惍悢寁夋朄儊儌

僩僢僾儁乕僕傊栠傞

杮儁乕僕偺URL偑 https://www.jskrtf.com/~miya/ipmemo.html 偐傜 https://web.jskrtf.com/~miya/ipmemo.html 傊曄峏偵側傝傑偟偨丏 偦傟偵墳偠偰丆杮儁乕僕偐傜儕儞僋偝傟偰偄傞僟僂儞儘乕僪壜擻側僼傽僀儖偵偮偄偰傕丆URL偑曄峏偵側偭偰偄傑偡丏

偼偠傔偵

丂偙偺儁乕僕偵偼丆惍悢嵟揔壔栤戣乮惍悢寁夋栤戣乯傪僜儖僶乕偱夝偔嵺偵丆抦偭偰偄傞偲栶偵棫偮偐傕偟傟側偄忣曬傪嶨懡偵婰偟偰偄傑偡丏 惍悢嵟揔壔乮惍悢寁夋朄乯偼嫮椡側嵟揔壔庤朄偺堦偮側偺偱偡偑丆乽幚嵺偵夝偒偨偄帪偵擔杮岅偺忣曬偑偁傑傝柍偄乿偲帹偵偟偨偺偑偙偺儁乕僕傪嶌偭偨偒偭偐偗偱偡丏

偍偙偲傢傝

丂偙偺儁乕僕偵彂偄偰偁傞忣曬偼柍曐徹偱偁傝丆昅幰偼堦愗偺愑擟傪帩偪傑偣傫丏 帺屓愑擟偱偛棙梡偔偩偝偄丏 傕偪傠傫丆側傞傋偔惓偟偄偙偲傪彂偔傛偆偵偟傑偡偑丆忣曬偑屆偔側偭偰偄偨傝娫堘偭偰偄偨傝偡傞偙偲傕偁傞偲巚傢傟傑偡丏 偍婥偯偒偺曽偼昅幰傑偱偛楢棈偄偨偩偗傞偲岾偄偱偡丏 摿偵丆儕儞僋愭乮URL/偦偺撪梕乯傗僜儖僶乕婡擻偺曄峏丆惍悢嵟揔壔傗寁嶼婡偺恑曕偵傛傞掕愇偺曄壔側偳偵偛拲堄偔偩偝偄丏

峏怴棜楌

丂偙偺儁乕僕偼2013擭4寧偵岞奐偟丆偦偺屻傕夵掶傪峴偭偰偒傑偟偨偑丆2020擭12寧偐傜峏怴棜楌傪偮偗傞傛偆偵偟傑偟偨丏 側偍丆岆帤扙帤傗偦偺懠偺旝嵶側掶惓偼峏怴棜楌偵婰偟傑偣傫丏

2024擭9寧 2024擭8寧 2024擭5寧 2024擭3寧 2024擭1寧 2023擭8寧 2023擭1寧 2022擭8寧 2022擭5寧 2022擭2寧 2021擭11寧 2021擭9寧 2021擭8寧 2021擭3寧 2020擭12寧
媨戙 棽暯乮搶嫗擾岺戝妛 岺妛晹 抦擻忣曬僔僗僥儉岺妛壢乯

嶲峫暥專

丂擖栧岦偗丆偍傛傃偙偺儁乕僕偵娭學偡傞傕偺偐傜偄偔偮偐傪徯夘偟傑偡丏 暥專 9 偼丆旕彜梡僜儖僶乕偺僟僂儞儘乕僪側偳傕娷傔丆僜儖僶乕傪弶傔偰巊偆嵺偺僈僀僪傪婰偟偰偄傑偡丏 LP 僼傽僀儖偺娙扨側夝愢傕娷傫偱偄傑偡丏 暥專 7 偼丆彮偟屆偔側偭偰偟傑偄傑偟偨偑丆惍悢嵟揔壔偵偮偄偰偛偔娙扨偵傑偲傔偰偁傝傑偡丏 暥專 15 偼丆惍悢嵟揔壔偲偼偳傫側傕偺偐丆嬤擭偺忣曬偑僗儔僀僪宍幃偱傢偐傝堈偔傑偲傔傜傟偰偄傑偡丏 暥專 18 傕丆摨偠偔惍悢嵟揔壔偺擖栧揑側抦幆傪僗儔僀僪宍幃偱傑偲傔偰偄傑偡丏 暥專 1, 5 偼丆惍悢嵟揔壔偵偍偗傞掕幃壔偺僥僋僯僢僋偑擔杮岅偱傑偲傑偭偰偄傑偡丏 暥專 5, 12 偼丆惍悢嵟揔壔傪巊偭偨掕幃壔偺椺偑懡悢婰弎偝傟偰偄傑偡丏 暥專 8 偼丆傗傗愱栧揑偵側傝傑偡偑丆嬤擭偺暘巬尷掕朄偺敪揥偵偮偄偰弎傋偰偄傑偡丏 壓婰偺乽傛偔偁傞幙栤乿偱摿掕偺僜儖僶乕傪懳徾偵偟偨崁偱偼丆僜儖僶乕偵墳偠偰暥專 2, 3, 13 偐傜偺忣曬傪尦偵偟偰偄傑偡丏 暥專 4, 6, 10, 11, 14, 16, 17, 19, 20, 21, 22 偵娭偟偰偼丆乽傛偔偁傞幙栤乿偺屄乆偺崁偐傜嶲徠偟偰偄傑偡丏

