Chapter 2 A Class of Robust Solution for Linear Bilevel Programming
|
|
- Martina Byrd
- 5 years ago
- Views:
Transcription
1 Chapter 2 A Class of Robust Soluton for Lnear Blevel Programmng Bo Lu, Bo L and Yan L Abstract Under the way of the centralzed decson-makng, the lnear b-level programmng (BLP) whose coeffcents are supposed to be unknown but bounded n box dsturbance set s studed. Accordngly, a class of robust soluton for lnear BLP s defned, and the orgnal uncertan BLP was converted to the determnstc trple level programmng, then a solvng process s proposed for the robust soluton. Fnally, a numercal example s shown to demonstrate the effectveness and feasblty of the algorthm. Keywords Box dsturbance Lnear blevel programmng Robust optmzaton Robust soluton 2.1 Introducton Blevel programmng (BLP) s the model wth leader-follower herarchcal structure, whch makes the parameter optmzaton problems as the constrants (Dempe 2002). In ts decson framework, the upper level programmng s connected wth not only the decson varables n ts level but also wth the optmal soluton n the lower level programmng, whle the optmal soluton n the lower lever programmng s affected by decson varables n the upper level B. Lu B. L Y. L School of Management, Tanjn Unversty, Tanjn, Chna B. Lu (&) School of Informaton Scence and Technology, Shhez Unversty, Xnjang, Chna e-mal: lubo.ce@163.com Y. L School of Scence, Shhez Unversty, Xnjang, Chna E. Q et al. (eds.), The 19th Internatonal Conference on Industral Engneerng and Engneerng Management, DOI: / _2, Ó Sprnger-Verlag Berln Hedelberg
2 12 B. Lu et al. programmng. Due to the leader-follower herarchcal structure problems wdely exst n the realstc decson-makng envronment, the scholars have been payng great attenton to BLP and have brought about good results on the theory and algorthms (Balas and Karwan 1982; Fortuny-Amat and McCarl 1981; Matheu et al. 1994; La 1996). Some degree of uncertanty exsts n realstc decsonmakng envronment, such as the nevtable error of measurng nstrument n data collecton, ncompleteness n data nformaton, the approxmate handle for the model and other factors; hence t s necessary to study on the uncertan Blevel programmng. For the uncertanty problem, the fuzzy optmzaton and stochastc optmzaton have been appled wdely. However, t s dffcult for decson-makers to gve the precse dstrbuton functons or membershp functons whch are requred n above methods. Thus, the robust optmzaton become an mportant method, because t can seek for the best soluton for the uncertan nput wthout consderng the parameter dstrbuton of uncertan parameters and s mmune from the uncertan data (Soyster 1973). For the uncertan BLP, the defnton of robust soluton s nfluenced by the dependent degree of the upper and lower levels n the decson-makng process. When the dependent degree s relatve ndependence, the robust soluton to the uncertan BLP s defned by the way of the decentralzed decson-makng (L and Du 2011); when the dependent degree s relatve dependence, the robust soluton to the uncertan BLP s defned by the way of the centralzed decson-makng, that s, when the lower level seeks ts own robust soluton, t consders the nfluence to the robust soluton of the upper level frstly. In the paper, the latter case wll be dscussed, and the coeffcents of BLP are supposed to be unknown but bounded n box dsturbance set. By the transform of the uncertan model, the robust soluton of BLP s obtaned. Fnally, a numercal example s shown to demonstrate the effectveness and feasblty of the algorthm. 2.2 The Defnton of Robust BLP The Model and the Defnton In ths paper we consder Lnear BLP formulated as follows: mn Fðx; yþ ¼c T x 1 x þ dt 1 y s:t: where y solves mn f ðx; yþ ¼c T y 2 x þ dt 2 y s:t: Ax þ By h x; y 0 ð2:1þ
