A Globally and Superlinearly Convergent Primal-dual Interior Point Method for General Constrained Optimization

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1 Numer. Math. Theor. Meth. Appl. Vol. 8, No. 3, pp do: /nmtma.2015.m1338 August 2015 A Globally and Superlnearly Convergent Prmal-dual Interor Pont Method for General Constraned Optmzaton Janlng L 1, Jan Lv 2 and Jnbao Jan 3, 1 College of Mathematcs and Informaton Scence, Guangx Unversty, Nannng, Guangx, , Chna. 2 School of Mathematcs Scence, Dalan Unversty of Technology, Dalan, , Chna. 3 School of Mathematcs and Informaton Scence, Yuln Normal Unversty; Guangx Colleges and Unverstes Key Lab of Complex System Optmzaton and Bg Data Processng, Yuln , Chna. Receved 14 December 2013; Accepted (n revsed verson) 28 June 2014 Abstract. In ths paper, a prmal-dual nteror pont method s proposed for general constraned optmzaton, whch ncorporated a penalty functon and a knd of new dentfcaton technque of the actve set. At each teraton, the proposed algorthm only needs to solve two or three reduced systems of lnear equatons wth the same coeffcent matrx. The sze of systems of lnear equatons can be decreased due to the ntroducton of the workng set, whch s an estmate of the actve set. The penalty parameter s automatcally updated and the unformly postve defnteness condton on the Hessan approxmaton of the Lagrangan s relaxed. The proposed algorthm possesses global and superlnear convergence under some mld condtons. Fnally, some prelmnary numercal results are reported. AMS subject classfcatons: 65K05, 90C30, 90C31 Key words: general constraned optmzaton, prmal-dual, actve set, global convergence, superlnear convergence. 1. Introducton Consder the followng general constraned optmzaton problem (P) mn f 0 (x) s.t. f (x) 0, I = {1,2,...,m}, f (x) = 0, L = {m+1,m+2,...,m+l}, Correspondng author. Emal addresses: janjb@gxu.edu.cn (J. B. Jan) janlngl@126.com (J.L. L), @qq.com (J. Lv), c 2015 Global-Scence Press

2 314 J.L. L, J. Lv and J.B. Jan where functons f 0,f ( I L) : R n R are all contnuously dfferentable. It s well known that sequental quadratc programmng (SQP) methods s one of effcent methods for constraned optmzaton. The nterested readers are referred to Boggs et al. [1] and Gll et al. [5] for more nformaton on SQP methods. However, SQP methods have to solve QP subproblems at each teraton, whch s computatonally expensve. Paner et al. n [10] presented a feasble QP-free algorthm (Algorthm PTH for short) for (P) only wth nequalty constrants. At each teraton, only three systems of lnear equatons (SLEs) need to be solved. Specfcally, for the current teratve pont x k, the master search drecton d k s generated by solvng two SLEs n (d,λ) wth the followng form ) ( f0 (x k ) ( Hk A k Z k A T k G k )( d λ = η ), (1.1) where H k s an approxmaton of the Hessan of the Lagrangan assocated wth (P), the vector η s chosen n dfferent ways, A k = ( f (x k ), I), G k = dag(f (x k ), I), Z k = dag(z k, I) and zk ( I) the current estmate of the KKT multplers. In order to avod the Maratos effect, the search drecton s modfed by solvng a least squares subproblem, whch s equvalent to an SLE. The global convergence of Algorthm PTH requres a strong assumpton,.e., the number of statonary ponts s fnte. Algorthm PTH was later mproved by Gao et al. n [6]. An extra SLE s solved and the algorthm globally converges to a KKT pont under some condtons ncludng a strong assumpton that the multpler sequence s bounded, whch s mpossble f the teraton matrx s ll-condtoned. Afterwards, based on the Fscher-Burmester functon and the KKT nonsmooth equaton system, Q and Q n [12] also mproved Algorthm PTH and proposed a new feasble QP-free algorthm. At each teraton, three SLEs, whch dffer from those n [10], are requred to solve. It s shown that the coeffcent matrx of the SLE s unformly nonsngular and the multpler approxmaton sequence s bounded, even f the strct complementarty s not satsfed. Wthout assumng the solatedness of the statonary ponts, the global convergence s proved under some sutable condtons. As we know, prmal-dual nteror pont methods are also one of effcent soluton methods for problem (P) (see, e.g., [2,11,13,15]). Especally, Bakhtar and Tts n [2] proceeded along the lnes of Algorthm PTH and proposed a smply feasble prmal-dual nteror-pont method (Algorthm BT for short). For completeness, we frst gve a short ntroducton about feasble prmal-dual nteror pont (PDIP) methods. The KKT system of problem (P) only wth nequalty constrants s gven as follows { x L(x,λ) = 0, (1.2) λ f (x) = 0, λ 0, f (x) 0, I, where the Lagrangan functon s gven byl(x,λ) = f 0 (x)+ I λ f (x). The key pont of PDIP teraton s a perturbed varant of (1.2) n (x,λ) as follows { x L(x,λ) = 0, (1.3) λ f (x) = µ, I.

