A Modified Limited Sqp Method for Constrained Optimization*

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 arithmetic operations. Under suitable conditions, the global convergence is established.


Introduction
Consider the constrained optimization problem where R R g h f  is an integer.Let the Lagrangian function be defined by where  and  are multipliers.Obviously, the Lagrangian function L is a twice continuously differentiable function.Let S be the feasible point set of the problem (1).We define  I to be the set of all the subscripts of those inequality constraints which are active at  x , i.e., }. 0 It is well known that the SQP methods for solving twice continuously differentiable nonlinear programming problems, are essentially Newton-type methods for finding Kuhn-Tucher points of nonlinear programming problems.These years, the SQP methods have been in vogue [1][2][3][4][5][6][7][8]: Powell [5] gave the BFGS-Newton-SQP method for the nonlinearly constrained optimization.He gave some sufficient conditions, under which SQP method 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 quasi-Newton-SQP method for the equality constrained nonlinear 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 convergent 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 function (2) available is neglected.
holds, then the problem (1) is called unconstrained optimization problem (UNP).There are many methods [9][10][11][12][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 information, while the information of function values available is neglected for UNP too.These years, lots of modified BFGS methods (see [14][15][16][17][18][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 convergence of this modified has been established for uniformly convex function.Its global convergence is established by Wei, Li, and Qi [20].Motivated by their ideas, Yuan and Wei [21] presented a modified BFGS method which can ensure that the update matrix are positive definite 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 } , , { being stored, where  and i  are the multipliers which are according to the Lagrangian objective function at i x , while  are the multipliers which are according to the Lagrangian objective function at x , and  i A is a scalar related to Lagrangian 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 convergence will be established in Section 4. In the last section, we give a conclusion.Throughout this paper,

|| || 
denotes the Euclidean norm of vectors or matrix.

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.

Modified BFGS Update
Quasi-Newton methods for solving UNP often need to update the iterate matrix k B .In tradition, } { k B satisfies the following quasi -Newton equation: where Let k H be the inverse of k B , then the inverse update formula of (4) method is represented as which is the dual form of the DFP update formula in the sense that , and It has been shown that the BFGS method is the most effective in quasi-Newton methods from computation point of view.The authors have studied the convergence of f and its characterizations for convex minimization [23][24][25][26][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][16][17]20,[28][29][30][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 nonconvergence 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 following form . Under 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: 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 formula differs from the standard BFGS update, and a higher order approximation of ) ( 2x f  can be obtained (see [18,20]).
It is well known that the matrix k B are very important for convergence if they are positive definite [24,25].It is not difficult to see that the condition . However this condition can be obtained only under the objective function is uniformly convex.If f is a general convex function, then 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 general 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

 and k
 for the general convex functions, Yuan and Wei [21] give a modified BFGS update, i. e., the modified update formula is defined by . Then the corresponding quasi-Newton equation is which can ensure that the condition holds for the general convex function f (see [21] in detail).
Therefore, the update matrix from (8) inherits the positive definiteness of k B for the general convex function.

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 obtained by updating the basic matrix ) 0 ( 0  m H times using BFGS formula with the previous m ~ iterations.The standard BFGS correction (5) has the following form where , I is the unit matrix.Thus, in the L-BFGS method has the following form: .

Modified SQP Method
In this section, we will state the normal SQP method and the modified limited memory SQP method, respectively.

Normal SQP Method
The first-order Kuhn-Tucker condition of ( 2) is The system (11) can be represented by the following system: , 0 ) (  z H (12) where Under the complementary condition, it is clearly that ) (z  is an index set of strongly active inequality constraints, and ) (z  is an index set of weakly active and inactive inequality constraints.In terms of these sets, the directional derivative along the direction ) , , ( where G is a matrix which elements are the partial derivatives of By (33) in [6], we know than the system where ) , , ( , define the Kuhn-Tucker condition of problem (2), which also defines the Kuhn-Tucker condition of the following quadratic programming : ) , (

where
). ( , where x .Particularly, when we use the update formula (20) to (19), the above quadratic programming problem can be written as : ) , ( is a Kuhn-Tucker triple of (2).

Modified Limited Memory SQP Method
The normal limited memory BFGS formula of quasi-Newton-SQP method with k H for constrained optimization ( 2) is defined by where , 1 I is the unit matrix.To maintain the positive definiteness of the limited memory BFGS matrix, some researchers suggested does not hold (e.g.[34]).Another technique was proposed by Powell [35] in which k y is defined by 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 definite for general convex Lagrangian objective function ) , , ( . This paper gives a positive answer.Let . Considering the discussion of the above section, we discuss in the following cases to state our motivation.
holds.Then we present our modified limited memory SQP formula where , 1 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 ) , , ( H is a symmetric and positive definite matrix, positive con- and obtain the unique optimal solution k d ; satisfies a prescribed termination criterion (18), stop.Otherwise, go to step 4; Step 4: by formula (25).
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 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 ) , , ( B is a symmetric and positive definite matrix, positive constants and obtain the unique optimal solution k d ; Step where Note that M-L-SQP-A1 and M-L-SQP-A2 are mathematically equivalent.In the next section, we will establish the global convergence of M-L-SQP-A2.

Convergence analysis of M-L-SQP-A2
Let  x be a local optimal solution and ) , , ( x z 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.
and i g are twice continuously differentiable functions for all S x  and S is bounded. 2 are positive linear independence. 3 is convex for all S z  .
Assumption A(vi) implies that there exists a constant Due to the strict complementary Assumption A(3), at a neighborhood of  z , the method ( 26) is equivalent to the following equality constrained quadratic programming: Without loss of generality for the locally convergent analysis, we may discuss that there are only active constraints in (2).Then (18) becomes the following system with In the case of only considering active constraints, we can suppose that Lemma 4.1 Let Assumption A hold.Then there exists a positive number Proof.By Assumption A, then there exists a positive number 0 M such that (see [36]) Using the definition of  k y , we get where the second inequality of (39) follows (37).Combining (38), (39), and (36), we obtain: , we get the conclusion of this lemma.The proof is complete.Lemma 4.2 Let k B is generated by (30).Then we have Proof.To begin with, we take the determinant in both sides of (20) where the third equality follows from the formula (see, e.g., [37] Lemma 7.6) ).
Therefore, there is also a simple expression for the determinant of (30) .) det( ) det( Then we complete the proof.Lemma 4.3 Let Assumption A hold.Then there exists a positive constant 1 Proof.By Assumption A, we have On the other hand, using (29), we get . The proof is complete.
Using Assumption A, it is not difficult to get the following lemma.
Lemma 4.4 Let Assumption A hold.Then the sequence )} ( { k z L monotonically decreases, and [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 Proof.We will get this result by contradiction.Assume that 0 sup lim  k k m , then, for , and the positive definiteness of will be generated by the update formula (30).Thus, the update matrix  1 k B will always be generated by the update formula (30).
Taking the trace operation in both sides of (30), we get , and Lemma 4.1, we obtain .
By (45), we obtain By the geometric-arithmetic mean value formula we get Using Lemma 4.2, (30), and (38), we have

Conclusion
For further research, we should study the properties of the modified limited memory SQP method under weak conditions.Moreover, numerical experiments for practically constrained problems should be done in the future.
and k  are the multipliers which are according to the Lagrangian objective function at k x according to the Lagrangian objective function at 1  k

4 . 1
using the geometric-arithmetic mean value formula again, we get .The proof is complete.Now we establish the global convergence theorem for M-L-SQP-A2.Theorem Let Assumption (i) hold and let the se-(54) can be obtained from (56) and Lemma 4.6 directly.