An interior point type QP-free algorithm with superlinear convergence for inequality constrained optimization

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1 Applied Mathematical Modelling 31 (2007) An interior point type QP-free algorithm with superlinear convergence for inequality constrained optimization Zhibin Zhu * Department of Computational Science and Mathematics, Guilin University of Electronic Technology, Guilin , PR China Received 1 May 2005; received in revised form 1 August 2005; accepted 12 April 2006 Available online 17 August 2006 Abstract A feasible interior point type algorithm is proposed for the inequality constrained optimization. Iterate points are prevented from leaving to interior of the feasible set. It is observed that the algorithm is merely necessary to solve three systems of linear equations with the same coefficient matrix. Under some suitable conditions, superlinear convergence rate is obtained. Some numerical results are also reported. Ó 2006 Elsevier Inc. All rights reserved. MSC: 90C30; 65K10 Keywords: Inequality constrained optimization; System of linear equations; Interior point; Global convergence; Superlinear convergence 1. Introduction In this paper, it is proposed to consider the following nonlinear mathematical programming problem: min f ðxþ s:t gðxþ 6 0; ð1:1þ where f : R n! R, g : R n! R m are continuously differentiable functions. To solve the problem (1.1), there are two type methods with superlinear convergence: Successive quadratic programming (SQP) type algorithms [1 6] and QP-free type algorithms [7]. In general, since SQP algorithms are necessary to solve one or more quadratic programming subproblems in single iteration, the computation effort is very large. In [8,9], QP-free algorithms were proposed to solve the problem (1.1), in which, an iteration similar to the following linear system was considered: * Tel.: ; fax: address: zhuzb@guet.edu.cn X/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi: /.apm

2 1202 Z. Zhu / Applied Mathematical Modelling 31 (2007) Hd 0 þr x Lðx; kþ ¼0; ð1:2þ l rg ðxþ T d 0 þ k 0 g ðxþ ¼0; ¼ 1 m; where Lðx; kþ ¼f ðxþþ P m ¼1 k g ðxþ is lagrangian, H an estimate of the Hessian of L, x the current estimate of a solution x *, d 0 the search direction, and k 0 the next estimate of the Kuhn Tucker multiplier vector associated with x *.In[8], l 2 R m and l > 0, but was not interpreted as the current multiplier estimate. At each iteration, the search direction d was computed in two stages: firstly, a descent direction d 0 is defined by solving (1.2). By modifying d 0, then a feasible descent direction d was obtained. An initial point was an interior point, and iteration points were prevented from leaving the interior of feasible set by using the line search with truncation of the step. Under some assumptions, global convergence to a KKT point was