implies . 1 )det( )det( 0 1 0 k j j k B B (49) By using the geometric-arithmetic mean value formula again, we get . )( )det( 1 1 n k kn BTr B (50) Using (47), (49) and (50), we obtain 1 3 10 0 10 0 1 10 0 10 0 0 1, ])([ )det( min }1, ])([ )det( min{ )exp( 1 ])([ )det( 1 1 ])1()([ )det( 1 k n n n n k n n n n k j j C MBTr nB MBTr nB n MBTr nB k MkBTr nB (51) where }1, ])([ )det( min{ )exp( 1 10 0 3n n MBTr nB n c . Let . ||||)(|| )( cos jjx j T jx jdzL dzL Multiplying (48) with (51), for all0k, we get 1 10 2 2 1 3 0 ] )1()( )1( [cos||)(|||||| kk k j jjxk MkBTr ck czLs .] )( [1 10 2 23 k MBTr cc (52) According to Lemma 4.4 and Assumption A we know that there exists a constant 0 2 M such that 2112|||||||||||||||| Mxxxxs kkkkk . (53) Combining the definition of k and (53), and noting that jjjx zL cos||)(|| , we get for all 0k, .] 2))(( [1 0 1 210 2 23 0 kk k j jMMBTr cc The proof is complete. Now we establish the global convergence theorem for M-L-SQP-A2. Theorem 4.1 Let Assumption (i) hold and let the se- quence }{ k z be generated by M-L-SQP-A2. Then we have 0||)(||inflim kx kzL . (54) Proof. By Lemma 4.3 and (28), we get .)( ||||)()( 2 1 1 kk kkkk zL szLzL (55) By (55), we have 0 2 k k , this implies that G. L. Yuan ET AL. Copyright © 2010 SciRes. AM 16 0lim k k . (56) Therefore, relation (54) can be obtained from (56) and Lemma 4.6 directly. 5. 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