3 2 A Class of Robust Soluton for Lnear Blevel Programmng 13 In model (2.1), x 2 R m1 ; y 2 R n1 ; c l 2 R m1 ; d l 2 R n1 ; l 2f1; 2g; A 2 R rm ; B 2 R rn ; h 2 R r1 ; there s some uncertanty or varaton n the parameters c 1 ; d 1 ; c 2 ; d 2 ; A; B; h. Let ðc 1 ; d 1 ; c 2 ; d 2 ; A; B; hþ 2l, l s a gven uncertanty set n Box dsturbance as follows: 8 >< l :¼ ðc l ; d l ; A; B; hþ >: j 2 c l ¼ c l þ ð u c l d lj ¼ dlj þ ð u d l a k ¼ a k þ ð u A b kj ¼ b kj þ ð u B h k ¼ h k þ ð u h Þ ; ðu cl Þ j ; ðu dl Þ k ; ðu A Þ kj ; ðu B Þ ð u c l Þ j ð u d l Þ ðu cl Þ Þ j ðu dl Þ j ð u AÞ k ðu A ð u BÞ kj ðu B Þ k ; ðu h Þ k ð u hþ k ðu h l ¼f1; 2g; 2 f1;...; mg; f1;...; ng; k ¼ f1;...; rg: Þ k 9 >= >; ð2:2þ For l ¼f1; 2g; 2 f1;...; mg; j 2 f1;...; ng; k ¼ f1;...; rg, c l ; d lj ; a k ; d kj ; h k are the gven data, and ðu cl Þ ; ð u d l Þ j ; ð u A ; ð u B ; ð u hþ k are the gven nonnegatve data. Under the way of the centralzed decson-makng, the robust soluton of uncertan BLP (1) s defned as follows: Defnton 1 (1) Constrant regon of the lnear BLP (1): X ¼ fðx; yþjax þ By h; x; y 0; ða; B; hþ 2lg (2) Feasble set for the follower for each fxed x XðxÞ ¼fyAxþ j By h; x; y 0; ða; B; hþ 2lg (3) Follower s ratonal reacton set for each fxed x ( ( MðxÞ ¼ yy2arg mn ct 2 x þ )) dt 2 y; y 2 XðxÞ; ða; B; hþ 2l (4) Inducble regon: IR ¼ fðx; yþjðx; yþ 2X; y 2 MðxÞg:
4 14 B. Lu et al. Defnton 2 Let ( F :¼ ðx; y; tþ 2R m R n c T 1 R x þ dt 1 y t ) ðx; yþ; ðc 1 ; d 1 Þ2l The programmng mnftjðx; y; tþ 2Fg ð2:3þ x;y;t s defned as robust counterpart of uncertan lnear BLP(1); F s defned as the robust feasble set of uncertan lnear BLP(1) The Transform of Uncertan BLP Model Under the way of the centralzed decson-makng, based on the orgnal dea of robust optmzaton that the objectve functon can get the optmal soluton even n the worst and uncertan stuaton, the transform theorem can be descrbed as followngs: Theorem The robust lnear BLP (1) wth ts coeffcents unknown but bounded n box dsturbance set l s equvalent to Model (2.4) wth certan coeffcents as followngs: mn x Fðx; yþ ¼ Xm s:t: where d 2 solves X m max c 1 þ l c d 1 2 c 1 þ l c 1 d1j þ l d 1 d1j þ l d 1 j y j s:t: d2j ð u d 2 Þ j d 2j d2j þ ð u d 2 Þ j ; j ¼f1;...; ng; where y solves mn f ðx; yþ ¼d T y 2 y s:t: Xm a k ð l A k ¼ f1;...; rg x; y 0: b kj ð l B j y j y j h k þðl hþ k ; ð2:4þ Proof (1) Frstly, the constrant regon X of the lnear BLP (1) s transformed nto the certan regon. Consder the constrant regon of the lnear BLP (1):
5 2 A Class of Robust Soluton for Lnear Blevel Programmng 15 X ¼ fðx; yþjax þ By h; x; y 0; ða; B; hþ 2lg Accordng to the process of the transformaton (Lobo et al. 1998), we can obtan Ax þ By h; ða; B; hþ 2l 8 ><, 0 mn l A ;l B ;l h, 0 Xm >< þ mn l A ;l B ;l h >: X m a k x þ Xn 8 >: X m a k x þ Xn b kj y j h k ðu A Þ k x þ Xn x;y 0, 0 Xm a k ð u A 8k2f1;...;rg, Xm a k ð u A a k ¼ a k þ u 9 ð AÞ k ; ðu A ð u AÞ k ðu A ; b kj ¼ b kj þ ð u BÞ kj ; ðu B b kj y j h ð u BÞ kj ðu B ; >= k h k ¼ h k þ ð u hþ k ; ðu h Þ k ð u hþ k ðu h Þ k ; 2f1;...; mg; j 2f1;...; ng; >; k 2f1;...; rg ðu B ðu A ð u AÞ k ðu A Þ 9 k ðu B ð u BÞ kj ðu B >= Þ k ðu h Þ k ð u hþ k ðu h Þ k 2f1;...; mg; j 2f1;...; ng; >; k 2f1;...; rg b kj ð u B y j h k þ ð u h Þ kj y j ðu h b kj ð u B Þ k y j h k þ ð u hþ k ; k 2f1;...; rg So the lnear BLP (1) s transformed nto the model (2.6) as followngs: ð2:5þ
6 16 B. Lu et al. mn Fðx; yþ ¼c T x 1 x þ dt 1 y s:t: where y solves mn f ðx; yþ ¼c T y 2 x þ dt 2 y s:t: Xm a k ð u A k ¼ f1;...; rg x; y 0 b kj ð u B y j h k þ ð u h Þ k ð2:6þ (2) Next, accordng the equvalent form (Lobo et al. 1998) mn f ðxþ x s:t: x 2 D, mn x;t t s:t: f ðxþt x 2 D and the K-T method, the model (2.6) can be transform-ed nto the model (2.7) (L and Du 2011): mn Fðx; yþ ¼ t x;t s:t: c T 1 x þ dt 1 y t where y solves mn f ðx; yþ ¼c T y 2 x þ dt 2 y s:t: Xm a k ðu AÞ k k ¼ f1;...; rg x; y 0 b kj ðu B y j h k þðu hþ k ð2:7þ (3) Smlar to the transformaton (2.5), c T 1 x þ dt 1 y t; ðc x;y 0 Pm 1; d 1 Þ2l, c 1 þ l c 1 x þ Pn d 1j þ l d 1 j So the model (2.7) can be transformed to the model (2.8) as follows: y j t