3 Globally and Superlnearly Convergent Prmal-dual Interor Pont Method 315 The vector µ R m s the barrer parameter vector wth component µ < 0, I. The man dea s then to attempt to solve (1.3) by means of Newton or quas-newton teratons, whle drvng µ to zero and enforcng prmal and dual (strct) feasble at each teraton. Specfcally, the followng SLE n ( x, λ) s consdered at par (x k,λ k ) { Hk x+ f (x k ) λ + f 0 (x k )+ (x I I f k )λ k = 0, λ k f (x k ) T x+f (x k ) λ +λ k f (x k ) = µ, I. The system above can be equvalently rewrtten as ( )( ) ( H k A k x f0 (x dag(λ k )A T k G k λ+λ k = k ) µ ), (1.4) whch s the same as (1.1) except that each component of µ s negatve. Now let us go back to [2]. The master search drecton of Algorthm BT s yelded by solvng only two SLEs smlar to (1.4), and a hgher correcton drecton s obtaned by solvng a least squares subproblem to avod the Maratos effect. The barrer parameter vector µ s carefully constructed to guarantee that Algorthm BT possesses some good propertes as follows: (1) the objectve functon value decreases at each teraton; (2) the postve defnte restrcton on the Hessan estmate s relaxed; (3) the ntal pont may le on the boundary of the feasble set, not necessary n the strct nteror of the feasble set; (4) Algorthm BT possesses global convergence under mlder condtons. The algorthms mentoned above share some common features: all the nequalty constrants and ther gradents are nvolved n the assocated SLEs. Consequently, the sze of the SLEs may be very large. Furthermore, the coeffcent matrx of the thrd SLE dffers from that of the frst two SLEs, whch leads to ncreasng computaton cost. Algorthm PTH and Algorthm BT are only two-step superlnear convergent. Tts et al. n [13] extended Algorthm PTH to solve problem (P) wth equalty and nequalty constrants by ncorporatng a modfed Mayne-Polak scheme (see [9]). In ther algorthm (Algorthm TWBUL for short), the postve defnteness assumpton on the Hessan estmate s relaxed and all the convergence results obtaned n [10] stll hold. Moreover, they developed a smpler penalty parameter update rule. Smlar to Algorthm PTH and Algorthm BT, Algorthm TWBUL also has the features descrbed above. Recently, based on the dea of PDIP and a new dentfcaton technque of the actve set, Jan et al. n [8] proposed a feasble QP-free algorthm for problem (P) only wth nequalty constrants. The sze of SLEs solved at each teraton s smaller than that of (1.1) and (1.4) due to the use of the workng set, whch s an estmate of the actve set. Moreover, the three SLEs solved at each teraton have the same coeffcent matrx. The algorthm possesses global and superlnear convergence under sutable condtons, ncludng the unformly postve defnteness of the Hessan estmate and the strct complementarty. In ths paper, motvated from the dea of [8], we propose a new prmal-dual method for problem (P) wth equalty and nequalty constrants, whch ncorporates a penalty

4 316 J.L. L, J. Lv and J.B. Jan functon and a new dentfcaton technque of the actve set. The man propertes of the proposed algorthm are as follows: the penalty functon s a sem-penalty functon,.e., only equalty constrants are penalzed and the penalty parameter s automatcally updated at each teraton; the sze of SLEs s smaller than that of (1.1) and (1.4) due to the ntroducton of the workng set; and the coeffcent matrces of the three SLEs solved at each teraton are the same. As a consequence, the computaton cost s decreased; the unformly postve defnteness assumpton on the Hessan estmate s weakened, whch s used n most of exstng algorthms, such as [8,10,12] ; the algorthm possesses global convergence and one-step superlnear convergence under some mld condtons. The remander of the paper s organzed as follows. The new algorthm s descrbed n detal and ts feasblty s dscussed n Secton 2. The global and superlnear convergence are analyzed n Secton 3 and Secton 4, respectvely. Numercal experments and results are stated n Secton 5 and Fnally, some concludng remarks are gven n Secton The algorthm In ths secton, we wll present our algorthm and then dscuss ts mportant propertes. For the sake of smplcty, we use the followng natatons throughout the paper: X = {x R n : f (x) 0, I; f (x) = 0, L}, X = {x R n : f (x) 0, I}, Xo = {x R n : f (x) < 0, I}, I(x) = { I : f (x) = 0}, f J (x) = (f (x), J I L), g (x) = f (x), {0} I L, A J (x) = (g (x), J I). Before proceedng, we state a basc assumpton throughout the paper. (A1) Gradent vectors {g (x), L} are lnearly ndependent for each x X o. Remark 2.1. From our algorthm gven n the sequel, all teraton ponts satsfy strct nequalty constrants, so the assumpton (A1) s weaker than the standard LICQ. In order to estmate the actve constrants, we defne a vector-valued functon by the way n [14]: x L(x,λ(x),γ(x)) Φ(x,λ(x),γ(x)) = mn{ f I (x),λ(x)}, f L (x) where L(x,λ,γ) s the Lagrange functon of problem (P),.e., L(x,λ,γ) = f 0 (x)+ I λ f (x)+ Lγ f (x).

5 Globally and Superlnearly Convergent Prmal-dual Interor Pont Method 317 Further, we defne an optmal dentfcaton functon for problem(p) asϕ: R n+m+l R : ϕ(x,λ(x),γ(x)) = Φ(x,λ(x),γ(x)) ζ, (2.1) where ζ (0,1), denotes the Eucldean norm. It s clear that (x,λ,γ ) s a KKT par of problem (P) f and only f ϕ(x,λ,γ ) = 0. Moreover, we know from [14] that the ndex set { I : f (x k ) + ϕ(x k,λ(x k ),γ(x k )) 0} s an exact dentfcaton set for the actve set I(x ) f the par (x k,λ(x k ),γ(x k )) s suffcently close to a KKT par (x,λ,γ ), and the Mangasaran-Fromovtz constrant qualfcaton (MFCQ) and the second order suffcent condtons are satsfed at a KKT par (x,λ,γ ). In ths paper, n order to reduce computaton cost, nspred from the dea of [14], for the current teraton pont x k X o, the multpler vectors λ(x k ) and γ(x k ) are chosen as follows: λ(x 0 ) = z 0, γ(x 0 ) = 1; λ(x k ) = λ k 1, γ(x k ) = γ k 1, k 1, where z 0 > 0, and λ k 1, γ k 1 are computed n the (k 1)-th teraton (see Algorthm 2.1 below). So we defne the workng set I k by I k = { I : f (x k )+ϕ(x k, λ(x k ), γ(x k )) 0}. (2.2) Wth the help ofi k, only two or three reduced SLEs wth the same coeffcent matrx are solved at each teraton n our algorthm, and the nactve constrants at a KKT pont x wll be neglected after fnte number of teratons. In vew of the equalty constrants, the coeffcent matrces of the SLEs n our algorthm are constructed wth the form ( ) Hk A k B k M(x k, z k, H I k ) = Z k A T k G k 0, (2.3) k Bk T 0 0 where H k s an approxmaton of the Hessan of the Lagrangan L(x,λ(x),γ(x)), and A k = A Ik (x k ) = (g (x k ), I k ), B k = (g (x k ), L), z k I k R I k, Z k = dag(z k I k ), G k = G Ik (x k ) = dag(f Ik (x k )), where I k means the cardnalty of I k. The search drectons of our algorthm are generated by solvng two or three SLEs of the form d g 0 (x k ) M(x k, z k, H I k ) λ Ik = µ k k γ ς k wth dfferent vectors µ k and ς k. In ths paper, the mert functon for arc search s a sem-penalty functon defned by Ψ(x,r) = f 0 (x)+r L where r > 0 s a penalty parameter. Now, the proposed algorthm s descrbed n detal. f (x), (2.4)

