Research Article An Improved Predictor-Corrector Interior-Point Algorithm for Linear Complementarity Problems with O nl -Iteration Complexity

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1 Joural of Applied Mathematics Volume 0, Article ID 3409, pages doi:0.55/0/3409 Research Article A Improved Predictor-Corrector Iterior-Poit Algorithm for Liear Complemetarity Problems with O L-Iteratio Complexity Debi Fag ad Qia Yu School of Ecoomics ad Maagemet, Wuha Uiversity, Wuha 43007, Chia School of Ecoomics, Wuha Uiversity of Techology, Wuha , Chia Correspodece should be addressed to Qia Yu, yuqia365@6.com Received 8 September 0; Revised 3 November 0; Accepted 4 November 0 Academic Editor: Chog Li Copyright q 0 D. Fag ad Q. Yu. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. This paper proposes a improved predictor-corrector iterior-poit algorithm for the liear complemetarity problem LCP based o the Mizuo-Todd-Ye algorithm. The modified corrector steps i our algorithm caot oly draw the iteratio poit back to a arrower eighborhood of the ceter path but also reduce the duality gap. It implies that the improved algorithm ca coverge faster tha the MTY algorithm. The iteratio complexity of the improved algorithm is proved to obtai O L which is similar to the classical Mizuo-Todd-Ye algorithm. Fially, the umerical experimets show that our algorithm improved the performace of the classical MTY algorithm.. Itroductio Sice Karmarkar published the first paper o iterior poit method i 984, the iterior poit methodologies have yielded rich theories ad algorithms i the fields of liear programmig LP, quadratic programmig QP, ad liear complemetarity problems LCP. Amog these iterior poit methods, predictor-corrector iterior-poit methods play a special role due to their best polyomial complexity ad superliear covergece. I 993, Mizuo et al. proposed the classical represetative of predictor-corrector method for liear programmig. The Mizuo-Todd-Ye MTY algorithm has O Literatio complexity which is the best iteratio complexity obtaied so far for all the iterior-poit method 3, 4. Moreover, Ye et al. 5 proved the duality gap of classical MTY algorithm coverges to zero quadratically, which dedicated that MTY algorithm has superliear covergece. The classical MTY algorithm is the first algorithm for LP has both polyomial complexity ad superliear covergece. So the classical MTY algorithm therefore was cosidered as the most efficiet iterior poit methods for LP.

2 Joural of Applied Mathematics Afterwards, the MTY algorithm was exteded to LCP 6, 7 for its excellet performace. I recet papers, Potra 8, 9 preseted several predictor-corrector methods for liear complemetarity problems actig i a wide eighborhood of the cetral path, ad Stoer et al. 0 proposed a predictor-corrector algorithm with arbitrarily high order of covergece to degeerate sufficiet liear complemetarity problems. Subsequetly, Potra proposed a geeralizatio of the MTY algorithm for ifeasible startig poits for mootoe LCP. Based o the so-called fast step-safe step strategy, Wright proposed a ifeasible iterior poit method with polyomial ad quadratic covergece for odegeerate LCP. Ad Ai ad Zhag preseted a O L-iteratio primal-dual path-followig method for mootoe LCP based o wide eighborhoods ad large updates 3. These workigs improved the MTY algorithm i various aspects. I this paper, the MTY algorithm for LCP will be improved by modifyig the corrector of the algorithm to obtai a larger iteratio reduce factor tha the origial, which ca guaratee a faster covergece. The typical iteratios of the MTY algorithm operate betwee two eighborhoods of the cetral path, ρ/4 ad ρ/. Before startig the MTY algorithm, a arbitrary iitial poit ω ρ/4 will be give. Ad the, the predictor produces a poit ω ρ/ by carefully choosig a steplegth alog the affie-scalig directio at ω ad reduces the primal dual gap by a factor of at least χ/ where χ / I the corrector steps, the corrector produces a poit ω ρ/4 by takig a uit steplegth alog the ceterig directio at ω. It is show that the corrector put the iteratio poits back to ρ/4 which is the arrower eighborhood of the cetral path ad maitais the same duality gap. Thus predictor-corrector reduces the duality gap μ k by a factor of at least χ/ ad holds the iteratio poits i the eighborhood ρ/4. It follows that the correspodig iterative procedure has O L-iteratio complexity. The modified algorithm i this paper has oly oe corrector step i eighborhood of the cetral path ρ/ ad oe predictor step i a arrow eighborhood ρ/4 same as the MTY algorithm. Moreover, the modified iteratio directio of corrector step ca make a reductio for duality gap. This improvemet attaied a larger iteratio reduce factor ad results i a faster covergece tha the classical MTY algorithm. Sectio describes the procedure of our predictor-corrector algorithm. Sectio 3 gives the covergece aalysis of our algorithm. It shows that the improved algorithm ca geerate a sequece {x k,y k } i the eighborhood ρ/4 from a arbitrary iitial poit x 0,y 0 ρ/4 ad coverge to optimal solutio x k,y k with O L-iteratio complexity. Fially, the performaces of our algorithm are evaluated by umerical experimets i Sectio 4.. Descriptio of the Algorithm I this paper, LCP is to fid a vector pair x, y R R such that y Mx h, x, y 0, 0 ad x T y 0, where h R ad M is a positive semidefiite matrix. Deote the set of all feasible poits of LCP by S {x, y Mx yh 0, x 0, y 0} ad the solutio set of LCP by S {x,y x,y S, x T y 0}. The relative iterior of S, S {x, y x, y S, x > 0, y > 0} is called the set of strictly feasible poits, or the set of iterior poits. We assume S /Φi this paper. It is kow that if S is oempty, the for ay parameter τ>0 the oliear system Mx y h 0, xy τe has a uique positive solutio 4. The set of all such solutios defies the cetral path C of LCP. So the assumptio is sufficiet to establish the existece of the cetral path ad the existece of a solutio to LCP.