7 2 A Class of Robust Soluton for Lnear Blevel Programmng 17 mn Fðx; yþ ¼ Xm c x;t 1 þ l c 1 s:t: where y solves mn f ðx; yþ ¼c T y 2 x þ dt 2 y s:t: Xm a k ð u A k 2f1;...; rg; x; y 0 b kj ð u B d1j þ l d 1 j y j y j h k þ ð u hþ k ; ð2:8þ (4) Next, because the optmal soluton of BLP (1) s not nfluenced by the value of c 2, we only consder how to choose the value of d 2. Based on the orgnal dea of robust optmzaton, the model (2.8) s transformed nto the model (2.4) above. 2.3 Solvng Process of the Model The determnstc trple level programmng (2.4) can be wrtten as the followng programmng (2.9) by the K-T method. mn x max d 2 ;y;u;v s:t: d 2j ¼ Xr Fðx; yþ ¼Xm u k b kj ð u B k¼1 c 1 þ l c 1 þ v j d1j þ l d 1 j y j d2j ð u d 2 Þ j d 2j d2j þ ð u d 2 Þ j " # X m u k a kj ð l AÞ x þ Xn b kj ð l B y j ðh k þ ð l hþ k Þ ¼ 0 ¼j v j y j ¼ 0 X m a k ð l A j ¼ f1;...; ng; k ¼ f1;...; rg; x; y; u; v 0: b kj ð l B y j h k þ ð l hþ k ; ð2:9þ Accordng to the lterature (Wang 2010), the model (2.9) can be transformed nto the model (2.10) as follows
8 18 B. Lu et al. mn t x;d 2 ;y;u;v s:t: Xm d 2j ¼ Xr c 1 þ l c 1 u k k¼1 b kj ð u B d1j þ l d 1 þ v j ; j y j t d2j ð u d 2 Þ j d 2j d2j þ ð u d 2 Þ j " ; # X m u k a k ð l A b kj ð l B y j h k þ ð l h ¼ 0; v j y j ¼ 0; X m a k ð l A j ¼ f1;...; ng; k ¼ f1;...; rg; x; y; u; v 0: b kj ð l B y j h k þ ð l hþ k ; Þ k ð2:10þ By ntroducng a large constant M, the model (2.10) above can be transformed nto a mxed nteger programmng as follows (Fortuny-Amat and McCarl 1981): mn t x;d 2 ;y;u;v;t;w s:t: Xm c 1 þ l c 1 d 2j ¼ Xr u k k¼1 b kj ð u B d1j þ l d 1 þ v j ; d2j ð u d 2 Þ j d 2j d2j þ ð u d 2 Þ j ; y j Mt j ; v j M 1 t j ; u k Mw k ; X m a k ð l A b kj ð l B X m a k ð l A h b kj ð l B j y j t y j h k þ ð l h Þ k y j h k þ ð l hþ k ; j ¼ f1;...; ng; k ¼ f1;...; rg; t j 2 f0; 1g; w k 2 f0; 1g; x; y; u; v 0: The model (2.11) can be solved by the software Lngo 9.0 M ð 1 wk Þ ð2:11þ
9 2 A Class of Robust Soluton for Lnear Blevel Programmng A Numercal Example We gve a numercal example to demonstrate the proposed approach as follows: mn F ¼ c 11 x þ d 11 y 1 þ d 12 y 2 x s:t: 2:5 x 8 where y 1 ; y 2 solve mn f ¼ d 21 y 1 þ d 22 y 2 y 1 ;y 2 s:t: a 11 x 1 þ b 11 y 1 þ b 12 y 2 h 1 a 21 x 1 þ b 21 y 1 þ b 22 y 2 h 2 a 31 x 1 þ b 31 y 1 þ b 32 y 2 h 3 a 41 x 1 þ b 41 y 1 þ b 42 y 2 h 4 y 1 0; y 2 0 where a 11 ¼ 0; b 21 ¼ 1; a 31 ¼ 0; a 41 ¼ 0; b 41 ¼ 0; b 42 ¼ 1: And the others are the uncertan data, the gven varables and dsturbances are c 11 ¼ 1:5; d 11 ¼ 1:5; d 12 ¼ 2; d 21 ¼ 1:5; d 22 ¼ 3:5; b 11 ¼ 1:75; b 12 ¼ 1:15; h 1 ¼ 2:5; a 21 ¼ 3:5; b 22 ¼ 1:5; h 4 ¼ 5:75: h 2 ¼ 11; b 31 ¼ 3:5; b 32 ¼ 1:25; h 3 ¼ 23; ðu c1 Þ 1 ¼ 0:5; ð u d 1 Þ 1 ¼ 0:5; ð u d 1 Þ 2 ¼ 3; ð u d 2 Þ 1 ¼ 0:5; ð u d 2 Þ 2 ¼ 0:5; ðu b ðu b Þ 11 ¼ 0:25; ð u bþ 12 ¼ 0:15; ð u hþ 1 ¼ 0:5; ðu a Þ 31 ¼ 0:5; ð u b Þ 21 ¼ 0:5; ð u b Þ 22 ¼ 0:5; ð u hþ 2 ¼ 1; Þ 32 ¼ 0:25; ð u hþ 3 ¼ 1; ð u hþ 4 ¼ 0:25: Accordng to the theorem and these data above, robust model transformed s demonstrated as
10 20 B. Lu et al. mn t x;y;u;v;g;z s:t: 2x y 1 þ y 2 t 2:5 x 8; y 2 5:5; 1:5y 1 1:3y 2 3; 4x þ y 1 2y 2 10; 4y 1 1:5y 2 22; 1 1:5u 1 þ u 2 4u 3 þ v 1 2; 4 1:3u 1 2u 2 1:5u 3 u 4 þ v 2 3; y 1 Mg 1 ; y 2 Mg 2 ; v 1 Mð1 g 1 Þ; v 2 Mð1 g 2 Þ; u k Mz k ; k ¼ 1; 2; 3; 4: 1:5y 1 1:3y 2 3 ð1 z 1 Þ; 4x þ y 1 2y 2 þ 10 ð1 z 2 Þ; 4y 1 1:5y 2 þ 22 ð1 z 3 Þ; y 2 þ 5:5 ð1 z 4 Þ; x; y; u; v; g; z 0; g 1 ; g 2 2 f0:1g; z k 2 f0:1g; k ¼ 1; 2; 3; 4: By the software Lngo 9.0, the robust soluton s obtaned as follows: ðx; y 1 ; y 2 Þ¼ð2:5201; 4:6443; 2:2819Þ; The robust optmal value s F mn ¼ 2: Concluson and Future Work Under the way of the centralzed decson-makng, a class of robust soluton for uncertan lnear BLP s defned, whch expands further the applcaton of BLP n dfferent crcumstances. And based on the orgnal dea of robust optmzaton, the uncertan BLP was converted to the determnstc trple level programmng. The solvng process s proposed to obtan the robust soluton of uncertan lnear BLP. Fnally, a numercal example s shown to demonstrate the effectveness and feasblty of the algorthm.