6 318 J.L. L, J. Lv and J.B. Jan Algorthm 2.1. (Prmal-dual Interor Pont Method) Parameters: α (0, 1 2 ), σ, β, θ, ζ (0,1), τ (2,3), r 0 > 0,ν > 2, ρ 1 > 0, ρ 2 > 0, M > 0, p > 0 and z max > z mn > 0. Data: z 0 R m wth z 0 j (z mn,z max ], j I,H 0 R n n s a symmetrc and postve defnte matrx; a startng pont x 0 satsfes x 0 X 0, or x 0 X such that vectors {g (x 0 ), I(x 0 ) L} are lnearly ndependent. Set k = 0. Step 1. Generate the workng set I k and matrx H k. Compute ϕ(x k,λ(x k ),γ(x k )) by (2.1), then generate the workng seti k by (2.2). Fork 1, compute a symmetrc matrx H k such that t s an approxmaton of the Lagrange Hessan of the problem (P) such that matrx def W k = H k z k f (x k ) g (x k )g (x k ) T I k s postve defnte. Step 2. Compute the search drectons. Denote M k = M(x k, z k I k, H k ). () Compute ( d k, λ k I k, γ k ) by solvng the frst SLE n (d,λ,γ): M k d λ Ik γ = g 0 (x k ) 0 f L (x k ). (2.5) Set λ k = ( λ k I k,0 I\Ik ). If d k = 0 and λ k I k 0, then x k s a KKT pont of (P), stop. () Compute (d k,λ k I k,γ k ) by solvng the second SLE n (d,λ,γ): where M k d λ Ik γ = g 0 (x k ) µ k I k f L (x k ), (2.6a) µ k = (1 β k)φ k +β k( d k ν ψ k ν )z k, I k; (2.6b) ψ k = g 0 (x k ) T dk λk z k φ k ( ) γ k f (x k )+ f (x k ) ; (2.6c) I k L { φ k = mn 0, (max{ λ k,0}) p Mf } (x k ), I k ; (2.6d) { 1, f bk 0; β k = mn{ (1 θ) ψ k (2.6e) b k,1}, f b k > 0; b k = ( ) λk z k ( d k ν + ψ k ν )z k +φ k. (2.6f) I k

7 Globally and Superlnearly Convergent Prmal-dual Interor Pont Method 319 Step 3. Update the penalty parameter. Denote η k = max {1 L + 2 γk γ k, γk }, and set r k = { rk 1, f r k 1 η k +ρ 1 ; max{r k 1,η k }+ρ 1, otherwse. (2.7) Step 4. Tral of unt step-length. If Ψ(x k +d k, r k ) Ψ(x k, r k )+αγ(x k,d k,r k ), f (x k +d k ) < 0, I, (2.8a) (2.8b) then set the step sze t k 1 and the hgher correcton drecton d k = 0, enter Step 7. Here, the functon Γ(x k,d k,r k ) s defned by Γ(x k,d k,r k ) = g 0 (x k ) T d k r k f (x k ). (2.9) Step 5. Generate correcton drecton. Compute ( d k, λ k I k, γ) by solvng the thrd SLE n (d,λ,γ): M k d λ Ik γ = 0 µ k I k ς k L, (2.10a) where µ k = { δ k z kf (x k +d k ) (zk )2 β λ k k ψ k ν, f λ k 0, I k; δ k z kf (x k +d k ), f λ k = 0, I k; ς k = d k τ f (x k +d k ), L; { δ k = max d k τ, d k 2 max { 1 zk σ } I k,λ k 0 λ k } (2.10b) (2.10c). (2.10d) If d k > d k, set d k = 0. Step 6. Perform arc search. Lett k be the frst numbertof the sequence{1, β, β 2,...} satsfyng the followng nequaltes: Ψ(x k +td k +t 2 dk, r k ) Ψ(x k,r k )+αtγ(x k,d k,r k ), f (x k +td k +t 2 dk ) < 0, I. (2.11a) (2.11b) Step 7. Update. Set x k+1 = x k + t k d k + t 2 k d k, z k+1 = mn{max{ d k 2 + z mn,λ k },z max}, I. Set k := k+1, go back to Step 1.

8 320 J.L. L, J. Lv and J.B. Jan Remark 2.2. (1) ψ k gven n Step 2() plays an mportant role n verfyng that Algorthm 2.1 s well defned. We can prove thatx k s a KKT pont whenψ k = 0 (see Lemma 2.2 below). (2) µ k I k defned n Step 2() s used to generate a sutable search drecton d k, whch can keep not only x k stayng n X o but also Ψ(x,r) decreasng. Remark 2.3. The updatng rule of z k n Step 7 plays a key role n ensurng that the coeffcent matrx M k s unformly nonsngular. Wth ths am, z k can be updated n dfferent ways, such as z k+1 = mn{max{z mn,λ k },z max}, I. Remark 2.4. Snce d k s the soluton to the SLE (2.5), Γ(x k,d k,r k ) defned n (2.9) s dentcal to the drectonal dervatve of Ψ(,r k ) at x k along the drecton d k. In what follows, we dscuss some features and feasblty of Algorthm 2.1. It s not dffcult to show that the followng concluson holds true. Lemma 2.1. Suppose that the assumpton (A1) holds and W k s postve defnte. Then the coeffcent matrx M k s nonsngular and the SLEs (2.5), (2.6a) and (2.10a) have a unque soluton, respectvely. The followng results play a crucal role n analyzng the feasblty of Algorthm 2.1. Lemma 2.2. Suppose that the stated assumptons n Lemma 2.1 hold. Then () g 0 (x k ) T dk γ kf (x k ) = ( d k ) T W k ( d k ); L () g 0 (x k ) T d k θψ k + (1+ 2 λ k λk ) f (x k ), θ (0,1); L () ψ k 0. If ψ k = 0, then x k s a KKT pont of the problem (P). Proof. () It follows from the SLE (2.5) that H k dk + λk g (x k )+ γ k g (x k ) = g 0 (x k ), I k L z k g (x k ) T dk + λ k f (x k ) = 0, I k, g (x k ) T dk = f (x k ), L. (2.12a) (2.12b) (2.12c) Further, t follows from the above equaltes that g 0 (x k ) T dk = ( d k ) T H k dk + I k whch together wth the defnton of W k gves g 0 (x k ) T dk L z k f (x k ) ( d k ) T g (x k )g (x k ) T dk + γ k f (x k ), L γ k f (x k ) = ( d k ) T W k ( d k ). That s, the result () holds.