  1. 摗峕揘栫丗 惍悢寁夋朄偵傛傞掕幃壔擖栧丏 僆儁儗乕僔儑儞僘丒儕僒乕僠丆57-4 (2012)丆pp. 190-197.
    PDF 僼傽僀儖 乮摗峕愭惗偺偛岤堄偵傛傝丆暥專僼傽僀儖偺宖嵹嫋壜傪捀偄偰偄傑偡乯
  2. Gurobi Optimization: Gurobi Optimizer Reference Manual Version 10.0. Gurobi Optimization, 2022.
  3. IBM ILOG: IBM ILOG CPLEX Optimization Studio 22.1.0 documentation. IBM ILOG, 2022.
  4. T. Koch, T. Achterberg, E. Andersen, O. Bastert, T. Berthold, R. E. Bixby, E. Danna, G. Gamrath, A. M. Gleixner, S. Heinz, A. Lodi, H. Mittelmann, T. Ralphs, D. Salvagnin, D. E. Steffy, K. Wolter: MIPLIB 2010. Mathematical Programming Computation, 3 (2011), pp. 103-163.
    doi:10.1007/s12532-011-0025-9
  5. 媣曐姴梇丆J. P. 儁僪儘僜丆懞徏惓榓丆A. 儗僀僗丗 怴偟偄悢棟嵟揔壔 乣Python 尵岅偲 Gurobi 偱夝偔乣丏 嬤戙壢妛幮丆2012丏 ISBN 978-4-7649-0433-0
  6. H. Mittelmann: Benchmarks for Optimization Software.
    https://plato.asu.edu/bench.html
  7. 媨戙棽暯丆徏堜抦屓丗 偙偙傑偱夝偗傞惍悢寁夋丏 僔僗僥儉/惂屼/忣曬丆50-9 (2006), pp. 363-368.
    乮塸戣丗R. Miyashiro, T. Matsui: Recent Progress in Integer Programming. Systems, Control and Information, 50-9 (2006), pp. 363-368.乯
    doi:10.11509/isciesci.50.9_363丂PDF 僼傽僀儖
  8. 媨戙棽暯丗 偙偙傑偱夝偗傞惍悢寁夋乗嬤擭偺敪揥乗丏 戞20夞RAMP僔儞億僕僂儉榑暥廤 (2008)丆擔杮僆儁儗乕僔儑儞僘丒儕僒乕僠妛夛丆pp. 1-21.
  9. 媨戙棽暯丗 惍悢寁夋僜儖僶乕擖栧丏 僆儁儗乕僔儑儞僘丒儕僒乕僠丆57-4 (2012)丆pp. 183-189.
    PDF 僼傽僀儖
  10. NEOS Server: NEOS Server for Optimization. University of Wisconsin-Madison, 2024.
    https://www.neos-server.org/neos/
  11. E. Rothberg: An Evolutionary Algorithm for Polishing Mixed Integer Programming Solutions. INFORMS Journal on Computing, 19 (2007), pp. 534-541.
    doi:10.1287/ijoc.1060.0189
  12. H. P. Williams: Model Building in Mathematical Programming (5th edition). Wiley, 2013. ISBN 978-1-118-44333-0
    乮戞 3 斉偺栿彂偼亀H. P. 僂僀儕傾儉僗挊丆彫椦塸嶰栿丗 悢棟寁夋儌僨儖偺嶌惉朄丏嶻嬈恾彂丆1995丆ISBN 978-4-782-84601-8亁偱偡乯
  13. Zuse Institute Berlin: SCIP (Version 9.1.1). Zuse Institute Berlin, 2024.
    https://www.scipopt.org/丂 https://www.scipopt.org/doc/html/
  14. H. D. Sherali, J. C. Smith: An Improved Linearization Strategy for Zero-one Quadratic Programming Problems. Optimization Letters, 1 (2007), pp. 33-47.
    doi:10.1007/s11590-006-0019-0
  15. 摗峕揘栫丗 偼偠傔傛偆惍悢寁夋朄丏 僠儏乕僩儕傾儖島墘丆擔杮僆儁儗乕僔儑儞僘丒儕僒乕僠妛夛 2014擭弔婫尋媶敪昞夛丆2014丏
    PDF 僼傽僀儖 乮摗峕愭惗偺偛岤堄偵傛傝丆僗儔僀僪偺宖嵹嫋壜傪捀偄偰偄傑偡乯
  16. E. Balas: Disjunctive Programming. 50 Years of Integer Programming 1958-2008, Chapter 10, pp. 283-340, Springer, 2010.
    doi:10.1007/978-3-540-68279-0_10
  17. 崅栰桽堦丗 ZIMPL尵岅偲SCIP偵傛傞悢棟嵟揔壔丏 愱廋僱僢僩儚乕僋仌僀儞僼僅儊乕僔儑儞丆 24 (2016)丆pp. 9-14.
    doi:10.34360/00005129
  18. 媨戙棽暯丗 惍悢嵟揔壔傾僾儘乕僠傊偺擖栧丏 僠儏乕僩儕傾儖島墘丆揹巕忣曬捠怣妛夛 2019擭憤崌戝夛丆2019丏
    PDF 僼傽僀儖
  19. A. Gleixner, G. Hendel, G. Gamrath, T. Achterberg, M. Bastubbe, T. Berthold, P. Christophel, K. Jarck, T. Koch, J. Linderoth, M. Lübbecke, H. D. Mittelmann, D. Ozyurt, T. K. Ralphs, D. Salvagnin, Y. Shinano: MIPLIB 2017: Data-Driven Compilation of the 6th Mixed-Integer Programming Library. Mathematical Programming Computation, 13 (2021), pp. 443-490.
    doi:10.1007/s12532-020-00194-3
  20. O. Günlük, J. Linderoth: Perspective Reformulations of Mixed Integer Nonlinear Programs with Indicator Variables. Mathematical Programming, Series B, 124 (2010), pp. 183-205.
    doi:10.1007/s10107-010-0360-z
  21. E. Balas: Disjunctive Programming. Springer, 2018. ISBN 978-3-030-00147-6
    doi:10.1007/978-3-030-00148-3
  22. MOSEK: MOSEK Modeling Cookbook 3.3.0.
    https://docs.mosek.com/modeling-cookbook/index.html
  23. 揷拞戝婤丗 楍嫇偺曽朄偵傛傞market split栤戣偺夝朄丏 擔杮僆儁儗乕僔儑儞僘丒儕僒乕僠妛夛 2024擭弔婫尋媶敪昞夛 傾僽僗僩儔僋僩廤丆pp. 230-231丏