11 2 A Class of Robust Soluton for Lnear Blevel Programmng 21 References Balas WF, Karwan MH (1982) On two-level optmzaton. IEEE Trans Autom control AC- 27(1): Dempe S (2002) Foundatons of blevel programmng. Kluwer Academc Publsher, Boston Fortuny-Amat J, McCarl B (1981). A representaton and economc nterpretaton of two-level programmng problem. J Oper Res Soc 32: Fortuny-Amat J, McCarl BA (1981) Representaton and economc nterpretaton of two-level programmng problem. J Oper Res Soc 32(7): La YJ (1996) Herarchcal optmzaton: a satsfactory soluton. Fuzzy Sets and Syst 77: L Y, Du G (2011) Robust lnear blevel programmng under ellpsodal uncertanty. Syst Eng 11: Lobo MS, Vandenberghe L, Boyd S, Lebret H (1998) Applcaton of second-order cone programmng. Lnear Algebra Appl 284: Matheu R, Pttard L, Anandalngam G (1994) Genetc algorthm based approach to b-level lnear programmng. Oper Res 28:1 21 Soyster AL (1973) Convex programmng wth set-nclusve constrants and applcatons to nexact lnear programmng. Oper Res 21: Wang J (2010) Research on the methods of nterval lnear b-level programmng, pp Tanjn Unversty, Tanjn
12
Interactive Bi-Level Multi-Objective Integer. Non-linear Programming Problem
Appled Mathematcal Scences Vol 5 0 no 65 3 33 Interactve B-Level Mult-Objectve Integer Non-lnear Programmng Problem O E Emam Department of Informaton Systems aculty of Computer Scence and nformaton Helwan
More informationFuzzy Approaches for Multiobjective Fuzzy Random Linear Programming Problems Through a Probability Maximization Model
Fuzzy Approaches for Multobjectve Fuzzy Random Lnear Programmng Problems Through a Probablty Maxmzaton Model Htosh Yano and Kota Matsu Abstract In ths paper, two knds of fuzzy approaches are proposed for
More informationVARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES
VARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES BÂRZĂ, Slvu Faculty of Mathematcs-Informatcs Spru Haret Unversty barza_slvu@yahoo.com Abstract Ths paper wants to contnue
More informationSome modelling aspects for the Matlab implementation of MMA
Some modellng aspects for the Matlab mplementaton of MMA Krster Svanberg krlle@math.kth.se Optmzaton and Systems Theory Department of Mathematcs KTH, SE 10044 Stockholm September 2004 1. Consdered optmzaton
More informationOn the Multicriteria Integer Network Flow Problem
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 5, No 2 Sofa 2005 On the Multcrtera Integer Network Flow Problem Vassl Vasslev, Marana Nkolova, Maryana Vassleva Insttute of
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationThe Study of Teaching-learning-based Optimization Algorithm
Advanced Scence and Technology Letters Vol. (AST 06), pp.05- http://dx.do.org/0.57/astl.06. The Study of Teachng-learnng-based Optmzaton Algorthm u Sun, Yan fu, Lele Kong, Haolang Q,, Helongang Insttute
More informationarxiv: v1 [math.oc] 3 Aug 2010
arxv:1008.0549v1 math.oc] 3 Aug 2010 Test Problems n Optmzaton Xn-She Yang Department of Engneerng, Unversty of Cambrdge, Cambrdge CB2 1PZ, UK Abstract Test functons are mportant to valdate new optmzaton
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationHeuristic Algorithm for Finding Sensitivity Analysis in Interval Solid Transportation Problems
Internatonal Journal of Innovatve Research n Advanced Engneerng (IJIRAE) ISSN: 349-63 Volume Issue 6 (July 04) http://rae.com Heurstc Algorm for Fndng Senstvty Analyss n Interval Sold Transportaton Problems
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More information10) Activity analysis
3C3 Mathematcal Methods for Economsts (6 cr) 1) Actvty analyss Abolfazl Keshvar Ph.D. Aalto Unversty School of Busness Sldes orgnally by: Tmo Kuosmanen Updated by: Abolfazl Keshvar 1 Outlne Hstorcal development