9 Globally and Superlnearly Convergent Prmal-dual Interor Pont Method 321 () One has from the SLE (2.6a) H k d k + I k λ k g (x k )+ Lγ k g (x k ) = g 0 (x k ), (2.13) z k g (x k ) T d k +λ k f (x k ) = µ k, I k; g (x k ) T d k = f (x k ), L. (2.14) Thus t follows from (2.13) and (2.12a) that g 0 (x k ) T d k g 0 (x k ) T dk = λ k g (x k ) T dk λk g (x k )d k + I k I k Lγ kg (x k ) T dk γ kg (x k )d k. L Further, combnng wth (2.12b) and (2.14), t follows that g 0 (x k ) T d k = g 0 (x k ) T dk I k λk z k = g 0 (x k ) T dk λk z k φ k +β λ k k I k z k I k ( ) (1 β k )φ k +β k( d k ν ψ k ν )z k + L( γ k γk )f (x k ) ( ) ( d k ν + ψ k ν )z k +φk + L( γ k γk )f (x k ) ψ k +β k b k + L(1+ 2 γ k γk ) f (x k ). (2.15) On the other hand, t follows from the concluson () and Step 7 that g 0 (x k ) T dk L γ kf (x k ) = ( d k ) T W k dk f (x k ) 2 ( d k ) T W k dk 0, L and t follows from the defnton (2.6d) of φ k that defnton (2.6c) of ψ k, we can conclude that λ k φ k z I k k 0. Therefore, from the ψ k 0. (2.16) Notng that β k = 1 for b k 0 and θ (0,1), t follows from (2.15) that g 0 (x k ) T d k θψ k + L(1+ 2γ k γ k ) f (x k ). Notng that β k = mn{ (1 θ) ψ k b k,1} for b k > 0, t follows that β k b k (θ 1)ψ k, whch together wth (2.15) gves g 0 (x k ) T d k θψ k + L(1+ 2 γ k γ k ) f (x k ). Hence, the result () holds true.

10 322 J.L. L, J. Lv and J.B. Jan () The concluson ψ k 0 s true from (2.16). If ψ k = 0, then t mples from (2.16) and (2.6c) that ( d k ) T W k dk = 0, λ k z k I k φ k = 0, f (x k ) = 0. (2.17) So we obtan that d k = 0 from the postve defnteness of W k and that f (x k ) = 0 ( L). Further, together wth (2.12b) and d k = 0, we have λ k f (x k ) = 0, I k. In vew of the second equalty of (2.17) and λ k φk 0, I k, we obtan L λ k φk = 0, I k. (2.18) We can show that λ k 0 for any I k. In fact, by contradcton, f there exsts some 0 I k such that λ k 0 < 0, then f 0 (x k ) = 0. Therefore, from (2.6d), we have φ k 0 < 0. It mples that λ k 0 φ k 0 > 0. It s a contradcton to (2.18). Hence, from d k = 0, λ k I k 0 and the SLE (2.5), we can conclude that x k s a KKT pont of (P). Based on the above dscusson, we now verfy the feasblty of Algorthm 2.1. Lemma 2.3. Suppose that the assumpton (A1) holds. Then () If x k s not a KKT pont of (P), then Γ(x k,d k,r k ) < 0; () Algorthm 2.1 s well defned. Proof. () From the defnton (2.9) of Γ(x k,d k,r k ) and Lemma 2.2(), we have Γ(x k,d k,r k ) θψ k + (1+ 2 γ k γk ) f (x k ) r k f (x) L L θψ k L(r k 1 2 γ k γ k ) f (x k ), whch together wth (2.7) and Lemma 2.2() gves Γ(x k,d k,r k ) θψ k < 0. (2.19) () In order to verfy the result (), t s suffcent to show that the nequaltes (2.11a) and (2.11b) hold for t > 0 suffcently small. Let s frst consder the nequalty (2.11a). From the defnton (2.4) of Ψ, we have Ψ(x k +td k +t 2 dk,r k ) Ψ(x k,r k ) = tg 0 (x k ) T d k +o(t)+r k ( f (x k )+tg (x k ) T d k +o( td k +t 2 dk ) f (x k ) ). L From the SLE (2.6a), we have g (x k ) T d k = f (x k ), L. Substtutng ths formula nto the above nequalty, we obtan Ψ(x k +td k +t 2 dk,r k ) Ψ(x k,r k ) = tg 0 (x k ) T d k +r k ( (1 t)f (x k ) f (x k ) )+o( td k +t 2 dk ). L

11 Globally and Superlnearly Convergent Prmal-dual Interor Pont Method 323 Note that 1 t > 0 for t suffcently small. It follows from the above nequalty that Ψ(x k +td k +t 2 dk,r k ) Ψ(x k,r k ) = tg 0 (x k ) T d k tr k f (x k ) +o( td k +t 2 dk ) L = tγ(x k,d k,r k )+o( td k +t 2 dk ) αtγ(x k,d k,r k )+o( td k +t 2 dk ), (2.20) the last nequalty s duo to (2.19) and α (0, 1 2 ). Therefore, there exsts a t k > 0 such that (2.11a) holds for all t (0, t k ). Next consder the nequalty (2.11b). Snce f (x k ) < 0 for I, the nequalty (2.11b) follows for t suffcently small due to the contnuty of f. By summarzng the above dscusson, there exsts a t k > 0 such that (2.11a) and (2.11b) hold for all t (0, t k ). Consequently, Algorthm 2.1 s well defned. 3. Global convergence In ths secton, we wll show that Algorthm 2.1 s globally convergent,.e., any accumulaton pont of the generated sequence {x k } s a KKT pont for (P). For ths goal, we need the followng assumptons. (A2) () The sequence {x k } generated by Algorthm 2.1 s bounded. () For each accumulaton pont x of {x k }, vectors {g (x ), I(x ) L} are lnearly ndependent. () There exst two postve constants a and b such that H k b, d T W k d a d 2, d R n, k. (3.1) In what follows, we frst present several Lemmas whch are used n the proof of global convergence. Lemma 3.1. Suppose that the assumptons (A1) and (A2) hold. Then () there exsts a c > 0 such that M 1 k c; () the sequences {( d k, λ k, γ k )}, {(d k,λ k,γ k )} and {( d k, λ k, γ k )} are all bounded. The proof s smlar to that of Lemma 3.1 n [8] and omtted here. The followng lemma can be shown from the updatng formula (2.7). Lemma 3.2. Suppose that the assumptons (A1) and (A2) hold. Then there exsts a postve nteger k 0 such that r k r k0 def = r for all k k 0. Based on Lemma 3.2, n the rest of ths paper we suppose, wthout loss of generalty, that r k r. By Lemma 3.1, (A2)() and Lemma 2.2(), we can show that the followng concluson s true, whch facltates the proof of global convergence.