傛偔偁傞幙栤

慜彂偒

丂惍悢嵟揔壔僜儖僶乕偵娭偡傞丆傛偔偁傞幙栤偲偦傟偵偮偄偰偺乮偲傝偁偊偢偺乯懳張曽朄傪彂偒傑偟偨丏 偙偙偵彂偐傟偰偄傞曽朄傗僥僋僯僢僋偺岠壥偼埖偆栤戣偺惈幙偵戝偒偔埶懚偡傞偨傔丆枩擻偺嶔偼柍偄偙偲偵偛棷堄偔偩偝偄丏 側偍丆偁傞僜儖僶乕忋偱摿掕偺婡擻傪徯夘偟偰偄傞応崌丆婰嵹偟偰偄側偄僜儖僶乕偵偦偺婡擻偑柍偄偙偲傪堄枴偡傞傕偺偱偼偁傝傑偣傫丏 忣曬偑摿掕偺僜儖僶乕偵曃偭偰偄傞偺偼扨偵杮儁乕僕嶌惉幰偺僜儖僶乕巊梡宱尡偺嵎偵傛傞傕偺偱丆懠偵傕懡悢偺僜儖僶乕偑偁傝傑偡丏 屄乆偺僜儖僶乕偵娭偡傞婰嵹偵偮偄偰偼 [CPLEX] [Gurobi] [SCIP] 偲偄偆儅乕僋傪偮偗傑偟偨丏 [CPLEX] 偼 IBM ILOG CPLEX 22.1乮 嶲峫暥專 3 乯丆[Gurobi] 偼 Gurobi Optimizer 10.0乮 嶲峫暥專 2 乯丆[SCIP] 偼 SCIP 9.1.1乮 嶲峫暥專 13 乯偵偍偗傞忣曬偱偡丏

丂壓婰偺乽傛偔偁傞幙栤乿偱偼丆婎杮揑偵乽夝偒偨偄栤戣傪 LP 僼傽僀儖偲偟偰嶌惉偟丆偦傟傪奺僜儖僶乕偺僐儅儞僪儔僀儞僀儞僞乕僼僃乕僗乮CLI乯偵擖椡偟偰巊偆曽朄乿傪擮摢偵偍偄偰偄傑偡乮[CPLEX]: Interactive Optimizer, [Gurobi]: Interactive Shell, [SCIP]: 僐儅儞僪儔僀儞乯丏 LP 僼傽僀儖偲 CLI 偺慻崌偣偼僜儖僶乕傪巊偆嵟傕扨弮側曽朄偱偡偑丆偙傟偩偗偱傕偐側傝偺偙偲偑偱偒傑偡丏 偨偩偟丆懠偺僾儘僌儔儉偲慻傒崌傢偣偨摦揑側嶌嬈傗丆楍惗惉朄側偳偵戙昞偝傟傞暘巬尷掕朄偺惂屼傪峴偆応崌偼丆奺僜儖僶乕偺 API 傪巊偆偙偲傪偍偡偡傔偟傑偡丏 偙偺偁偨傝偵偮偄偰偼丆乽嫄戝側 LP 僼傽僀儖傪偳偆傗偭偰惗惉偡傟偽傛偄偐乿傕嶲徠偟偰偔偩偝偄丏

丂側偍丆杮儁乕僕偵婰嵹偟偨僷儔儊乕僞柤側偳偼奺僜儖僶乕偺 CLI 偵弨嫆偟偰偄傞偺偱丆API 傪夘偟偰巊偆側偳懠偺曽朄偺帪偵偼婰嵹偟偨僷儔儊乕僞柤偱儅僯儏傾儖傪専嶕偟偰偔偩偝偄丏 傑偨丆僜儖僶乕偺僶乕僕儑儞傾僢僾偵傛偭偰僷儔儊乕僞柤側偳偑偟偽偟偽曄峏偵側傝傑偡偺偱丆偛拲堄偔偩偝偄丏