More informationInternational Journal of Pure and Applied Sciences and Technology
Int. J. Pure Appl. Sc. Technol., 6( (0, pp. 5-3 Internatonal Journal of Pure and Appled Scences and Technology ISS 9-607 Avalable onlne at www.jopaasat.n Research Paper Goal Programmng Approach to Lnear
More informationSystem in Weibull Distribution
Internatonal Matheatcal Foru 4 9 no. 9 94-95 Relablty Equvalence Factors of a Seres-Parallel Syste n Webull Dstrbuton M. A. El-Dacese Matheatcs Departent Faculty of Scence Tanta Unversty Tanta Egypt eldacese@yahoo.co
More informationFUZZY GOAL PROGRAMMING VS ORDINARY FUZZY PROGRAMMING APPROACH FOR MULTI OBJECTIVE PROGRAMMING PROBLEM
Internatonal Conference on Ceramcs, Bkaner, Inda Internatonal Journal of Modern Physcs: Conference Seres Vol. 22 (2013) 757 761 World Scentfc Publshng Company DOI: 10.1142/S2010194513010982 FUZZY GOAL
More informationUncertain Models for Bed Allocation
www.ccsenet.org/ghs Global Journal of Health Scence Vol., No. ; October 00 Uncertan Models for Bed Allocaton Lng Gao (Correspondng author) College of Scence, Guln Unversty of Technology Box 733, Guln 54004,
More informationPower law and dimension of the maximum value for belief distribution with the max Deng entropy
Power law and dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy Bngy Kang a, a College of Informaton Engneerng, Northwest A&F Unversty, Yanglng, Shaanx, 712100, Chna. Abstract Deng
More informationGeneral viscosity iterative method for a sequence of quasi-nonexpansive mappings
Avalable onlne at www.tjnsa.com J. Nonlnear Sc. Appl. 9 (2016), 5672 5682 Research Artcle General vscosty teratve method for a sequence of quas-nonexpansve mappngs Cuje Zhang, Ynan Wang College of Scence,
More informationYong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )
Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often
More informationA PROBABILITY-DRIVEN SEARCH ALGORITHM FOR SOLVING MULTI-OBJECTIVE OPTIMIZATION PROBLEMS
HCMC Unversty of Pedagogy Thong Nguyen Huu et al. A PROBABILITY-DRIVEN SEARCH ALGORITHM FOR SOLVING MULTI-OBJECTIVE OPTIMIZATION PROBLEMS Thong Nguyen Huu and Hao Tran Van Department of mathematcs-nformaton,
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationCS : Algorithms and Uncertainty Lecture 17 Date: October 26, 2016
CS 29-128: Algorthms and Uncertanty Lecture 17 Date: October 26, 2016 Instructor: Nkhl Bansal Scrbe: Mchael Denns 1 Introducton In ths lecture we wll be lookng nto the secretary problem, and an nterestng
More informationAssortment Optimization under MNL
Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.
More informationThe lower and upper bounds on Perron root of nonnegative irreducible matrices
Journal of Computatonal Appled Mathematcs 217 (2008) 259 267 wwwelsevercom/locate/cam The lower upper bounds on Perron root of nonnegatve rreducble matrces Guang-Xn Huang a,, Feng Yn b,keguo a a College
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationLecture 10 Support Vector Machines. Oct
Lecture 10 Support Vector Machnes Oct - 20-2008 Lnear Separators Whch of the lnear separators s optmal? Concept of Margn Recall that n Perceptron, we learned that the convergence rate of the Perceptron
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More informationA New Algorithm for Finding a Fuzzy Optimal. Solution for Fuzzy Transportation Problems
Appled Mathematcal Scences, Vol. 4, 200, no. 2, 79-90 A New Algorthm for Fndng a Fuzzy Optmal Soluton for Fuzzy Transportaton Problems P. Pandan and G. Nataraan Department of Mathematcs, School of Scence
More informationThe Expectation-Maximization Algorithm
The Expectaton-Maxmaton Algorthm Charles Elan elan@cs.ucsd.edu November 16, 2007 Ths chapter explans the EM algorthm at multple levels of generalty. Secton 1 gves the standard hgh-level verson of the algorthm.