proven, but superlinear convergence was lost due to truncation of the step. In [9], l > 0 was viewed as the current estimate of the KKT multiplier vector associated with x *. At each iteration, based on the direction d 0, which is the solution of (1.2), a revised direction d 1 is obtained by solving the following linear system: Hd þr x Lðx; kþ ¼0; ð1:3þ l rg ðxþ T d þ k g ðxþ ¼ l kd 0 k m ; ¼ 1 m; where m > 2, the search direction d was computed by making a convex combination with d 0 and d 1. In order to avoid Maratos effect, a high-order correction direction d ~ is obtained by solving the following linear least squares problem: 1 min 2 kdk2 ð1:4þ s:t: g ðx þ dþþrg ðxþ T d ¼ w; 2 IðxÞ; where I(x) is a suitable approximate active set at x, and w is a scalar variable. Obviously, (1.4) is equivalent to a linear system, but has a different coefficient matrix, comparing with linear systems (1.2) and (1.3). At the same time, an arc search was presented to prevent the step from being truncated and iteration points from leaving the interior of feasible set. Under some assumptions, global convergence was proven. Unlike [8], it was obtained locally superlinear convergence. Recently, in [10], this type QP-free method is further modified for solving the problem (1.1), in which the method is based on a nonsmooth equation reformulation of the KKT optimality condition by means of the Fischer Burmeister NCP function. Under some weaker assumptions than those in [9], it is obtained the superlinear convergence rate. But, in single iteration, it is necessary to solve three systems of linear equation and a linear squares problem to obtain the search direction. In this paper, a new QP-free algorithm is proposed to improve some facts ust pointed out. Unlike [9], the high-order correction direction d ~ is obtained by solving the following linear system: Hd þr x Lðx; kþ ¼0; ð1:5þ l rg ðxþ T d þ k g ðxþ ¼~g ; ¼ 1 m; where ~g ¼ð~g ; 2 IÞ is a suitable vector (Please see the algorithm). Thereby, the search direction is obtained by solving three systems of linear equations with the same coefficient matrix as (1.2). Thereby, the computational effort of the proposed algorithm is reduced further. Under some suitable assumptions, global convergence is obtained as well as the superlinear convergence rate. In the end, some limited numerical experiments are given to show that the algorithm is effective. 2. Description of algorithm For the sake of simplicity, denote I ¼f1;...; mg; X ¼fx 2 R n gðxþ 6 0g; X 0 ¼fx2 R n g ðxþ < 0; 2 Ig; IðxÞ ¼fg ðxþ ¼0; 2 Ig: The following algorithm is proposed for solving problem (1.1).