栚師

  1. 僜儖僶乕偺擖庤丆栤戣偺婰弎丆婎杮揑側僜儖僶乕偺憖嶌
    1. 僜儖僶乕傪擖庤偟偨偄
    2. 栤戣傪僜儖僶乕偵擖椡偡傞偨傔偵偼
    3. 僜儖僶乕偍傛傃 LP 僼傽僀儖偼弨旛偑偱偒偨丏崱偡偖夝偒偨偄
    4. 寁嶼傪搑拞偱掆巭偟偨偄
    5. 栤戣偺撉傒崬傒帪偵拲堄偡傞偙偲
    6. 僜儖僶乕傊偺僐儅儞僪擖椡偵偮偄偰
    7. 僜儖僶乕偺僿儖僾僐儅儞僪傪昞帵偟偨偄
  2. 栤戣傪夝偔嵺偵僄儔乕偑弌傞
    1. 惂栺幃偑挿偡偓偰堦峴偵擖傜側偄丏 傑偨偼 general, binary 愰尵偺屻偵堦峴偵廂傑傜側偄
    2. LP 僼傽僀儖偺撉傒崬傒偱僄儔乕偑弌傞
    3. 惂栺幃 5 x + y = 7 傪昞偡偺偵丆LP 僼傽僀儖偱 a x + y = 7 偍傛傃 a = 5 (傑偨偼 a * x + y = 7 偍傛傃 a = 5) 偲彂偄偨傜僄儔乕偵側偭偨
    4. 暘巬尷掕朄偺搑拞偱儊儌儕偑懌傝側偔側傞
    5. 悢抣攋抅傪婲偙偟偰僄儔乕偵側傞
    6. 栚揑娭悢傑偨偼惂栺幃偵擇師崁傪彂偄偨偑僄儔乕偵側偭偨
    7. [CPLEX] ILOG CPLEX Optimization Studio 偑幚峴偱偒側偄
  3. 夝偄偰傒偨偑寢壥偑偍偐偟偄
    1. 夝偵抣偑昞帵偝傟側偄曄悢偑偁傞
    2. x < 7 偲偄偆惂栺幃傪擖傟偨偺偵丆寁嶼偡傞偲 x 偑 7 偺夝偑弌傞
    3. LP 僼傽僀儖傪嶌偭偨傜丆彑庤偵旕晧曄悢偵側偭偰偄傞丏儅僀僫僗偺抣傪庢傝偆傞曄悢乮帺桼曄悢乯傪嶌傝偨偄
    4. 夝偗傞偵偼夝偗偨偑丆堄恾偟偨栤戣偲偟偰撉傒崬傑傟偰偄側偄傛偆偩
    5. 帺暘偱 LP 僼傽僀儖傪惗惉偟偨偑丆嫋梕夝偑偁傞偼偢偺栤戣側偺偵晄擻 (infeasible) 偵側傞
    6. 暿偺僜儖僶乕偱夝偔偲嵟揔夝偑堘偆
    7. 惍悢曄悢側偺偵丆0.000001 傑偨偼 0.999999 偺傛偆側夝偑弌傞
    8. 暘巬尷掕朄傪恑傔偨傜丆嵟揔惈僊儍僢僾偑憹偊偨
    9. [CPLEX]丂ILOG CPLEX Optimization Studio 偱 LP 僼傽僀儖傪惗惉偟偨偑丆揧帤偑偢傟傞
  4. LP 僼傽僀儖偵偮偄偰
    1. LP 僼傽僀儖偺徻嵶側暥朄偑抦傝偨偄
    2. 栚揑娭悢偑柍偄栤戣傪埖偄偨偄
    3. LP 僼傽僀儖偱僔僌儅婰崋傗攝楍偵憡摉偡傞傕偺傪梡偄偨偄
    4. 嫄戝側 LP 僼傽僀儖傪偳偆傗偭偰惗惉偡傟偽傛偄偐
    5. LP 僼傽僀儖傪帺摦揑偵惍宍偟偨偄
    6. MPS 僼傽僀儖偲偼壗偐丏MPS 僼傽僀儖偲 LP 僼傽僀儖傪曄姺偟偨偄
    7. LP 僼傽僀儖傪 MPS 僼傽僀儖偵曄姺偟偨傜丆栚揑娭悢抣偑儅僀僫僗 1 攞偵側偭偨
    8. LP 僼傽僀儖偱擇師偺栚揑娭悢傪昞偟偨偄
    9. LP 僼傽僀儖偱擇師偺惂栺幃傪昞偟偨偄
    10. LP 僼傽僀儖偱栚揑娭悢偵掕悢崁傪娷傔偨偄
    11. LP 僼傽僀儖偵彂偔曄悢傗惂栺幃偺弴斣傪曄偊偨傜寁嶼帪娫偑曄傢偭偨
    12. Lazy Constraints 傪巊偄偨偄
  5. 僜儖僶乕偺挷惍偲寁嶼懍搙偵偮偄偰
    1. 嵟彫壔栤戣偱側偐側偐壓奅偑忋偑傜側偄丏嵟戝壔栤戣偱側偐側偐忋奅偑壓偑傜側偄
    2. 嵟揔夝偵偙偩傢傜側偄偺偱丆側傞傋偔崅懍偵幙偺椙偄夝偑傎偟偄
    3. 寁嶼帪娫傪惂尷偟偰丆暘巬尷掕朄傪帺摦揑偵掆巭偡傞傛偆偵偟偨偄
    4. 掕傔偨鑷抣偲摨偠偐傛傝椙偄栚揑娭悢抣偺嫋梕夝偑弌偨傜丆暘巬尷掕朄傪帺摦揑偵掆巭偡傞傛偆偵偟偨偄
    5. 嫋梕夝傪梌偊偨忬懺偱暘巬尷掕朄傪僗僞乕僩偟偨偄
    6. 僜儖僶乕傪僸儏乕儕僗僥傿僋僗偲偟偰巊偄偨偄
    7. 暘巬弴彉傪庤摦偱巜掕偟偨偄
    8. 暲楍暘巬尷掕朄偱僗儗僢僪悢傪巜掕偟偨偄
    9. 暲楍暘巬尷掕朄偵偍偄偰丆僗儗僢僪悢傪憹傗偟偨傜抶偔側偭偰偟傑偭偨
    10. 僜儖僶乕偺僆僾僔儑儞 A 偺傒傪僆儞偵偟偨傜寁嶼偑懍偔側傝丆僆僾僔儑儞 B 偺傒傪僆儞偵偟偨傜寁嶼偑懍偔側偭偨偺偱丆僆僾僔儑儞 A 偲 B 傪椉曽僆儞偵偟偨偲偙傠寁嶼偑抶偔側偭偰偟傑偭偨
    11. 寁嶼傪堦搙拞抐偟偰偐傜嵞奐偟偨傜丆拞抐偟側偄応崌偲寁嶼夁掱偑堎側偭偨
    12. 寁嶼帪娫傪惂尷偟偨傜丆惂尷偟側偄応崌偲寁嶼夁掱偑堎側偭偨
    13. 慄宍嵟揔壔栤戣傪暋悢夞夝偄偰傒偨傜丆嵟揔夝偑堎側偭偨
  6. 僜儖僶乕傪傕偭偲曋棙偵巊偄偨偄
    1. 僜儖僶乕偑慜張棟傪偟偨 LP 僼傽僀儖傪尒偨偄
    2. 偳偺傛偆偵暘巬尷掕朄偑恑傫偱偄傞偐傪徻嵶偵娤嶡偟偨偄
    3. 惍悢曄悢偺抣偑堎側傞嵟揔夝偺屄悢傪悢偊偨偄
    4. 儘僌僼傽僀儖偺弌椡愭傪曄峏偟偨偄
    5. 暋悢偺栤戣傪帺摦揑偵夝偐偣偨偄
    6. 暋悢偺栤戣娫偱忣曬傪傗傝偲傝偟偨偄丏懠偺僾儘僌儔儉偲楢実偟偨偄
    7. 寁嶼傪拞抐偡傞偙偲側偔丆巄掕夝偦偺傕偺傪帺摦揑偵婰榐偟偨偄
    8. 曄峏偟偨僷儔儊乕僞愝掕傪昞帵丒曐懚丒暅尦偟偨偄
    9. 栤戣偺摑寁忣曬偑尒偨偄
    10. 梌偊偨惍悢嵟揔壔栤戣偺楢懕娚榓栤戣傪摼偨偄
    11. 梌偊偨慄宍嵟揔壔栤戣偺憃懳栤戣傪摼偨偄
    12. [CPLEX] 巄掕夝偑摼傜傟偨偲偒偺宱夁帪娫傪帺摦揑偵婰榐偟偨偄
    13. [CPLEX] Benders 暘夝傪峴偄偨偄
  7. 掕幃壔偵偮偄偰
    1. 幚嵺偺尰応偱偼巊偊側偄傛偆側摎偊偑弌偰偔傞
    2. 慄宍幃偱彂偗側偄偲巚傢傟傞惂栺忦審傑偨偼栚揑娭悢偑偁傞
    3. 0-1 曄悢 x, y 偵懳偟偰丆乽z = x * y乿偵懳墳偡傞 0-1 曄悢 z 傪巊偄偨偄
    4. 惂栺幃傪懠偺曄悢偵墳偠偰僆儞丒僆僼偟偨偄丏乽傕偟亊亊側傜偽仮仮乿偲偄偆傛偆側惂栺幃偑彂偒偨偄
    5. 曄悢偺屄悢偑彮側偄栤戣側偺偵夝偗側偄丏曄悢偺屄悢偑彮側偄掕幃壔偵曄偊偨傜抶偔側偭偨
    6. 慄宍傗擇師偱側偄栚揑娭悢傪埖偄偨偄
  8. 偦偺懠偺忣曬
    1. 惍悢嵟揔壔栤戣偺儀儞僠儅乕僋栤戣廤偑抦傝偨偄
    2. 彜梡僜儖僶乕偲旕彜梡僜儖僶乕偼寁嶼帪娫偑偳偺掱搙堎側傞偺偐
    3. 偳偺彜梡僜儖僶乕偑懍偄偺偐
    4. 彜梡僜儖僶乕傪帋偟偰傒偨偄