More informationA New Refinement of Jacobi Method for Solution of Linear System Equations AX=b
Int J Contemp Math Scences, Vol 3, 28, no 17, 819-827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,
More informationA Local Variational Problem of Second Order for a Class of Optimal Control Problems with Nonsmooth Objective Function
A Local Varatonal Problem of Second Order for a Class of Optmal Control Problems wth Nonsmooth Objectve Functon Alexander P. Afanasev Insttute for Informaton Transmsson Problems, Russan Academy of Scences,
More informationFuzzy Boundaries of Sample Selection Model
Proceedngs of the 9th WSES Internatonal Conference on ppled Mathematcs, Istanbul, Turkey, May 7-9, 006 (pp309-34) Fuzzy Boundares of Sample Selecton Model L. MUHMD SFIIH, NTON BDULBSH KMIL, M. T. BU OSMN
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationFeature Selection in Multi-instance Learning
The Nnth Internatonal Symposum on Operatons Research and Its Applcatons (ISORA 10) Chengdu-Juzhagou, Chna, August 19 23, 2010 Copyrght 2010 ORSC & APORC, pp. 462 469 Feature Selecton n Mult-nstance Learnng
More informationDETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM
Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for S111 DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM by Davood Domr GANJI
More informationChapter 8 Indicator Variables
Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n
More informationNeryškioji dichotominių testo klausimų ir socialinių rodiklių diferencijavimo savybių klasifikacija
Neryškoj dchotomnų testo klausmų r socalnų rodklų dferencjavmo savybų klasfkacja Aleksandras KRYLOVAS, Natalja KOSAREVA, Julja KARALIŪNAITĖ Technologcal and Economc Development of Economy Receved 9 May
More informationAn Application of Fuzzy Hypotheses Testing in Radar Detection
Proceedngs of the th WSES Internatonal Conference on FUZZY SYSEMS n pplcaton of Fuy Hypotheses estng n Radar Detecton.K.ELSHERIF, F.M.BBDY, G.M.BDELHMID Department of Mathematcs Mltary echncal Collage
More informationECE559VV Project Report
ECE559VV Project Report (Supplementary Notes Loc Xuan Bu I. MAX SUM-RATE SCHEDULING: THE UPLINK CASE We have seen (n the presentaton that, for downlnk (broadcast channels, the strategy maxmzng the sum-rate
More informationOrientation Model of Elite Education and Mass Education
Proceedngs of the 8th Internatonal Conference on Innovaton & Management 723 Orentaton Model of Elte Educaton and Mass Educaton Ye Peng Huanggang Normal Unversty, Huanggang, P.R.Chna, 438 (E-mal: yepeng@hgnc.edu.cn)
More informationA Network Intrusion Detection Method Based on Improved K-means Algorithm
Advanced Scence and Technology Letters, pp.429-433 http://dx.do.org/10.14257/astl.2014.53.89 A Network Intruson Detecton Method Based on Improved K-means Algorthm Meng Gao 1,1, Nhong Wang 1, 1 Informaton
More informationSolving Nonlinear Differential Equations by a Neural Network Method
Solvng Nonlnear Dfferental Equatons by a Neural Network Method Luce P. Aarts and Peter Van der Veer Delft Unversty of Technology, Faculty of Cvlengneerng and Geoscences, Secton of Cvlengneerng Informatcs,
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationSolutions to exam in SF1811 Optimization, Jan 14, 2015
Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O -5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable
More informationTRAPEZOIDAL FUZZY NUMBERS FOR THE TRANSPORTATION PROBLEM. Abstract
TRAPEZOIDAL FUZZY NUMBERS FOR THE TRANSPORTATION PROBLEM ARINDAM CHAUDHURI* Lecturer (Mathematcs & Computer Scence) Meghnad Saha Insttute of Technology, Kolkata, Inda arndam_chau@yahoo.co.n *correspondng
More informationDouble Layered Fuzzy Planar Graph
Global Journal of Pure and Appled Mathematcs. ISSN 0973-768 Volume 3, Number 0 07), pp. 7365-7376 Research Inda Publcatons http://www.rpublcaton.com Double Layered Fuzzy Planar Graph J. Jon Arockaraj Assstant
More informationControl of Uncertain Bilinear Systems using Linear Controllers: Stability Region Estimation and Controller Design
Control of Uncertan Blnear Systems usng Lnear Controllers: Stablty Regon Estmaton Controller Desgn Shoudong Huang Department of Engneerng Australan Natonal Unversty Canberra, ACT 2, Australa shoudong.huang@anu.edu.au
More informationINAk-out-of-n: G system, there are totally n components,
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 26, NO. 5, OCTOBER 2018 2663 Relablty Analyss of Uncertan Weghted k-out-of-n Systems Jnwu Gao,KaYao, Jan Zhou,andHuaKe Abstract The uncertan varable s used to model
More informationHongyi Miao, College of Science, Nanjing Forestry University, Nanjing ,China. (Received 20 June 2013, accepted 11 March 2014) I)ϕ (k)
ISSN 1749-3889 (prnt), 1749-3897 (onlne) Internatonal Journal of Nonlnear Scence Vol.17(2014) No.2,pp.188-192 Modfed Block Jacob-Davdson Method for Solvng Large Sparse Egenproblems Hongy Mao, College of
More informationA New Scrambling Evaluation Scheme based on Spatial Distribution Entropy and Centroid Difference of Bit-plane
A New Scramblng Evaluaton Scheme based on Spatal Dstrbuton Entropy and Centrod Dfference of Bt-plane Lang Zhao *, Avshek Adhkar Kouch Sakura * * Graduate School of Informaton Scence and Electrcal Engneerng,
More informationSupporting Information
Supportng Informaton The neural network f n Eq. 1 s gven by: f x l = ReLU W atom x l + b atom, 2 where ReLU s the element-wse rectfed lnear unt, 21.e., ReLUx = max0, x, W atom R d d s the weght matrx to
More informationFREQUENCY DISTRIBUTIONS Page 1 of The idea of a frequency distribution for sets of observations will be introduced,
FREQUENCY DISTRIBUTIONS Page 1 of 6 I. Introducton 1. The dea of a frequency dstrbuton for sets of observatons wll be ntroduced, together wth some of the mechancs for constructng dstrbutons of data. Then
More informationSolutions HW #2. minimize. Ax = b. Give the dual problem, and make the implicit equality constraints explicit. Solution.