3 Z. Zhu / Applied Mathematical Modelling 31 (2007) Algorithm Step 0. Initialization and date: Given a starting point x 0 2 X 0, H 0 2 R n n, an initial symmetric positive definite matrix. h, r 2 (0,1), a 2 0; 1 2,2<s < d <3,l > 0; 0 < l 0 6 l, =1 m, k =0; Step 1. Computation of the Newton direction d k 0 : Let N k ¼ðrg ðx k Þ; 2 IÞ; G k ¼ diagðg ðx k Þ; 2 IÞ; M k ¼ diagðl k ; 2 IÞ: ð2:1þ Solve the following system of linear equations: H k N k d ¼ rf ðxk Þ : ð2:2þ M k N T k G k k 0 Let ðd k 0 ; pk Þ be the solution. If d k 0 ¼ 0, STOP. Step 2. Computation of the search direction: 2.1. Computation of the descent direction d k 1 : Solve the following system of linear equations: H k N k d ¼ rf! ðxk Þ ; M k N T k G k k kd k 0 kd e ð2:3þ where e = (1,...,1) T 2 R m. Let ðd k 1 ; ~pk Þ be the solution Computation of the main search direction d k : Establish a convex combination of d k 0 and dk 1 : d k ¼ð1 b k Þd k 0 þ b kd k 1 ; kk ¼ð1 b k Þp k þ b k ~p k ; ð2:4þ where b k ¼ max b 2ð0; 1Šð1 bþrf ðx k Þ T d k 0 þ brf ðxk Þ T d k 1 6 hrf ðxk Þ T d k 0 : ð2:5þ 2.3. Computation of the high-order corrected direction d ~ k : Let L k ¼f2Ig ðx k Þ P k k g: ð2:6þ Solve the following system of linear equations:! H k N k d rf ðx k Þ ¼ ; ð2:7þ M k N T k G k k kd k 0 ks e þ M k ~g k where ( ~g k ¼ð~g k ; 2 IÞ; ~gk ¼ g ðx k þ d k Þ; 2 L k ð2:8þ 0; 2 I n L k : Let ð d ~ k þ d k ; ~ k k þ k k Þ be the solution. If k d ~ k k > kd k k, set d ~ k ¼ 0. Step 3. The line search: Let J k ¼f2Ig ðx k Þ P k k g: ð2:9þ Compute t k, the first number t in the sequence fl; 1 ; 1 ; 1 ;...g satisfying f ðx k þ td k þ t 2 d ~ k Þ 6 f ðx k Þþatrfðx k Þ T d k ; ð2:10þ g ðx k þ td k þ t 2 d ~ k Þ 6 g ðx k Þ; 2 J k : ð2:11þ g ðx k þ td k þ t 2 d ~ k Þ < 0; 2 I n J k : ð2:12þ

4 1204 Z. Zhu / Applied Mathematical Modelling 31 (2007) Step 4. Update: Obtain H k+1 by updating the positive definite matrix H k using some quasi-newton formulas. Set l kþ1 ¼ minfmaxfp k ; kdk 0kg; lg; 2 I; ð2:13þ and x kþ1 ¼ x k þ t k d k þ t 2~ k d k. Let k = k + 1. Go back to step Global convergence of algorithm In this section, it is first shown that Algorithm A is well defined. The following assumptions are true throughout the paper. H3.1 The interior X 0 of the feasible set X is nonempty. H3.2 The functions f, g are continuously differentiable. H3.3 The set X \ {x 2 R n f(x) 6 f(x 0 )} is compact. H3.4 For all x 2 X, the vectors fr gi ðxþ; 2 IðxÞg are linearly independent. H3.5 There exist two constants 0 < a 6 b, such that akdk 2 < d T H k d < bkdk 2, for all k, for all d 2 R n. Lemma 3.1. Given any vector, x 2 X, any positive definite matrix H 2 R n n, and any nonnegative vector l =(l, 2 I) 2 R m such that l > 0, 2 I(x), the matrix F(x, H, l) defined by H NðxÞ F ðx; H; lþ ¼ MNðxÞ T GðxÞ is nonsingular, where NðxÞ ¼ðrg ðxþ; 2 IÞ; M ¼ diagðl; 2 IÞ; GðxÞ ¼diagðg i ðxþ; 2 IÞ: Proof. The proof of this lemma is similar to that of Lemma 3.1 in [9]. h In view of Lemma 3.1, linear systems (2.2), (2.3) and (2.7) are consistent. Lemma 3.2 (1) If d k 0 ¼ 0, then xk is a KKT point of (1.1). (2) If d k 0 6¼ 0, then dk computed according to (2.4) is well