12 324 J.L. L, J. Lv and J.B. Jan Lemma 3.3. Suppose that the assumptons (A1) and (A2) hold. Let x be an accumulaton pont of the sequence {x k } generated by Algorthm 2.1 and suppose that subsequence {x k } K x. If {ψ k } K 0, then x s a KKT pont of (P) and {( λ k, γ k )} K converges to the unque multpler vector (λ,γ ) correspondng to x. At the end of ths secton, based on the above results, we present a concluson of global convergence of Algorthm 2.1. Theorem 3.1. Suppose that the assumptons (A1) and (A2) hold and x s an accumulaton pont of the sequence {x k } generated by Algorthm 2.1. Then x s a KKT pont of (P),.e., Algorthm 2.1 s globally convergent. Proof. Let sequence {x k } K converge to x. Wthout loss of generalty, we suppose that {ψ k } K ψ. It s clear from Lemma 2.2() that ψ 0. We suppose, by contradcton, that x s not a KKT pont of (P). Then, by Lemma 3.3, one obtansψ < 0. The remnant of the proof s dvded nto two steps as follows. Step A. We show the nequaltes (2.11a) and (2.11b) hold for k K suffcently large and t > 0 suffcently small. We frst prove the nequalty (2.11b),.e., f (x k +td k +t 2 dk ) < 0, I. () For I \ I(x ),.e., f (x ) < 0. The nequalty (2.11b) follows from the contnuty of f and the boundedness of {(d k, d k )} K. () For I(x ). In vew of Lemma 3.1(), wthout loss of generalty, we can suppose that λ k 1 λ, γ k 1 γ, k K. Thus, (x, λ, γ ) s not a KKT par of problem (P), further, we have ϕ(x, λ, γ ) > 0. Snce I(x ) I k, from the Taylor expanson, (2.6a), (2.6b), Lemma 3.1() and (2.6e), one gets for t suffcently small f (x k +td k +t 2 dk ) = f (x k )+t µk λk f (x k ) z k = (1 t λk z k +o(t) )f (x k )+t 1 β k z k φ k β k t( d k ν + ψ k ν )+o(t). (3.2) The boundedness of {λ k } and {zk } mples 1 tλk 0. It s clear that z k ( ) 1 t λk z k f (x k ) 0, t 1 β k z k φ k 0, (3.3) whch shows the followng nequalty: f (x k +td k +t 2 dk ) β k t( d k ν + ψ k ν )+o(t). (3.4) In vew of (2.6e), (2.6f) and ψ < 0, there exsts a scalar β > 0 such that β k β > 0. From Lemma 3.1 and Lemma 3.3, there exsts a subsequence K K and a scalar > 0 such that nf k K ( d k ν + ψ k ν ) > 0.

13 Globally and Superlnearly Convergent Prmal-dual Interor Pont Method 325 So, t follows, for k suffcently large and t > 0 suffcently small, that f (x k +td k +t 2 dk ) 1 β t+o(t) < 0. (3.5) 2 Hence, the nequalty (2.11b) holds true. Next, we consder the nequalty (2.11a). From (2.20) and the boundedness of {d k, d k }, we have Ψ(x k +td k +t 2 dk,r) Ψ(x k,r) αtγ(x k,d k,r) = tγ(x k,d k,r) αtγ(x k,d k,r)+o( td k +t 2 dk ) (1 α)tθψ k +o(t) (1 α)tθ αψ +o(t), (3.6) where α (0, 1). The above nequalty mples that the nequalty (2.11a) holds for k K suffcently large and t > 0 suffcently small. By summarzng the above dscusson, there exsts a constant t > 0 such that the step sze t k t for all k K. Step B. Brng a contradcton based on t k t > 0 (k K ). Notng that lm k K Ψ(xk,r) = Ψ(x,r) and the monotonc property of {Ψ(x k,r)}, one obtans lm k Ψ(xk,r) = Ψ(x,r). In vew of Step 6 n Algorthm 2.1 and (2.19), t follows for k K suffcently large that Ψ(x k+1,r) Ψ(x k,r) αt k Γ(x k,d k,r) αθψ k t. (3.7) Passng to the lmt k K n the nequalty above, one gets αθψ t 0, whch s a contradcton. Hence, x s a KKT pont of (P). The whole proof s completed. 4. Strong and superlnear convergence In ths secton, we frst analyze the strong convergence of Algorthm 2.1 and then analyze the acceptance of a full step of one for all k suffcently large. Fnally, we establsh the superlnear convergence under some mld assumptons. In the analyss of ths secton, the followng assumptons are necessary. (A3) Functons f 0, f ( I L) are all twce contnuously dfferentable.