  1. 僜儖僶乕偺擖庤丆栤戣偺婰弎丆婎杮揑側僜儖僶乕偺憖嶌

丂偙偙偱偼丆旕彜梡僜儖僶乕偺擖庤偐傜丆僜儖僶乕偺娙扨側憖嶌傑偱傪徯夘偟傑偡丏 嶲峫暥專 9 偵傕丆堦捠傝偺棳傟偑傑偲傔偰彂偄偰偁傝傑偡偺偱嶲峫偵偟偰偔偩偝偄丏

  1. 僜儖僶乕傪擖庤偟偨偄

    丂傑偢偼丆旕彜梡僜儖僶乕 SCIP 傪帋偟偰傒傑偟傚偆丏 懠偵傕奺庬偺僜儖僶乕偼偁傝傑偡偑丆偲傝偁偊偢偼 SCIP 偐傜僗僞乕僩偡傞偺偑偍庤寉偐偲巚傢傟傑偡丏 SCIP 偼 2023 擭枛偺帪揰偱丆旕彜梡偱偼嵟傕崅懍側僜儖僶乕偺堦偮偱偡乮彜梡僜儖僶乕偱偼 SCIP 傛傝崅懍側傕偺傕懚嵼偟傑偡乯丏 偝傜偵丆2022擭 12 寧偵岞奐偝傟偨 SCIP 8.0.3 埲崀偐傜偼 Apache 2.0 儔僀僙儞僗 偑揔梡偝傟偰偄傑偡丏 偙傟偵傛傝丆塩棙梡搑偵偍偄偰傕儔僀僙儞僗偺斖埻撪偱柍彏棙梡偑壜擻偵側偭偰偄傑偡丏
    丂SCIP乮 嶲峫暥專 13 乯偺僟僂儞儘乕僪儁乕僕乮 https://www.scipopt.org/ 乯偐傜丆庤帩偪偺寁嶼婡娐嫬偵偁偭偨僼傽僀儖傪僟僂儞儘乕僪偟傑偡丏 側偍丆Apache 2.0 儔僀僙儞僗偑揔梡偝傟偰偄傞偺偼 SCIP 8.0.3 埲崀偱偡偺偱偛拲堄偔偩偝偄丏
    丂傾僇僨儈僢僋娐嫬偵強懏偟偰偄傞曽偼丆NEOS 僒乕僶乕偵偍偗傞彜梡僜儖僶乕偺棙梡傕壜擻偱偡丏 乽彜梡僜儖僶乕傪帋偟偰傒偨偄乿傪嶲徠偟偰偔偩偝偄丏