defined, and rf ðx k Þ T d k 0 6 ðdk 0 ÞT H k d k 0 < 0; rf ðxk Þ T d k 6 hrf ðx k Þ T d k 0 < 0; rg ðx k Þ T d k ¼ kk g l k ðx k Þ b k kd k 0 kd ; 2 I: ð3:1þ Proof (1) It is obvious according to the definition of the KKT point of (1.1). (2) If d k 0 6¼ 0, from (2.2), we have rf ðx k Þ T d k 0 ¼ ðdk 0 ÞT H k d k 0 ðn kp k Þ T d k 0 ¼ ðd k 0 ÞT H k d k 0 þ X 2I ðp k Þ2 l k g ðx k Þ 6 ðd k 0 ÞT H k d k 0 < 0; rg ðx k Þ T d k 0 ¼ pk g l k ðx k Þ; 2 I:

5 Thereby, from (2.5), there exists some b 2 (0,1], such that b k = b 2 (0,1], i.e., d k is well-defined. In addition, from (2.3), it follows that: rg ðx k Þ T d k 1 ¼ ~pk g l k ðx k Þ kd k 0 kd ; 2 I: Thus, from (2.4), it is clear to see that Z. Zhu / Applied Mathematical Modelling 31 (2007) rg ðx k Þ T d k ¼ð1 b k Þrg ðx k Þ T d k 0 þ b krg ðx k Þ T d k 1 ¼ kk g l k ðx k Þx k b k kd k 0 kd ; 2 I; rf ðx k Þ T d k ¼ð1 b k Þrf ðx k Þ T d k 0 þ b krf ðx k Þ T d k 1 6 hrf ðxk Þ T d k 0 < 0: The claim holds. h Lemma 3.3. The line search in step 3 yields a step t k ¼ 1 i for some finite i = i(k). 2 Proof. Firstly, for (2.10), we have a k, f ðx k þ td k þ t 2 d ~ k Þ fðx k Þ atrfðx k Þ T d k ¼rf ðx k Þ T ðtd k þ t 2 d k Þ atrf ðx k Þ T d k oðtþ ¼ð1 aþtrfðx k Þ T d k þ oðtþ: Since f is continuously differentiable, $f(x k ) T d k <0, a 2 0; 1 2, there exists some t > 0, such that a k 6 0, 8t 2½0;tŠ. Secondly, for (2.11), from (2.9) and (3.1), we obtain, for 2 J k, that b k, g ðx k þ td k þ t 2 ~ d k Þ g ðx k Þ ¼ trg ðx k Þ T d k þ oðtþ ¼ kk tg l k ðx k Þ tb k kd k 0 kd þ oðtþ 6 1 tg 2 l k ðxk Þ tb k kd k 0 kd þ oðtþ: So, there exists some t > 0, 2 J k, such that b k 6 0; 8t 2½0; t Š. Thirdly, for (2.12), since g is continuous and g (x k ) < 0, there exists some t > 0, 62 J k, such that g ðx k þ td k þ t 2 d ~ k Þ g ðx k Þ < 0: 1 i Let ¼ minft;t 2 ; 2 Ig, then the claim holds. h In the sequel, we will prove that the algorithm is globally convergent. From H3.2, H3.3 and H3.5, we might as well assume that there exists a subsequence K, such that x k! x ; H k! H ; d k 0! d 0 ; pk! p ; l k! l ; k 2 K: ð3:2þ Lemma 3.4. If x k! x *,k2k, then d k 0! 0; k 2 K. Proof. Since {f(x k )} is monotonically decreasing, the facts {x k } k2k! x * and continuity of f simply that f ðx k Þ!fðx Þ; k!1: ð3:3þ Suppose by contradiction that d 0 6¼ 0. Then, from (2.13), wehave l k! l > 0; 2 I; k 2 K:

6 1206 Z. Zhu / Applied Mathematical Modelling 31 (2007) So, it is easy to prove that ðd 0 ; p Þ is the unique solution of the following linear system: H d þrfðx ÞþN p ¼ 0; l rg ðx Þ T d þ p g ðx Þ¼0; 2 I; where N * =($g (x * ), 2 I). Thereby, rf ðx Þ T d 0 < 0: Similar to (2.4), we define d *, and, by imitating the proof of Lemma 3.2, it follows that: rf ðx Þ T d < 0; rg ðx Þ T d 6 k g l ðx Þ b kd 0 kd ; 2 I: ð3:6þ From (3.5), (3.6) and the proof of Lemma 3.3, we can conclude that the step-size t k obtained by the linear search in step 3 is bounded away from zero on K, i.e., t k P t ¼ infft k ; k 2 Kg > 0; k 2 K: So, from (2.10), (3.3) and (3.6), we get 0 ¼ limðf ðx kþ1 Þ fðx k ÞÞ 6 lim at k rf ðx k Þ T d k 6 1 k2k k2k 2 at rf ðx Þ T d < 0: It is a contradiction, which shows that d k 0! 