14 326 J.L. L, J. Lv and J.B. Jan (A4) The sequence {x k } generated by Algorthm 2.1 possesses an accumulaton pont x, at whch the second order suffcent condtons and the strct complementarty hold,.e., the Lagrange multpler vector λ E m at x satsfes λ > 0 ( I(x )) and the Hessan matrx 2 xx L(x,λ,γ ) s postve defnte on the subspace {y R n y 0, g (x ) T y = 0, I(x ) L}. Lemma 4.1. Suppose that the stated assumptons (A1)-(A4) hold. For any subset K of {1,2, } such that lm K x k = x, then () lm K ( d k, λ k, γ k ) = (0,λ,γ ); () lm K (d k,λ k,γ k ) = (0,λ,γ ). Proof. () Snce {( d k, λ k, γ k )} and {Z k } are bounded, n vew of (A4) and I k I and the fnteness of I, wthout loss of generalty, suppose that there exsts an ndex set Ī such that I k Ī, k K, and there exsts an nfnte subset K K such that H k H, ( λ k, λ k 1 ) ( λ, λ ), ( γ k, γ k 1 ) ( γ, γ ), d k d, Z Ik Z Ī, G I k GĪ def = dag(fī(x )), k K. If (x, λ, γ ) s a KKT par of (P), then λ = λ, γ = γ, and t follows from Theorem 2.3 and Theorem 3.7 n [4] that I k I(x ). If (x, λ, γ ) s not a KKT par of (P), then ϕ(x, λ, γ ) > 0. Ths mples that I(x ) I k, k K. Takng the lmt as k, k K n (2.5), we deduce that( d, λ, γ) solves the followng SLE n(u,v,v ) Ī 1 2 H AĪ(x ) B L (x ) Z Ī A Ī (x ) T GĪ(x ) 0 B L (x ) T 0 0 u v 1 v 2 = g 0 (x ) 0 f L (x ). (4.1) Further, smlar to the proof of Lemma 3.1 n [8], we can show that the coeffcent matrx of (4.1) s nonsngular. So the soluton of (4.1) s unque. On the other hand, notng that (x,λ,γ ) s a KKT par of (P) from Theorem 3.4, we know that (0,λ Ī,γ ) s a soluton of (4.1), too. So we conclude that ( d, λī, γ) = (0,λ Ī,γ ) due to the unqueness. Hence, lm K ( d k, λ k, γ k ) = (0,λ,γ ). () For any subsetk such thatx k x ask,k K, n vew ofµ k I k 0(k K), smlar to the proof of the concluson (), we can get lm K (d k,λ k,γ k ) = (0,λ,γ ). Next we present the strong convergence of Algorthm 2.1 and some assocated conclusons. Theorem 4.1. Suppose that the stated assumptons (A1)-(A4) hold. Then the followng conclusons hold true: () the entre sequence{x k } yelded by Algorthm 2.1 converges tox,.e., Algorthm 2.1 s strongly convergent;

15 Globally and Superlnearly Convergent Prmal-dual Interor Pont Method 327 () ( d k, λ k, γ k ) (0,λ,γ ) as k ; () φ k I k = 0, µ k I k = β k ( d k ν ψ k ν )z k I k and I k I(x ) for k suffcently large; (v) (d k,λ k,γ k ) (0,λ,γ ) as k ; (v) z k mn{max{z mn,λ },z max } as k, where = (1,1,,1) T R m. The proof s smlar to that of Theorem 4.1 n [8] and omtted here. We now present some features of the soluton of the SLE (2.10a), whch wll be used n the analyss of a full step of one n the sequel. Lemma 4.2 Suppose that the stated assumptons (A1)-(A4) hold. If ρ 2 mn{λ, I(x )} and z max max{λ, I(x )}, then the soluton of the SLE (2.10a) satsfes the followng relatons: ( ) d k = O max 1 d k, d k 2 } = o( d k ); (4.2) I(x ) { zk λ k ( λ k = O max I(x ) { zk λ k ) 1 d k, d k 2 } ; (4.3) d k d k = o(δ k ), λ k d k = o(δ k ). (4.4) Proof. It follows from the SLEs (2.5) and (2.6a) that z k g (x k ) T dk = λ k f (x k ), I k, g (x k ) T d k = f (x k ), L. (4.5a) (4.5b) So t s clear that f (x k ) = O( d k ), I k ; f (x k ) = O( d k ), L. In vew of φ Ik 0 for k suffcent large, t follows from (2.6c) that ψ k = g 0 (x k ) T dk L γ k f (x k ) L f (x k ). (4.6) From Lemma 2.2() and (A1), we obtan g 0 (x k ) T dk L γ k f (x k ) = O( d k ρ ), (ρ [1,2]), so (4.6) s equvalently rewrtten as ψ k = O(max{ d k, d k ρ }). (4.7) Agan, from the SLEs (2.6a) and (2.5), one obtans d k d k 0 M k λ Ik λ Ik γ γ = µ k I k 0. (4.8)

16 328 J.L. L, J. Lv and J.B. Jan Together wth µ k I k = β k ( d k ν ψ k ν )z k I k, (4.7) and Lemma 3.1, the above equalty mples that there exsts a postve constant c such that d k d k c( d k ν + ψ k ν ) = c d k ν +O(max{ d k ν, d k ρν }). (4.9) Hence, one obtans d k d k. In vew of the strct complementarty,.e., λ > 0, I(x ) = I k, t follows from (2.10b) for any I(x ), that µ k = δ k z k [f (x k )+g (x k ) T d k] (zk )2 β k ψ k ν +O( d k 2 ). (4.10) λ k Agan, t follows from the SLE (2.6a) that or equvalently, z k g (x k ) T d k +f (x k )λ k = µ k = β k( d k ν ψ k ν )z k = O( d k 2 ) β k ψ k ν z k, f (x k ) = zk λ k g (x k ) T d k 1 λ k β k ψ k ν z k +O( d k 2 ). Substtutng ths equalty nto (4.10), we obtan µ k = δ k z k (1 zk λ k )g (x k ) T d k + zk λ k = δ k z k (1 zk )g (x k ) T d k +O( d k 2 ) λ k ( = δ k +O max{ zk λ k β k ψ k ν z k (zk )2 λ k β k ψ k ν +O( d k 2 ) ) 1 d k, d k 2, I(x )} +O( d k 2 ), whch together wth (2.10d) shows that ( ) µ k = O max{ zk I k λ k 1 d k, d k 2, I(x )} = o( d k ). On the other hand, t follows from (2.10c) and the Taylor expanson that for any L ς k = d k τ f (x k ) g (x k ) T d k +O( d k 2 ) = d k τ +O( d k 2 ) = O( d k 2 ). The second equalty and the thrd equalty are due to (4.5a) andτ (2,3), respectvely. Thus, together wth Lemma 3.1(), the relatons (4.2) and (4.3) are mmedate. Further, the relaton (4.4) follows from (2.10b) and (2.10d). The proof s completed.