  2. 栤戣傪僜儖僶乕偵擖椡偡傞偨傔偵偼

    丂偄傠偄傠側曽朄偑偁傝傑偡偑丆嵟弶偼 LP 僼傽僀儖傪棙梡偡傞偺偑娙扨偩偲巚傢傟傑偡丏 LP 僼傽僀儖偲偼丆悢棟嵟揔壔栤戣傪昞偡僼傽僀儖宍幃偺堦庬偱丆奼挘巕偑 .lp 偺僥僉僗僩僼傽僀儖偱偡丏 悢幃傪傎傏偦偺傑傑僥僉僗僩偵偟偨宍偵側偭偰偄傑偡丏 LP 僼傽僀儖偼丆杮儁乕僕偱摿偵徻偟偔徯夘偟偰偁傞 CPLEX丆Gurobi丆SCIP 偺偄偢傟偺僜儖僶乕偵偍偄偰傕巊梡壜擻偱丆懠偵傕懳墳偟偰偄傞僜儖僶乕偑偄偔偮偐懚嵼偟傑偡丏 LP 僼傽僀儖偺椺傪埲壓偵嵹偣傑偡丏 娙扨側暥朄偵偮偄偰偼丆嶲峫暥專 9 傪嶲徠偟偰偔偩偝偄丏 
    \ LP 僼傽僀儖偺僒儞僾儖
\ 僐儊儞僩偼 \ 偺屻傠偑堦峴僐儊儞僩偵側傝傑偡

minimize                      \ minimize 偼梊栺岅丏嵟戝壔偺応崌偼 maximize
- 3 x + 4.5 y - 2 z(1) + f    \ 栚揑娭悢乮暋悢峴偵側偭偰傕傛偄乯

subject to                    \ subject to 偼梊栺岅丏偙偙偐傜屻傠偵惂栺幃
c1: - g(1,1) + g(1,2) <= 5    \ 惂栺幃偵偼愭摢偵柤慜傪偮偗傜傟傞乮柤慜亄僐儘儞乯
c2: 3 g(1,1) - 7 g(1,2)       \ 弴偵 c1, c2, ... 偲斣崋傪怳偭偰偍偔偲傛偄
    + z(2) >= -10             \ 摨椶崁偼堦偮偵傑偲傔丆曄悢崁偼嵍曈偵丆掕悢崁偼塃曈偵抲偔
c3: 2 f - g(1,1)              \ 惂栺幃偼暋悢峴偵傑偨偑偭偰傕傛偄
    = 6                       \ 偨偩偟斾妑墘嶼巕偲掕悢崁偼摨偠峴偵彂偔
c4: + 1 x + 0.5 y = -4.6      \ 愭摢偺 + 傗學悢偺 1 偼偁偭偰傕柍偔偰傕傛偄
                              

bounds                        \ bounds 偼梊栺岅丏偙偙偐傜帺桼曄悢偺愰尵側偳傪峴偆
x free                        \ 堦峴偵堦偮偺曄悢丆偦傟偵懕偗偰 free 偲偡傞偲丆
g(1,1) free                   \ 偦偺曄悢偼帺桼曄悢偵側傞
                              \ free 愰尵傪偟側偄曄悢偼旕晧忦審偑壽偝傟傞

general                       \ general 偼梊栺岅丏惍悢曄悢偵偡傞曄悢傪彂偔
g(1,1) g(1,2)                 \ free 愰尵傪偟偰偄側偗傟偽帺摦揑偵旕晧忦審偑偮偔偙偲偵拲堄
                              \ 偙偺応崌丆g(1,2) 偼旕晧偺惍悢曄悢

binary                        \ binary 偼梊栺岅丏0-1 曄悢偵偡傞曄悢傪彂偔
z(1)                          \ 偙偺応崌丆z(1) 偲 z(2) 偼 0-1 曄悢
z(2)                          \ general, binary 愰尵偲傕暋悢峴偵搉偭偰峔傢側偄

end                           \ end 偼梊栺岅丏僼傽僀儖偺嵟屻偵彂偔

  3. 僜儖僶乕偍傛傃 LP 僼傽僀儖偼弨旛偑偱偒偨丏崱偡偖夝偒偨偄

    丂LP 僼傽僀儖偺柤慜傪偙偙偱偼壖偵 test1.lp 偲偟傑偡丏 埲壓偺僐儅儞僪偱丆栤戣偺撉傒崬傒丆寁嶼偺幚峴丆嵟揔夝偺昞帵偑偱偒傑偡丏

  4. 寁嶼傪搑拞偱掆巭偟偨偄

    丂擄偟偄惍悢嵟揔壔栤戣傪嵟屻傑偱夝偒偒傟側偄偲偒偵丆寁嶼傪庤摦偱掆巭偡傞曽朄偱偡丏 偁傜偐偠傔寁嶼帪娫傪惂尷偡傞応崌偼丆乽寁嶼帪娫傪惂尷偟偰丆暘巬尷掕朄傪帺摦揑偵掆巭偡傞傛偆偵偟偨偄乿傪嶲徠偟偰偔偩偝偄丏