3 Joural of Applied Mathematics 3 The mea value of x T y is deoted as μ x T y/ throughout this paper. The ρβ {x, y S Xy μe βμ, μ x T y/} is cosidered as the eighborhood of cetral path, where β>0, e deotes the vector of oes, X diagx ad express the Euclidea orm. The improved predictor-corrector iterior-poit algorithm IPCIP is as follows... Mai Steps of IPCIP Algorithm Step 0. Choose a iitial pair of iterior poit x 0,y 0 with x 0,y 0 ρ/4, ad set the accuracy parameter ε>0. Let k 0. Step. If x k T y k / ε, the stop. Step. Compute the predictor directio Δx p, Δy p by solvig the system., MΔx p Δy p 0, Y k Δx p X k Δy p 3 μk e X k y k.. Step 3. Set x k, ŷ k x k,y k Q k Δx p, Δy p where Q k mi{/8, μ k /8 ΔX p Δy p }. We will show later that X k ŷ k μ k e /μ k, where μ k Q k μ k holds for every x k, ŷ k. Step 4. Set r /4, ad compute the corrector directio Δx c, Δy c by solvig the system., MΔx c Δy c 0, Ŷ k Δx c X k Δy c rμ k e X k ŷ k.. Step 5. Set x k,y k x k, ŷ k Δx c, Δy c,updatek k, ad retur to Step. 3. Covergece ad Complexity Aalysis Lemma 3.. If b, c R ad b c h, b T c 0, B diagb,the i Bc /4 h ; ii b T c / h ; iii b h. Proof. The proof of i ca be see i Lemma of. From h b c b c b T c ad b T c 0, we ca obtai b T c / h ad b h. Thus iequalities ii ad iii hold. This completes the proof. Lemma 3.. If the poit x k, ŷ k was geerated by the algorithm, the oe has μ k x k T ŷ k / μ k, where μ k Q k μ k.

4 4 Joural of Applied Mathematics Proof. From Step of our algorithm ad M is a positive semidefiite matrix, we have Δx p T Δy p Δx p T MΔx p 0. Hece x ) k Tŷk μ k x k Q k Δx p) T y k Q K Δy p) x k ) T y k Q k[ x k ) T Δy p Δx p T y k] Q k) Δx p T Δy p ) x k T y k Qk 3 μk x k) ) T y k 3. μ k 3 Qk μ k Q k μ k Q k) μ k μ k. This completes the proof. Lemma 3.3. If x k,y k ρ/4, the the poit x k, ŷ k geerated by the algorithm satisfies x k > 0, ŷ k > 0, X k ŷ k μ k e /μ k, ad x k, ŷ k ρ/. Proof. Let us defie xq,yq x k QΔx p,y k QΔy p, where 0 Q Q k mi 8, μ k 8 ΔX p Δy p. 3. Note that 0 Q Q k mi{/8, μ k /8 ΔX p Δy p }, the we have /3 Q / from Q /8 ad Q ΔX p Δy p /8μ k from Q μ k /8 ΔX p Δz p. Furthermore X k y k μ k e /4μ k holds sice x k,y k ρ/4. Therefore we have XQyQ Qμ k e X k y k Q X p Δy p ΔX p y p) Q ΔX p Δy p Qμ k e ) Xk y k Q 3 μk e X k y k Q ΔX p Δy p Qμ k e QX k y k μ k e Qμ 3 k Q ΔX p Δy p