17 Globally and Superlnearly Convergent Prmal-dual Interor Pont Method 329 To ensure the step sze t k 1 for k suffcently large, the followng assumpton on the Hessan approxmaton H k s necessary. (A5) Assume P k ( 2 xxl(x k,λ k,γ k ) H k )d k = o( d k ), where P k = E (A k,b k )[(A k,b k ) T (A k,b k )] 1 (A k,b k ) T. (4.11) In what follows, we present an mportant result,.e., a full step of one s accepted. Theorem 4.2. Suppose that the stated assumptons (A1)-(A5) hold and z mn mn{λ, I(x )}, z max max{λ, I(x )}. Then, the arc search of Algorthm 2.1 eventually accepts a full step of one,.e., t k 1 for k suffcently large. Proof. In vew of the nequaltes (2.11a) and (2.11b), t s suffcent to verfy the followng nequaltes hold true. Ψ(x k +d k + d k,r) Ψ(x k,r)+αγ(x k,d k,r); (4.12) f (x k +d k + d k ) < 0, I. (4.13) (1) We frst prove the nequalty (4.13) holds. For I\I(x ),.e., f (x ) < 0, usng the contnuty of f and (x k,d k, d k ) (x,0,0) as k, the nequalty (4.13) s mmedate for k suffcently large. For I(x ),.e.,f (x ) = 0. We know from the strct complementarty thatλ > 0. It follows from the SLE (2.10a) and (2.10b) that z k g (x k ) T dk + λ k f (x k ) = δ k z k f (x k +d k ) (zk )2 λ k β k ψ k ν, whch combnng wth λ k > 0,β k > 0 and z k > 0 gves f (x k +d k )+g (x k ) T dk δ k z k λ k z k f (x k ) = δ k z k +o(δ k ). Therefore, by the Taylor expanson and the nequalty above, one has f (x k +d k + d k ) = f (x k +d k )+g (x k ) T dk +O( d k d k ) δ k z k +o(δ k ), whch together wth z k z mn > 0 shows that the nequalty (4.13) holds true. (2) In what follows, we analyze the nequalty (4.12). By the Taylor expanson and Lemma 4.2, we obtan w k := Ψ(x k +d k + d k,r) Ψ(x k,r) αγ(x k,d k,r) = g 0 (x k ) T (d k + d k )+ 1 2 (dk ) T 2 xxf 0 (x k )d k +r L f (x k )+g (x k ) T (d k + d k ) (dk ) T 2 xxf (x k )d k r L f (x k ) αγ(x k,d k,r)+o( d k 2 ). (4.14)

18 330 J.L. L, J. Lv and J.B. Jan From (2.6a), (2.10a) and (2.10c), one has g (x k ) T d k = f (x k ), g (x k ) T dk = d k τ f (x k +d k ), L, further, by the Taylor expanson, one obtans equvalently, g (x k ) T dk = d k τ f (x k ) g (x k ) T d k 1 2 (dk ) T 2 xxf (x k )d k +o( d k 2 ), L, f (x k )+g (x k ) T (d k + d k )+ 1 2 (dk ) T 2 xx f (x k )d k = o( d k 2 ), L. (4.15) On the other hand, t follows from (2.6a) and Theorem 4.1() that g 0 (x k )+H k d k + γ k g (x k ) = 0, whch together wth Lemma 4.2 gves g 0 (x k ) T d k = (d k ) T H k d k g 0 (x k ) T (d k + d k ) = I(x ) I(x )λ k g (x k )+ L I(x )λ k g (x k ) T d k L λ k g (x k ) T (d k + d k ) γ k g (x k ) T d k, (4.16a) Lγ k g (x k ) T (d k + d k ) (d k ) T H k d k +o( d k 2 ).(4.16b) Agan, from the SLE (2.10a) and (2.10b), t follows that for I(x ) and λ k 0 z k g (x k ) T dk + λ k f (x k ) = δ k z k f (x k +d k ) (zk )2 λ k β k ψ k υ. By Lemma 4.2, the above equalty can be rewrtten as f (x k +d k )+g (x k ) T dk = λ k z k f (x k ) δ k z k zk λ k β k ψ k ν = o( d k 2 ), whch together wth the Taylor expanson of f (x k +d k ) shows that g (x k ) T (d k + d k ) = f (x k ) 1 2 (dk ) T 2 xx f (x k )d k +o( d k 2 ), I(x ). (4.17) Agan, from the SLE (2.10a) and (2.10c), we easly obtan g (x k ) T (d k + d k ) = f (x k ) 1 2 (dk ) T 2 xx f (x k )d k +o( d k 2 ), L. (4.18)

19 Globally and Superlnearly Convergent Prmal-dual Interor Pont Method 331 Thus, n vew of (2.9), (4.15) and (4.17) (4.19), the equalty (4.14) s rewrtten as w k = I(x )λ k f (x k )+ γ k f (x k )+ 1 2 (dk ) T ( 2 xxl(x k,λ k,γ k ) H k )d k L 1 2 (dk ) T H k d k αg 0 (x k ) T (d k )+(α 1)r f (x k ) +o( d k 2 ). L It follows from the defnton (4.11) of the projecton matrx P k that d k = P k d k +d k 0, where d k 0 = (A k,b k )[(A k,b k ) T (A k,b k )] 1 (A k,b k ) T d k. Furthermore, t follows from the SLE (2.6a) and Theorem 4.1() that ( ) (A k,b k ) T d k Z 1 k = µk Z 1 I k k G kλ k I k f L (x k, d k 0 = O( f ) Ik (x k ) )+o( d k 2 ). Based on the above relatons, t follows from (A5) and (4.16a) that w k = (α 1 2 )(dk ) T W k d k +(α 1 2 ) + I(x ) +(α 1)r L I(x ) z k f (x k ) )(dk ) T g (x k )g (x k ) T d k λ k (1 αλk z k )f (x k )+ Lγ k (1 α)f (x k ) f (x k ) +o( f Ik )+o( d k 2 ), further, t follows from A2() and the SLE (2.6a) that ( w k α 1 ) ( a d k 2 + α 1 ) (λ k 2 2 )2f (x k ) z k I(x ) + ( ) λ k 1 α λk z k f (x k )+ Lγ k (1 α)f (x k ) I(x ) +(α 1)r L f (x k ) +o( f Ik )+o( d k 2 ). (4.19) Notng that α (0, 1 2 ) and r η k > γ k, we obtan whch together wth (4.19) shows ( w k α 1 ) a d k Lγ k (1 α)f (x k )+(α 1)r L I(x ) λ k f (x k ) < 0, ( 1 1 2λ k ) z k f (x k )+o( f Ik )+o( d k 2 ). (4.20)