  5. 栤戣偺撉傒崬傒帪偵拲堄偡傞偙偲

    丂LP 僼傽僀儖傪撉傒崬傓嵺偺拲堄偱偡丏

  6. 僜儖僶乕傊偺僐儅儞僪擖椡偵偮偄偰

    丂僐儅儞僪擖椡偺徣棯曽朄偵偮偄偰偱偡丏

  7. 僜儖僶乕偺僿儖僾僐儅儞僪傪昞帵偟偨偄

    丂奺僜儖僶乕偺徻嵶偼儅僯儏傾儖偵婰嵹偝傟偰偄傑偡偑丆僜儖僶乕忋偱傕娙堈僿儖僾傪嶲徠偱偒傑偡丏
栚師傊栠傞

  1. 栤戣傪夝偔嵺偵僄儔乕偑弌傞

丂僜儖僶乕偱栤戣傪夝偔嵺偵僄儔乕偑弌傞応崌偺懳張曽朄偱偡丏

  1. 惂栺幃偑挿偡偓偰堦峴偵擖傜側偄丏 傑偨偼 general, binary 愰尵偺屻偵堦峴偵廂傑傜側偄

    丂堦峴偵廂傔傞昁梫偼偁傝傑偣傫丏 LP 僼傽僀儖偺堦峴偺暥帤悢偑偁傑傝偵懡偡偓傞偲丆僜儖僶乕偵傛偭偰偼愗傝幪偰偰偟傑偄傑偡偺偱揔媂夵峴偑昁梫偱偡丏 梊栺岅傗曄悢柤偺搑拞偱側偗傟偽丆堦偮偺惂栺幃偺搑拞偱夵峴偟偰峔偄傑偣傫乮偨偩偟丆斾妑墘嶼巕偲塃曈崁偼摨偠峴偵彂偒傑偡乯丏 堦峴偵曄悢偑 1 屄偟偐側偄傛偆側廲挿偺 LP 僼傽僀儖偱傕栤戣偁傝傑偣傫乮LP 僼傽僀儖偺惍宍 偼僜儖僶乕偱峴偊傑偡乯丏

  2. LP 僼傽僀儖偺撉傒崬傒偱僄儔乕偑弌傞

    丂傑偢偼丆梊栺岅乮minimize 摍乯偺僗儁儖儈僗偑柍偄偐妋擣偟偰偔偩偝偄丏 戝晹暘偺梊栺岅偼婎杮揑偵怴偟偄峴偐傜巒傔丆偦偺屻傠偱夵峴偟傑偡丏 慄宍惂栺幃傪彂偔偺偵 [ ] 傗 / 傗 * 側偳傪巊偭偰偄側偄偱偟傚偆偐乮偙傟傜偺婰崋偼擇師崁傪昞偡偺偵巊偄傑偡乯丏 掕悢學悢偲曄悢偺愊偼丆僗儁乕僗傪巊偭偰昞偟傑偡丏 傑偨丆e 傗 E 偼 1e4 y 側偳偺巜悢昞婰偵傕巊傢傟傞偨傔丆曄悢柤偺偮偗曽偵拲堄偟傑偟傚偆丏

  3. 惂栺幃 5 x + y = 7 傪昞偡偺偵丆LP 僼傽僀儖偱 a x + y = 7 偍傛傃 a = 5 (傑偨偼 a * x + y = 7 偍傛傃 a = 5) 偲彂偄偨傜僄儔乕偵側偭偨

    丂a 偑曄悢偲傒側偝傟偰丆惂栺幃偵曄悢偺愊偑娷傑傟偰偟傑偆偨傔偱偡丏 LP 僼傽僀儖偱偼丆掕悢學悢偼悢帤傪捈愙彂偒丆偦偺屻傠偵僗儁乕僗傪嬻偗偰學傞曄悢傪抲偒傑偡丏 偨偩偟丆掕悢傪曄悢偵偟偰傕惂栺幃偑慄宍偵側傞傑傑側傜丆掕悢傪曄悢偵抲偒偐偊偰彂偄偰偍偔偙偲偼壜擻偱偡丏 椺偊偽丆惂栺幃 5 x + y = 7 傪 LP 僼傽僀儖偱 5 x + y - b = 0 偍傛傃 b = 7 偲彂偄偰傕丆曄悢偺愊偵偼側偭偰偄側偄偺偱慄宍惂栺幃偲傒側偝傟傑偡丏 尗偄僜儖僶乕側傜偽丆抣偑屌掕偝傟偨曄悢偼慜張棟偱彍嫀偟偰偟傑偆偨傔丆幚嵺偺岠棪傕曄傢傝偁傝傑偣傫丏

  4. 暘巬尷掕朄偺搑拞偱儊儌儕偑懌傝側偔側傞

    丂栤戣偺僒僀僘偑偦傟傎偳戝偒偔側偄偺偵儊儌儕晄懌偵側傞応崌偼丆栤戣偺惈幙傕偟偔偼掕幃壔乮偁傞偄偼偦偺椉曽乯偑偁傑傝椙偔側偄偙偲傪帵嵈偟偰偄傑偡丏 壜擻側傜偽椙偄掕幃壔偵曄偊傞偺偑朷傑偟偄偱偡偑丆偲傝偁偊偢僜儖僶乕偺僆僾僔儑儞偱懳嶔傪偲傞偙偲偼偱偒傑偡丏 側偍丆暘巬尷掕朄偱悢 GB 偺儊儌儕傪怘偄偮傇偟偰偄傞傛偆側応崌偵偼丆寁嶼傪懕偗傞偲偡偖偵偦偺攞偺儊儌儕傪徚旓偟偰偟傑偆偙偲傕懡偔丆儊儌儕傪懡彮憹愝偟偰傕偁傑傝岠壥偑側偄偙偲偑偁傝傑偡丏

  5. 悢抣攋抅傪婲偙偟偰僄儔乕偵側傞

    丂旕忢偵戝偒側學悢傪帩偮惂栺幃偑崿嵼偡傞栤戣側偳丆悢抣揑偵晄埨掕偵側傞働乕僗偑偄偔偮偐偁傝傑偡丏 摿偵丆惂栺幃偵擇師崁傪娷傓栤戣偼悢抣揑偵晄埨掕偵側傝偑偪偱偡丏 偙偺傛偆側応崌丆惂栺幃傪慜傕偭偰僗働乕儕儞僌偟偰偍偔偲椙偄寢壥偵側傞偙偲傕偁傝傑偡丏 丂傑偨丆栚揑娭悢偑撌擇師娭悢乮嵟戝壔栤戣偺応崌偼墯擇師娭悢乯偺偼偢側偺偵丆悢抣岆嵎偺塭嬁偱乽撌偱側偄乿偲偄偆僄儔乕偑弌傞応崌偑偁傝傑偡丏 偙偺応崌丆栚揑娭悢偺學悢傪慡偰惍悢偵僗働乕儕儞僌偡傞丆懳妏崁偺學悢傪旝彫偵憹壛偝偣傞丆怴偟偄曄悢偲惂栺幃傪摫擖偟偰學悢峴楍傪慳偵偡傞側偳偺懳嶔傪帋偟偰傒偰偔偩偝偄丏