5 Joural of Applied Mathematics 5 Q 4 μk μk 8 μk [ 4 Q )] Qμ k ) Qμ k Qμk. 3.3 So x i Qy i Q / Qμ k > 0 i,,..., for ay Q 0 Q Q K. Sice xq,yqx k QΔx p,y k QΔy p is cotiuous with respect to Q, x k > 0 ad y k > 0, it is easily see that x k xq k > 0, ŷ k yq k > 0. Note that XQ k X k,yq k Ŷ k, Q k μ k μ k, the above argumet implies X k ŷ k μ k e /μ k. It s well kow that the iequality Xz x T z/e Xz ce holds for arbitrary vector x, z ad arbitrary costat c. So we have X k ŷ k μ k e X k ŷ k μ k e /μ k / μ from the proof above ad Lemma 3.,thus x k, ŷ k ρ/. This completes the proof. Remark 3.4. So Lemma 3.3 dedicates that the predictors of our algorithm operate i a wide eighborhood of the cetral path ρ/. Lemma 3.5. If X k ŷ k μ k e /μ k the X k Ŷ k / rμ k e X k ŷ k /4 r μ k Proof. X k Ŷ k) /rμ k e X k ŷ k) mi i xk i ŷk i rμ k e X k ŷ k X k ŷ k μ k e rμ k e /μ k X k ŷ k μ k e rμ k e ) μ k 3.4 ) 4 r μ k. This completes the proof. Lemma 3.6. If > ad the poit x k,y k was geerated by the algorithm, the μ k Q k μ k always hold.

6 6 Joural of Applied Mathematics Proof. Deote D X k / Ŷ k / the D Δx c DΔy c X k Ŷ k) /rμ k e X k ŷ k) D Δx c) T DΔy c ) Δx c T Δy c Δx c T MΔx c From Lemmas 3. ad 3.5 we have Δx c T Δy c X k ŷ k) /rμ k e X k ŷ k) ) 4 r μ k ) x k T y k μ k x k Δx c) Tŷk Δy c) x ) k Tŷk x k) T Δy c Δx c T ŷ k Δx c T Δy c rμk Δx c T Δy c ) rμ k r μ k Note that r /4, we have r ) r It is obvious that r /4 r < whe >, so μ k μ k Q k μ k. This completes the proof. Remark 3.7. From Lemmas 3. ad 3.6, we ca obtai immediately that μ k μ k μ k.that meas the corrector reduces μ k to μ k. Because the corrector i the classical MTY algorithm just maitais the same duality gap, the improvemet of the corrector i our algorithm will make a faster reductio of μ k tha MTY algorithm. The the polyomial covergece of the algorithm could be established. Theorem 3.8. Suppose that the sequece {x k,y k } geerated by the algorithm, the oe has x k,y k ρ/4. Proof. Let us defie that x k α x k αδx c,y k α ŷ k αδy c, ad deote x k μ k α ) T y k α α. 3.8

7 Joural of Applied Mathematics 7 As discussed i Lemma 3.6, we also have D Δx c T DΔy c Δx c T Δy c Δx c T MΔx c 0. From Lemmas 3. ad 3.5 ad r /4, the ΔX c Δy c 4 X k Ŷ k) /rμ k e X k ŷ k) ) 4 4 r μ k ) 4 r μ k 4 ) μ k μk, x k μ k α ) T y k α α x k αδx c) Tŷk αδy c) 3.9 k x ) Tŷk α x ) k T Δy c Δx c T ŷ k) α Δx c T Δy c x k ) Tŷk α x k ) T Δy c Δx c T ŷ k) k x ) Tŷk αrμ k x k) ) Tŷk μ k α rμ k μ k) α μ k αrμ k rαμ k. So we ca write μ k α rαμ k,thatis,μ k μ k α/ rα. Hece x X k αy k α μ k αe X k αδx c) ŷ k αδy c) k αδx c) Tŷk αδy c) e X k ŷ k α X k Δy c ΔX c ŷ k) α ΔX c Δy c k x ) Tŷk α x ) k T Δy c Δx c T ŷ k) α Δx c T Δy c e