Solutons HW #2 Dual of general LP. Fnd the dual functon of the LP mnmze subject to c T x Gx h Ax = b. Gve the dual problem, and make the mplct equalty constrants explct. Soluton. 1. The Lagrangan s L(x,
More informationErrors in Nobel Prize for Physics (7) Improper Schrodinger Equation and Dirac Equation
Errors n Nobel Prze for Physcs (7) Improper Schrodnger Equaton and Drac Equaton u Yuhua (CNOOC Research Insttute, E-mal:fuyh945@sna.com) Abstract: One of the reasons for 933 Nobel Prze for physcs s for
More informationAGGREGATION OF FUZZY OPINIONS UNDER GROUP DECISION-MAKING BASED ON SIMILARITY AND DISTANCE
Jrl Syst Sc & Complexty (2006) 19: 63 71 AGGREGATION OF FUZZY OPINIONS UNDER GROUP DECISION-MAKING BASED ON SIMILARITY AND DISTANCE Chengguo LU Jbn LAN Zhongxng WANG Receved: 6 December 2004 / Revsed:
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationIrene Hepzibah.R 1 and Vidhya.R 2
Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 3, March-204 374 ISSN 2229-558 INTUITIONISTIC FUZZY MULTI-OBJECTIVE LINEAR PROGRAMMING PROBLEM (IFMOLPP) USING TAYLOR SERIES APPROACH
More informationColor Rendering Uncertainty
Australan Journal of Basc and Appled Scences 4(10): 4601-4608 010 ISSN 1991-8178 Color Renderng Uncertanty 1 A.el Bally M.M. El-Ganany 3 A. Al-amel 1 Physcs Department Photometry department- NIS Abstract:
More informationDynamic Slope Scaling Procedure to solve. Stochastic Integer Programming Problem
Journal of Computatons & Modellng, vol.2, no.4, 2012, 133-148 ISSN: 1792-7625 (prnt), 1792-8850 (onlne) Scenpress Ltd, 2012 Dynamc Slope Scalng Procedure to solve Stochastc Integer Programmng Problem Takayuk
More informationInvestigation of the Relationship between Diesel Fuel Properties and Emissions from Engines with Fuzzy Linear Regression
www.esc.org Internatonal Journal of Energ Scence (IJES) Volume 3 Issue 2, Aprl 2013 Investgaton of the Relatonshp between Desel Fuel Propertes and Emssons from Engnes wth Fuzz Lnear Regresson Yuanwang
More informationA Robust Method for Calculating the Correlation Coefficient
A Robust Method for Calculatng the Correlaton Coeffcent E.B. Nven and C. V. Deutsch Relatonshps between prmary and secondary data are frequently quantfed usng the correlaton coeffcent; however, the tradtonal
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationComplement of Type-2 Fuzzy Shortest Path Using Possibility Measure
Intern. J. Fuzzy Mathematcal rchve Vol. 5, No., 04, 9-7 ISSN: 30 34 (P, 30 350 (onlne Publshed on 5 November 04 www.researchmathsc.org Internatonal Journal of Complement of Type- Fuzzy Shortest Path Usng
More informationCOS 521: Advanced Algorithms Game Theory and Linear Programming
COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton
More informationCalculation of time complexity (3%)
Problem 1. (30%) Calculaton of tme complexty (3%) Gven n ctes, usng exhaust search to see every result takes O(n!). Calculaton of tme needed to solve the problem (2%) 40 ctes:40! dfferent tours 40 add
More informationA CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS
Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs
More informationA Bayes Algorithm for the Multitask Pattern Recognition Problem Direct Approach
A Bayes Algorthm for the Multtask Pattern Recognton Problem Drect Approach Edward Puchala Wroclaw Unversty of Technology, Char of Systems and Computer etworks, Wybrzeze Wyspanskego 7, 50-370 Wroclaw, Poland
More informationx i1 =1 for all i (the constant ).
Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by
More informationSingle-Facility Scheduling over Long Time Horizons by Logic-based Benders Decomposition
Sngle-Faclty Schedulng over Long Tme Horzons by Logc-based Benders Decomposton Elvn Coban and J. N. Hooker Tepper School of Busness, Carnege Mellon Unversty ecoban@andrew.cmu.edu, john@hooker.tepper.cmu.edu
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models
More informationResearch Article Global Sufficient Optimality Conditions for a Special Cubic Minimization Problem
Mathematcal Problems n Engneerng Volume 2012, Artcle ID 871741, 16 pages do:10.1155/2012/871741 Research Artcle Global Suffcent Optmalty Condtons for a Specal Cubc Mnmzaton Problem Xaome Zhang, 1 Yanjun
More informationDetermine the Optimal Order Quantity in Multi-items&s EOQ Model with Backorder
Australan Journal of Basc and Appled Scences, 5(7): 863-873, 0 ISSN 99-878 Determne the Optmal Order Quantty n Mult-tems&s EOQ Model wth Backorder Babak Khabr, Had Nasser, 3 Ehsan Ehsan and Nma Kazem Department
More informationProceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3,
roceedngs of the 0th WSEAS Internatonal Confenrence on ALIED MATHEMATICS, Dallas, Texas, USA, November -3, 2006 365 Impact of Statc Load Modelng on Industral Load Nodal rces G. REZA YOUSEFI M. MOHSEN EDRAM
More informationTHIS paper considers the following generalized linear
Engneerng Letters, 25:3, EL_25_3_06 A New Branch and Reduce Approach for Solvng Generalzed Lnear Fractonal Programmng Yong-Hong Zhang, and Chun-Feng Wang Abstract In ths paper, for solvng generalzed lnear
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationAmiri s Supply Chain Model. System Engineering b Department of Mathematics and Statistics c Odette School of Business