20 332 J.L. L, J. Lv and J.B. Jan It follows from Theorem 4.1(v) that ( lm k λk 1 1 λ k ) 2z k = 1 2 λ > 0, I(x ), (4.21) and notng that f (x k ) < 0, I(x ), so t follows from (4.20), (4.21) and α (0, 1 2 ) that for k suffcent large w k (α 1 2 )a dk 2 +o( d k 2 ) < 0, whch means that the nequalty (4.12) holds for k suffcent large. The proof s fnshed. Based on Theorem 4.2, smlar to Theorem 5.2 n [16], we can obtan the concluson of the superlnear convergence of Algorthm 2.1. Theorem 4.3 Suppose that the stated assumptons (A1)-(A5) hold. Then x k+1 x = o( x k x ),.e., Algorthm 2.1 s superlnearly convergent. 5. Numercal experments In ths secton, we test Algorthm 2.1 on some problems whch are chosen from [7]. Algorthm 2.1 was coded by MATLAB and ran on a computer wth Wndows 7 and 2.20 GHz CPU. The detals about the mplementaton are descrbed as follows. (a) The Hessan estmate H k s updated as follows: H k = h k E n + 2 xxl(x k,λ k,γ k ), where 0, f kmn > 10 5 ; h k = kmn +10 5, f kmn 10 5 ; 2 kmn, otherwse, and kmn s the smallest egenvalue of the matrx U k def = 2 xxl(x k,λ k,γ k ) I k z k f (x k ) g (x k )g (x k ) T. It can be shown that the matrx W k defned by Step 7 satsfes the assumpton (A2)(). (b) The parameters n Algorthm 2.1 are chosen as follows: α = 0.45, σ = 0.01, β = 0.5, θ = 0.99, ξ = 0.5, τ = 2.5, ν = 3, z max = 10 5,

21 Globally and Superlnearly Convergent Prmal-dual Interor Pont Method 333 Table 1: The numercal results. Problem n (m, l) Algorthm 2.1 Algorthm n [17] Itr Nf Ng Aset Tme f fnal Itr Nf Ng Svanberg (30, 0) e Svanberg (90, 0) e Svanberg (150, 0) e Svanberg (240, 0) e Svanberg (300, 0) e Svanberg (1500, 0) e ε 0 = 10 5, M = 1, p = 1, ρ 1 = 0.1, ρ 2 = (c) Termnaton crtera. Algorthm 2.1 stops f one of the followng termnaton crtera s satsfed: () Φ(x k,λ(x k )) 10 5 ; () d k 10 5 and max{ λ k, I} In the numercal experment we compared our algorthm,.e., Algorthm 2.1, wth Algorthm n [17] and Algorthm n [13], respectvely. We use the ntal ponts gven n [7] f they are feasble. The numercal results are summarzed n Table 1 and Table 2, n whch the followng notatons are used: n: the number of varables; m: the number of nequalty constrants; l: the number of equalty constrants; Itr: the number of teratons; N f : the number of functon evaluatons for f; N g : the number of functon evaluatons for g ; Aset: the number of ndces n the fnal workng set; f fnal : the objectve functon value at the fnal teraton. We can know from tables 1 and 2 that the cardnalty of the fnal workng set Aset s generally smaller than the number of constrants, whch mples that subproblems of Algorthm 2.1 are generally smaller than that of the full dmensonal methods. Our workng set s very useful. Table 1 shows that the numbers of Itr and Nf n Algorthm 2.1 are much smaller than those n [17]. The numbers of Ng n Algorthm 2.1 are larger than those of Ng n [17] because we compute the number of g whch s needed durng the teratve procedure. Remark 5.1. The numercal results of Algorthm TWBUL are obtaned drectly from [13]. The number of teratons for Algorthm 2.1 s less than that of Algorthm n [13] for half of the test problems. Especally, for problem HS66 and problem HS111, the numbers of teratons of Algorthm TWBUL exceed 1000, whereas the numbers of teratons of Algorthm 2.1 are only 10 and 298, respectvely.

22 334 J.L. L, J. Lv and J.B. Jan Table 2: The numercal results. Problem n (m, l) Itr N f Algorthm 2.1 Algorthm n [13] N g Aset f fnal Itr f fnal HS2 2 (1, 0) e+00 HS3 2 (1, 0) e e 09 HS4 2 (1, 1) e e+00 HS8 2 (0, 2) e e+00 HS12 2 (1, 0) e e+01 HS24 2 (5, 0) e e+00 HS29 3 (1, 0) e e+01 HS33 3 (6, 0) e e+00 HS34 3 (8, 0) e e 01 HS35 3 (4, 0) e e 01 HS36 3 (7, 0) e e+03 HS37 3 (1, 7) e e+03 HS38 4 (8, 0) e e 24 HS43 4 (3, 0) e e+01 HS48 5 (0, 2) e e+00 HS49 5 (0, 2) e e 010 HS50 5 (0, 3) e e 017 HS66 3 (8, 0) e e 01 HS93 6 (8, 0) e e+02 HS100 7 (4, 0) e e+02 HS (20, 3) e e+01 HS (8, 0) e e+01 HS (20, 0) e e Concludng remarks Wth the help of a penalty functon and a new dentfcaton technque of actve constrants, we proposed a new prmal-dual nteror pont method for general constraned optmzaton. Because of the ntroducton of workng set, the sze of SLEs s smaller than that of SLEs n some exstng algorthms. At each teraton, the master search drecton and hgher-correcton drecton are generated by only solvng two or three reduced SLEs wth the same coeffcent matrx and the penalty parameter s automatcally updated. The unformly postve defnteness assumpton on the Hessan estmate s weaken, whch s used n most of exstng algorthms. The global and superlnear convergence are shown under some mld condtons. The prelmnary numercal results ndcates the feasblty and effcency of the proposed algorthm for small and medum sze test problems. Acknowledgments The authors thank the two anonymous referees and the Assocate Edtor, Xaoq Yang, for ther nsghtful comments and suggestons on the orgnal verson. Ths work s supported by the Natonal Natural Scence Foundaton No , the Natural Scence Foundaton of Guangx Nos. 2012GXNSFAA053007

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