8 8 Joural of Applied Mathematics X k ŷ k α rμ k e X k ŷ k) α ΔX c Δy c x ) k Tŷk α rμ k x k) ) Tŷk α Δx c T Δy c e x ) k Tŷk ) α X k ŷ k e α ΔX c Δy c Δxc T Δy c e. 3.0 Sice the iequality Xz x T z/e Xz ce holds for arbitrary vector x, z ad arbitrary costat c,thus X k αy k α μ k x ) k Tŷk αe α X k ŷ k e α ΔXc Δy c Δxc T Δy c e α X k ŷ k μ k e α ΔX c Δy c α μk α 7 3 μk / α 7/3α μ k α. rα 3. Let us defie fα / α7/3α / rα, the f α /r 7/6α 7/3α r/ rα. Whe >, we have r 7/8, this results i f α 0 ad fα decreases mootoously, whe 0 α. Because fα f0 / holds for every α 0, thus X k αy k α μ k αe μk α. 3. Sice μ k >ε>0 otherwise the algorithm will be termiated, it follows that X k i αy k α μk α i rαμk 3.3 rα Q k) μ k > 0. We have proved that x k xq k > 0, ŷ k yq k > 0iLemma 3.3. So we have x k 0 x k > 0, y k 0 ŷ k > 0, whe α

9 Joural of Applied Mathematics 9 Note that x k α ad y k α are cotiuous with respect to α. It implies that x k α > 0ady k α > 0 hold for ay α 0,. Let α, the we have x k > 0ady k > 0. Furthermore, whe α, fα 7/3r thus X k y k μ k e 7/3rμ k /4μ k. Cosider that we ca obtai Mx k y k h 0 directly from the steps i our algorithm, so x k,y k ρ/4 holds for every x k,y k geerated by the algorithm. This completes the proof. Theorem 3.9. The iteratio complexity of the algorithm is O L. Proof. Let D X k / Y k / the D ) Δx p D Δy p X k Y k) ) / 3 μk e X k y k, D ) ) T Δx p D Δy p) Δx p T Δy p Δx p T MΔy p From Lemma 3., we have ΔX p Δy p D ) ΔX p D Δy p 4 X k y k) ) / 3 μk e X k y k 4 ) ) ) ) 3 μk x k i y k 4 i 3 μk x k i yk i. i 3.6 From X k y k μ k e /4μ k, we have x k i yk i 3/4μ k, the x k i y k i 4/3μ k, ΔX p Δy p 4 i μk 7 08 μk μk 4 ) 3 μk μ k 3.7 Hece μ k /8 ΔX p Δy p 7/4, ad we ca obtai mi{/8, μ k /8 ΔX p Δy p } 7/4. So it implies that μ k Q k μ k 7/4 μ k for each k. This meas that μ k will decrease at least by a costat factor of 7/4 at each iteratio, which guaratees a O L-iteratio complexity for the algorithm. This completes the proof.

10 0 Joural of Applied Mathematics Remark 3.0. Note that the reduce factor i the MTY algorithm is χ/ where χ / , but the reduce factor i our algorithm is 7/4 λ/ where λ 7/4.68, which is larger tha χ. It implies that our algorithm will coverge faster tha the MTY algorithm, although both have the same iteratio complexity. 4. Examples ad Numerical Results Fially, the umerical experimets were carried out to evaluate the performace ad practical efficiecy of the algorithm, Test Problem Fid a vector pair x, y R 3 R 3 such that y M x h, x, y 0, 0, x T y 0, where h R 3 ad M is a 3 3 positive semidefiite matrix, 3 M 3 0, 5 h Test Problem Fid a vector pair x, y R 5 R 5 such that y M x h, x, y 0, 0, x T y 0, where h R 5 ad M is a 5 5 positive semidefiite matrix, Test Problem M , h This example is a geeral test problem used by Noor et al. 5. The problem is to fid a vector pair x, y R R such that y M 3 x h 3, x, y 0, 0, x T y 0, where h 3 R ad M 3 is a positive semidefiite matrix. To test the efficiecy of our algorithm by large-scale problem, we set the dimesio of test problem 3 as 00, that is, 00, M , h