0, k 2 K. h In order to obtain the global convergence of the algorithm, we assume the following condition. H3.6 The number of stationary points of (1.1) is finite. ð3:4þ ð3:5þ Theorem 3.5. The algorithm A either stops at the KKT point x k of (1.1) in finite iteration, or generates an infinite sequence {x k } whose all accumulation points are KKT points of (1.1). Proof. The first statement is obvious, the only stopping point being in step 1. Thus, suppose that {x k } k2k! x *, d k 0! 0, k 2 K. From (3.4), we have rf ðx ÞþN p ¼ 0; p g ðx Þ¼0; 2 I ð3:7þ If g (x * )<0," 2 I, then p * =0,$f(x * ) = 0, it is obvious that x * is a KKT point of (1.1). Without loss of generality, we suppose that there exists some 0 2 I, such that g 0 ðx Þ¼0. If p 0 P 0, then it is easy to see that x * is a KKT point of (1.1). Suppose that p 0 < 0. Since there are only finitely many choices for sets J k I, we might as well assume, for k 2 K, k large enough, that J k J, where J is a constant set. Obviously, the fact d k k2k 0! 0 and the definition of k k imply that 0 2 J 5 ;. From Lemma 3.4 and the condition H3.6, according to Theorem 3.11 in [9], it holds that x k! x *, k!1. Thereby, Lemma 3.4 shows that d k 0! 0, k k! u *, k!1. So, it holds that k k 0 6 g 0 ðx k Þ; k k 0! k 0 < 0; g 0 ðx k Þ!g 0 ðx Þ¼0: ð3:8þ While, from (2.11), there exists some k 0 such that, for k P k 0 g 0 ðx k Þ 6 g 0 ðx k 1 Þ 6 6 g 0 ðx k0þ1 Þ 6 g 0 ðx k 0 Þ < 0: It is in contradiction with (3.8), which shows that x * is a KKT point of (1.1). h 4. Rate of convergence Now we strengthen the regularity assumptions on the functions involved. Assumption H3.2 are replaced by H4.1 The functions f, g are twice continuously differentiable. In order to obtain superlinear convergence, we also make the following additional assumptions, where H4.3 may supersede H3.6.

7 Z. Zhu / Applied Mathematical Modelling 31 (2007) H4.2 The sequence generated by the algorithm possesses an accumulation point x * (in view of Theorem 3.5,a KKT point). H4.3 The second-order sufficiency conditions with strict complementary slackness are satisfied at the KKT point x * and the corresponding multiplier vector u *. u * satisfies that u 6 l; ¼ 1 m. Lemma 4.1. The entire sequence {x k } converges to x *, i.e., x k! x *,k!1. Proof. From (2.10) and (3.1), it holds that f ðx kþ1 Þ 6 f ðx k Þþat k rf ðx k Þ T d k 6 f ðx k Þ aht k ðd k 0 ÞT H k d k 0 : from (3.3) and H3.5, we have t k kd k 0 k!0. Thereby, from (2.3) and (2.4), it holds that t kkd k 0k!0. So, t k kd k kþt 2 k k~ d k k 6 2t k kd k k!0, that is to say, kx k+1 x k k!0. According to H4.3 and proposition 4.1 in [5], we have lim k!1 x k = x *. h Lemma 4.2. For k large enough, it holds that L k I(x * ), and ðd k 0 Þ!0; pk! u ; ~p k! u ; k k! u ; l k! u : Proof. According to Lemma 3.4, and x k! x *, it holds that d k 0! 