Amr s Supply Chan Model by S. Ashtab a,, R.J. Caron b E. Selvarajah c a Department of Industral Manufacturng System Engneerng b Department of Mathematcs Statstcs c Odette School of Busness Unversty of
More informationGeneralized Linear Methods
Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set
More informationChapter 2 Robust Covariance Intersection Fusion Steady-State Kalman Filter with Uncertain Parameters
Chapter 2 Robust Covarance Intersecton Fuson Steady-State Kalman Flter wth Uncertan Parameters Wenjuan Q, Xueme Wang, Wenqang Lu and Zl Deng Abstract For the lnear dscrete tme-nvarant system wth uncertan
More informationIntegrals and Invariants of Euler-Lagrange Equations
Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
More informationParameter Estimation for Dynamic System using Unscented Kalman filter
Parameter Estmaton for Dynamc System usng Unscented Kalman flter Jhoon Seung 1,a, Amr Atya F. 2,b, Alexander G.Parlos 3,c, and Klto Chong 1,4,d* 1 Dvson of Electroncs Engneerng, Chonbuk Natonal Unversty,
More informationGroup Analysis of Ordinary Differential Equations of the Order n>2
Symmetry n Nonlnear Mathematcal Physcs 997, V., 64 7. Group Analyss of Ordnary Dfferental Equatons of the Order n> L.M. BERKOVICH and S.Y. POPOV Samara State Unversty, 4430, Samara, Russa E-mal: berk@nfo.ssu.samara.ru
More informationCommunication Complexity 16:198: February Lecture 4. x ij y ij
Communcaton Complexty 16:198:671 09 February 2010 Lecture 4 Lecturer: Troy Lee Scrbe: Rajat Mttal 1 Homework problem : Trbes We wll solve the thrd queston n the homework. The goal s to show that the nondetermnstc
More informationon the improved Partial Least Squares regression
Internatonal Conference on Manufacturng Scence and Engneerng (ICMSE 05) Identfcaton of the multvarable outlers usng T eclpse chart based on the mproved Partal Least Squares regresson Lu Yunlan,a X Yanhu,b
More informationMEM 255 Introduction to Control Systems Review: Basics of Linear Algebra
MEM 255 Introducton to Control Systems Revew: Bascs of Lnear Algebra Harry G. Kwatny Department of Mechancal Engneerng & Mechancs Drexel Unversty Outlne Vectors Matrces MATLAB Advanced Topcs Vectors A
More informationOn the Global Linear Convergence of the ADMM with Multi-Block Variables
On the Global Lnear Convergence of the ADMM wth Mult-Block Varables Tany Ln Shqan Ma Shuzhong Zhang May 31, 01 Abstract The alternatng drecton method of multplers ADMM has been wdely used for solvng structured
More informationUncertainty and auto-correlation in. Measurement
Uncertanty and auto-correlaton n arxv:1707.03276v2 [physcs.data-an] 30 Dec 2017 Measurement Markus Schebl Federal Offce of Metrology and Surveyng (BEV), 1160 Venna, Austra E-mal: markus.schebl@bev.gv.at
More informationGlobal Sensitivity. Tuesday 20 th February, 2018
Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values
More informationBeyond Zudilin s Conjectured q-analog of Schmidt s problem
Beyond Zudln s Conectured q-analog of Schmdt s problem Thotsaporn Ae Thanatpanonda thotsaporn@gmalcom Mathematcs Subect Classfcaton: 11B65 33B99 Abstract Usng the methodology of (rgorous expermental mathematcs
More informationThe Analytical Solution of a System of Nonlinear Differential Equations
Int. Journal of Math. Analyss, Vol. 1, 007, no. 10, 451-46 The Analytcal Soluton of a System of Nonlnear Dfferental Equatons Yunhu L a, Fazhan Geng b and Mnggen Cu b1 a Dept. of Math., Harbn Unversty Harbn,
More informationInternational Conference on Advanced Computer Science and Electronics Information (ICACSEI 2013) equation. E. M. E. Zayed and S. A.
Internatonal Conference on Advanced Computer Scence and Electroncs Informaton (ICACSEI ) The two varable (G'/G/G) -expanson method for fndng exact travelng wave solutons of the (+) dmensonal nonlnear potental
More informationSome Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)
Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998
More informationFeature Selection: Part 1
CSE 546: Machne Learnng Lecture 5 Feature Selecton: Part 1 Instructor: Sham Kakade 1 Regresson n the hgh dmensonal settng How do we learn when the number of features d s greater than the sample sze n?
More informationLecture 3. Ax x i a i. i i
18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest
More informationOn an Extension of Stochastic Approximation EM Algorithm for Incomplete Data Problems. Vahid Tadayon 1
On an Extenson of Stochastc Approxmaton EM Algorthm for Incomplete Data Problems Vahd Tadayon Abstract: The Stochastc Approxmaton EM (SAEM algorthm, a varant stochastc approxmaton of EM, s a versatle tool
More informationCHAPTER-5 INFORMATION MEASURE OF FUZZY MATRIX AND FUZZY BINARY RELATION
CAPTER- INFORMATION MEASURE OF FUZZY MATRI AN FUZZY BINARY RELATION Introducton The basc concept of the fuzz matr theor s ver smple and can be appled to socal and natural stuatons A branch of fuzz matr
More informationFixed point method and its improvement for the system of Volterra-Fredholm integral equations of the second kind
MATEMATIKA, 217, Volume 33, Number 2, 191 26 c Penerbt UTM Press. All rghts reserved Fxed pont method and ts mprovement for the system of Volterra-Fredholm ntegral equatons of the second knd 1 Talaat I.
More informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More information