11 Joural of Applied Mathematics Table : Computatioal results of test problem. MTY IPCIP x 0,, T,, T y 0 5, 5, 5 T 5, 5, 5 T ε e 06 e 06 Iteratios μ k 3.76e e 07 x k 0.0, , T 0.0, , T y k 0.00, , 9.9e 07 T 0.004, 0.008,.8e 07 T Ru time s Table : Computatioal results of test problem. MTY IPCIP x 0.5,.5,.5,.5,.5 T.5,.5,.5,.5,.5 T y 0 4, 4, 4, 4, 4 T 4, 4, 4, 4, 4 T ε e 06 e 06 Iteratios 0 9 μ k.6e 08.8e 07 x k 0.840,.376, ,.8094,.630 T 0.840,.376, ,.8094,.630 T y k 0, 0, 0, 0, 0 T 0, 0, 0, 0, 0 T Ru time s Table 3: Computatioal results of test problem 3. MTY IPCIP x 0,,,...,, T,,,...,, T y 0 3, 4, 4,...,4,6 T 3, 4, 4,...,4,6 T ε e 05 e 05 Iteratios 8 7 μ k.00e 08.9e 08 x k 0, 0, 0,...,0,0 T 0, 0, 0,...,0,0 T y k,,,...,, T,,,...,, T Ru time s These test problems are solved by our IPCIP algorithm ad the classical MTY algorithm. The experimets are ru o a PC.6 GHz CPU, G DDR RAM usig MATLAB 7. The results icludig the correspodig umbers of iteratios, μ k, the approximate solutios, ad computig times are show i Tables,, ad3. 5. Coclusio This paper modified the MTY algorithm for solvig mootoe LCPs to stregthe the covergece results. Although the iteratio complexity of the improved algorithm was proved to be O L which is similar to the classical MTY algorithm, but the reduce factor of duality gap was ehaced to.68 ad results i a faster covergece tha the classical MTY algorithm.

12 Joural of Applied Mathematics The umerical results show that our IPCIP algorithm is more efficiet tha the MTY algorithm. The umber of iteratios was decreased, ad the computig times could be reduced by early 0% to 50%. That idicated IPCIP algorithm has a better performace tha the MTY algorithm. Ackowledgmets This paper was supported by the Humaities ad Social Sciece Research Projects of the Miistry of Educatio of Chia o. YJCZH ad the Natioal Natural Sciece Foudatio of Chia o Refereces N. Karmarkar, A ew polyomial-time algorithm for liear programmig, Combiatorica, vol. 4, o. 4, pp , 984. S. Mizuo, M. J. Todd, ad Y. Ye, O adaptive-step primal-dual iterior-poit algorithms for liear programmig, Mathematics of Operatios Research, vol. 8, o. 4, pp , T. Illés, M. Nagy, ad T. Terlaky, A polyomial path-followig iterior poit algorithm for geeral liear complemetarity problems, Joural of Global Optimizatio, vol. 47, o. 3, pp , Y. Q. Bai, G. Lesaja, ad C. Roos, A ew class of polyomial iterior-poit algorithms for liear complemetary problems, Pacific Joural of Optimizatio. A Iteratioal Joural, vol. 4, o., pp. 9 4, Y. Ye, O. Güler, R. A. Tapia, ad Y. Zhag, A quadratically coverget O L-iteratio algorithm for liear programmig, Mathematical Programmig, vol. 59, o., pp. 5 6, Y. Ye ad K. Astreicher, O quadratic ad O L covergece of a predictor-corrector algorithm for LCP, Mathematical Programmig, vol. 6, o. 3, pp , J. Ji, F. A. Potra, ad S. Huag, Predictor-corrector method for liear complemetarity problems with polyomial complexity ad superliear covergece, Joural of Optimizatio Theory ad Applicatios, vol. 85, o., pp , F. A. Potra, The Mizuo-Todd-Ye algorithm i a larger eighborhood of the cetral path, Europea Joural of Operatioal Research, vol. 43, o., pp , F. A. Potra, A superliearly coverget predictor-corrector method for degeerate LCP i a wide eighborhood of the cetral path with O L-iteratio complexity, Mathematical Programmig A, vol. 00, o., pp , J. Stoer, M. Wechs, ad S. Mizuo, High order ifeasible-iterior-poit methods for solvig sufficiet liear complemetarity problems, Mathematics of Operatios Research, vol. 3, o. 4, pp , 998. F. A. Potra, A O L ifeasible-iterior-poit algorithm for LCP with quadratic covergece, Aals of Operatios Research, vol. 6, pp. 8 0, 996. S. J. Wright, A path-followig iterior-poit algorithm for liear ad quadratic problems, Aals of Operatios Research, vol. 6, pp , W. Ai ad S. Zhag, A O L iteratio primal-dual path-followig method, based o wide eighborhoods ad large updates, for mootoe LCP, SIAM Joural o Optimizatio, vol.6,o., pp , M. Kojima, N. Megiddo, T. Noma, ad A. Yoshise, A uified approach to iterior poit algorithms for liear complemetarity problems, Lecture Notes i Computer Sciece, vol. 538, M. A. Noor, Y. J. Wag, ad N. Xiu, Projectio iterative schemes for geeral variatioal iequalities, Joural of Iequalities i Pure ad Applied Mathematics, vol. 3, o. 3, pp. 8, 00.

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