0, k!1. Since x* is a KKT point of (1.1), it follows that: rf ðx ÞþN u ¼ 0; g ðx Þu ¼ 0; 2 I: Denote D k ¼ diagðg 2 ðxk Þ; 2 IÞ; D ¼ diagðg 2 ðx Þ; 2 IÞ. From H3.4, it is clear that ðn T N þ D Þ is nonsingular. Thereby, we have u ¼ ðn T N þ D Þ 1 N T rf ðx Þ: ð4:1þ In addition, from (2.2), we get H k d k 0 þrfðxk ÞþN k p k ¼ 0; D k p k,d k ¼ðD k ; 2 IÞ; Dk ¼ lk g ðx k Þrg ðx k Þ T d k 0 ; 2 I: So, ðn T k N k D k Þp k ¼ N T k ðrf ðxk ÞþH k d k 0 ÞþDk : Since ðn T k N k þ D k Þ!ðN T N þ D Þ; it is obvious, for k large enough, that ðn T k N k þ D k Þ is nonsingular; and ðn T k N k þ D k Þ 1!ðN T N þ D Þ 1 : So, from d k 0! 0; Dk! 0, we have p k!ðn T N þ D Þ 1 N T rf ðx Þ¼u : With the same reason, we can prove that ~p k! u. Thereby, from the definition of k k, it is easy to see that k k! u *. Thereby, from H4.3 and the definition of L k, it holds that L k I(x * ). In the end, from the definition of l k and H4.3, we know that l k! u *. h Lemma 4.3. For k large enough, it holds that kd k kkd k 1 kkdk 0k; k!1;

8 1208 Z. Zhu / Applied Mathematical Modelling 31 (2007) and rf ðx k ÞþH k d k þ A k p k Iðx Þ ¼ oðkdk kþ; rf ðx k ÞþH k d k þ A k ~p k Iðx Þ ¼ oðkdk kþ; rf ðx k ÞþH k d k þ A k k k Iðx Þ ¼ oðkdk kþ; g ðx k Þþrg ðx k Þ T d k ¼ oðkd k k 2 Þ; 2 Iðx Þ; p k ¼ oðkdk kþ; ~p k ¼ oðkdk kþ; k k ¼ oðkdk kþ; 62 Iðx Þ; where A k ¼ðrg ðx k Þ; 2 Iðx ÞÞ; p k Iðx Þ ¼ðpk ; 2 Iðx ÞÞ; ~p k Iðx Þ ¼ð~pk ; 2 Iðx ÞÞ; k k Iðx Þ ¼ðkk ; 2 Iðx ÞÞ: ð4:2þ Proof. From (2.2), we have rf ðx k ÞþH k d k 0 þ N kp k ¼ 0; p k g ðx k Þþl k rg ðx k Þ T d k 0 ¼ 0; 2 I: ð4:3þ Obviously, the fact that d k 0! 0, pk! u *, l k! u *, k!1and the definition of l k imply, for k; large enough, that l k ¼ pk, 2 I(x* ), and rf ðx k ÞþH k d k 0 þ A kp k Iðx Þ ¼ oðkdk 0 kþ; g ðx k Þþrg ðx k Þ T d k 0 ¼ 0; g ðx k Þ¼Oðkd k 0 kþ; 2 Iðx Þ; ð4:4þ p k ¼ oðkdk 0 kþ; 62 Iðx Þ: From (2.3), we have that rf ðx k ÞþH k d k 1 þ N k~p k ¼ 0; ~p k g ðx k Þþl k rg ðx k Þ T d k 1 ¼ ðkdk 0 kþd ¼ oðkd k 0 k2 Þ; 2 Iðx Þ: With the same reason, we have that rf ðx k ÞþH k d k 1 þ A k~p k Iðx Þ ¼ oðkdk 0 kþ; ~p k g ðx k Þþl k rg ðx k Þ T d k 1 ¼ oðkdk 0 k2 Þ; 2 Iðx Þ; ~p k ¼ oðkdk 0 kþ; 62 Iðx Þ: Denote Dd k ¼ d k 1 dk 0 ; Dpk Iðx Þ ¼ pk Iðx Þ ~pk Iðx Þ ¼ lk Iðx Þ ~pk Iðx Þ ; ð4:5þ ð4:6þ then, according to (4.4) and (4.6), it holds that H k Dd k þ A k Dp k Iðx Þ ¼ oðkdk 0 kþ; ð4:7þ According to H3.4, we know, for k large enough, that A T k A k is nonsingular. Furthermore, denote P k ¼ I n A k ða T k A kþ 1 A T k ; Ddk ¼ Dd k 1 þ Ddk 2 ; Ddk 1 ¼ P kdd k ; then, from (4.7), the fact that P k A k = 0 implies that ðdd k 1 ÞT H k Dd k ¼ oðkd k 0 kkddk 1 kþ; i.e., OðkDd k 1 kkddk kþ ¼ oðkd k 0 kkddk 1 kþ; kddk k¼oðkd k 0 kþ: Thereby, it holds that kd k kkd k 0 kkdk 1 k:

9 To prove that (4.2) is true. First of all, from (4.7), the fact that kdd k k¼oðkd k 0 kþ ¼ oðkdk kþ implies that Dp k Iðx Þ ¼ oðkdk kþ: ð4:8þ So, according to (4.4), (4.6) and (4.8) and the definition of d k, k k,wehave rf ðx k ÞþH k d k þ A k p k Iðx Þ ¼ oðkdk kþ; rf ðx k ÞþH k d k þ A k ~p k Iðx Þ ¼ oðkdk kþ; rf ðx k ÞþH k d k þ A k k k Iðx Þ ¼ oðkdk kþ; g ðx k Þþrg ðx k Þ T d k 0 ¼ 0; 2 Iðx Þ; and, for 2 I(x * ), it holds that g ðx k Þþrg ðx k Þ T d k 1 ¼ ~pk 1 l k! g ðx k Þþoðkd k k 2 Þ¼ 1 Dp k l k g ðx k Þþoðkd k k 2 Þ¼oðkd k k 2 Þ: So, it holds that g ðx k Þþrg ðx k Þ T d k ¼ oðkd k k 2 Þ; 2 Iðx Þ: While, from the definition of k k, it is easy to see that k k ¼ oðkdk kþ; 62 Iðx Þ: The claim holds. h Lemma 4.4. For k large enough, ~ d k obtained by step 3 satisfies k ~ d k k¼oðkd k k 2 Þ: Z. Zhu / Applied Mathematical Modelling 31 (2007) Proof. According to (2.7) and Lemma 4.3, it is easy to that ~g k ¼ Oðkdk k 2 Þ; 2 I. So, we have rf ðx k ÞþH k ð d ~ k þ d k ÞþN k ð ~ k k þ k k Þ¼0; ð ~ k k þ k k Þg ðx k Þþl k rg ðx k Þ T ð d ~ k þ d k Þ¼ kd k 0 ks l k ~gk ¼ Oðkdk k 2 Þ; 2 I: So, from (4.3), (4.5), (2.4) and (4.9), it holds that i.e., H k d ~ k þ N k ~ k k ¼ 0; l k rg ðx k Þ T d ~ k þ ~ k k g ðx k Þ¼ kd k 0 ks þ b k kd k 0 kd l k ~gk ¼ Oðdk k 2 Þ; 2 I:! H k N k ~d k 0 ¼ M k N T k G k ~k k Oðkd k k 2 ; Þ ð4:9þ ð4:10þ which shows that k d ~ k k¼oðkd k k 2 Þ; k ~ k k k¼oðkd k k 2 Þ: The claim holds. h In order to obtain superlinear; convergence, a crucial requirement is that a unit step size be used in a neighborhood of the solution. This can be achieved thanks to the following assumption: H4.4 The sequence of matrices {H k } satisfies kp k ðh k r 2 xx Lðxk ; k k ÞÞd k k¼oðkd k kþ;

10 1210 Z. Zhu / Applied Mathematical Modelling 31 (2007) where P k ¼ I n A k ða T k A kþ 1 A T k ; r2 xx Lðxk ; k k Þ¼r 2 f ðx k Þþ X 2IðX Þ k k r2 g ðx k Þ: Lemma 4.5. For k large enough, inequalities in step 4.1 are all true, and x kþ1 ¼ x k þ d k þ ~ d k ; t k 1. Proof. According to H4.3, it is easy to see that J k = ;. So, it is only necessary to prove that f ðx k þ d k þ d ~ k Þ 6 f ðx k Þþarfðx k Þ T d k ; ð4:11þ g ðx k þ d k þ d ~ k Þ < 0; 2 I: ð4:12þ With the same reason, we can prove that (4.10) holds in this case. So, the facts that d > s 2 (2,3), k ~ k k k¼oðkd k k 2 Þ and g (x k )=O(kd k k), 2 I(x * ) imply that rg ðx k Þ T d ~ k ¼ 1 kd k l k 0 ks b k kd k 0 kd þ g ðx k Þ ~ k k g ðx k þ d k Þ ¼ 1 kd k l k 0 ks g ðx k þ d k Þþoðkd k 0 ks Þ: ð4:13þ So, it holds, for 2 I(x * ), that g ðx k þ d k þ d ~ k Þ¼g ðx k þ d k Þþrg ðx k þ d k Þ T d ~ k þ Oðk d ~ k k 2 Þ ¼ g ðx k þ d k Þþrg ðx k Þ T d ~ k þ Oðkd k k 3 Þ ¼ 1 kd k l k 0 ks þ oðkd k 0 ks Þ: ð4:14þ Thereby, for k large enough, for 2 I(x * ), (4.12) holds. For 2 I n I(x * ), (4.12) holds from the facts x k! x *, d k! 0, g (x * ) < 0 and the continuity of g. According to (2.2), (4.2) and (4.14) and the proof of Lemma 4.4 in [11], the fact that k d ~ k k¼oðkd k k 2 Þ implies that (4.11) holds. h In view of Lemma 4.5, (4.2), H4.4 and the way of Theorem 5.2 in [2] or Theorem 4.1 in [12], it is easy to get the convergence theorem as follows: Theorem 4.6. Under all stated assumptions, the algorithm is superlinearly convergent, i.e., the sequence {x k } generated by the algorithm satisfies kx k+1 x * k = O(kx k x * k). 5. Numerical experiments In this section, we carry out some limited numerical experiments based on the algorithm. The code of the proposed algorithm is written by using C++ programming language, and run on Windows In the implementation, H k is updated by the BFGS formula [13]. The stopping criterion of step 1 is changed to If kd k k ; STOP: This algorithm has been tested on some problems from [14], where no equality constraints are present, and a feasible initial point is provided for each problem. The results are summarized in Table 1. For each test problem, No. is the number of the test problem in [14], NIT the number of iterations, NF the number of evaluations of the obective functions, NG the number of evaluations of scalar constraint functions, FV the final value of the obective function, and CPU the total time taken by the process (unit: millisecond). A CPU-time of 0 simply means execution time below 10 ms (or 0.01 s). During numerical experiments, in order to obtain some better numerical results, we make some small modifications in the algorithm. When the strict complementarity conditions are not satisfied near to a solution of

11 Z. Zhu / Applied Mathematical Modelling 31 (2007) Table 1 The detail information of numerical experiments No. NIT/NF/NG FV kd k 0 k CPU 12 8/8/ e [14] 24 28/28/ e [14] 31 11/11/ e [14] 33 53/53/2646 p ffiffiffi e [14] 43 16/16/ e [14] 66 17/17/ e [14] 76 20/20/ e [14] 93 17/17/ e [14] /32/ e [14] 110 2/2/ e [14] /43/ e [14] /55/ e [14] the problem (1.1), some parameters l corresponding to a nearly active constraint g become very small, thus linear systems (2.2), (2.3) and (2.7) may become ill-conditioned. Here, replace the coefficient matrix of those linear systems by the following matrix H k A k A T k 0 ; A k ¼ðr g ðx k Þ; 2 L k 1 Þ: and replace those right hands by the following corresponding vectors: rf!!! ðxk Þ rf ðx k Þ rf ðx k Þ ; ; ; g Lk 1 ðx k Þ g Lk 1 ðx k Þþkd k 0 kd ~e g Lk 1 ðx k Þþkd k 0 ks ~e þ ~g k g Lk 1 ðx k Þ¼ðg ðx k Þ; 2 L k 1 Þ; ~e ¼ð1;...; 1Þ T 2 R L k 1 ; ~g k ¼ðg ðx k þ d k Þ; 2 L k 1 Þ: respectively. Acknowledgements The author would like to thank two anonymous referees, whose constructive comments led to a considerable revision of the original paper. This work was supported in part by the NNSF ( , ) of China.

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