  6. 栚揑娭悢傑偨偼惂栺幃偵擇師崁傪彂偄偨偑僄儔乕偵側偭偨

    丂偄偔偮偐偺椺奜傪彍偄偰丆栚揑娭悢偼慄宍娭悢傑偨偼撌擇師娭悢乮嵟戝壔栤戣偺応崌偵偼墯擇師娭悢乯偱側偔偰偼側傝傑偣傫丏 偝傜偵丆惂栺幃偺応崌偼惍悢忦審傪彍偄偨嫋梕椞堟偑撌偱偁傞偙偲偑昁梫偱偡丏 傑偨擇師崁偑偁傞応崌偵偼丆栚揑娭悢偐惂栺幃偐偱 LP 僼傽僀儖偺彂偒曽偑庒姳曄傢傝傑偡丏 乽LP 僼傽僀儖偱丆擇師偺栚揑娭悢傪昞偟偨偄乿乽LP 僼傽僀儖偱丆擇師偺惂栺幃傪昞偟偨偄乿 偺崁傕嶲徠偟偰偔偩偝偄丏 側偍丆0-1 曄悢偺愊偵娭偟偰偼丆摿暿側慜張棟傪巤偟偰擟堄偺宍偺擇師崁偑埖偊傞僜儖僶乕偑偁傞傛偆偱偡丏 乽0-1 曄悢 x, y 偵懳偟偰丆乽z = x * y乿偵懳墳偡傞 0-1 曄悢 z 傪巊偄偨偄乿傕嶲峫偵偟偰偔偩偝偄丏 丂傑偨丆栚揑娭悢偑撌擇師娭悢乮嵟戝壔栤戣偺応崌偼墯擇師娭悢乯偺偼偢側偺偵丆悢抣岆嵎偺塭嬁偱乽撌偱側偄乿偲偄偆僄儔乕偑弌傞応崌偑偁傝傑偡丏 偙偺応崌丆栚揑娭悢偺學悢傪慡偰惍悢偵僗働乕儕儞僌偡傞丆懳妏崁偺學悢傪旝彫偵憹壛偝偣傞丆怴偟偄曄悢偲惂栺幃傪摫擖偟偰學悢峴楍傪慳偵偡傞側偳偺懳嶔傪帋偟偰傒偰偔偩偝偄丏

  7. [CPLEX]丂ILOG CPLEX Optimization Studio 偑幚峴偱偒側偄

    丂CPLEX 偵晅懏偟偰偄傞 CPLEX Optimization Studio 傪巊梡偟偰偄偰丆栤戣傪 OPL 尵岅偱惓偟偔婰弎偟偨偵傕偐偐傢傜偢丆幚峴偡傞偲乽慖戰傪婲摦偱偒傑偣傫丏傑偨丆嵟嬤婲摦偝傟偰偄傑偣傫乿偲昞帵偝傟偰恑傑側偄丆偁傞偄偼幚峴偱偒偰傕乽晄柧側僄儔乕乿偑弌偰巭傑偭偰偟傑偆応崌偵偼丆僨僼僅儖僩偱乽峔惉1乿偵側偭偰偄傞幚峴峔惉偺柤慜傪敿妏塸悢帤偵曄峏偟偰偔偩偝偄丏
    丂傑偨偼丆CPLEX Optimization Studio 偺僀儞僞乕僼僃乕僗傪慡偰塸岅偵偟偰偟傑偆曽朄偑偁傝傑偡丏 Windows 偺応崌偼丆僔儑乕僩僇僢僩偺僾儘僷僥傿偺儕儞僋愭傪 "C:\...\oplide.exe" -nl en_US 偺傛偆偵捈偡偙偲偱塸岅斉偑婲摦偟傑偡乮-nl en_US 偼 " " 偺奜懁偵彂偔乯丏 僐儅儞僪儔僀儞偐傜偺婲摦偺応崌偼丆堷悢偵 -nl en_US 傪偮偗偰婲摦偟偰偔偩偝偄丏
栚師傊栠傞

  1. 夝偄偰傒偨偑寢壥偑偍偐偟偄

丂栤戣偼夝偗偨偑丆偳偆傕寢壥偑曄偩偲偄偆偲偒偵偼偙偪傜傪嶲峫偵偟偰偔偩偝偄丏

  1. 夝偵抣偑昞帵偝傟側偄曄悢偑偁傞

    丂戝晹暘偺僜儖僶乕偱丆抣偑 0 偺曄悢偼昞帵偝傟傑偣傫乮戝婯柾側嵟揔壔栤戣偺夝偱偼丆曄悢偺抣偺偆偪傎偲傫偳偑 0 偱偁傞偙偲偑懡偄偨傔乯丏

  2. x < 7 偲偄偆惂栺幃傪擖傟偨偺偵丆寁嶼偡傞偲 x 偑 7 偺夝偑弌傞

    丂LP 僼傽僀儖偱偼丆摍崋柍偟偺晄摍崋偼丆摍崋晅偒偺晄摍崋偲摨偠堄枴偵側傝傑偡丏

  3. LP 僼傽僀儖傪嶌偭偨傜丆彑庤偵旕晧曄悢偵側偭偰偄傞丏儅僀僫僗偺抣傪庢傝偆傞曄悢乮帺桼曄悢乯傪嶌傝偨偄

    丂LP 僼傽僀儖偱偼丆掕媊偟偨曄悢偼僨僼僅儖僩偱旕晧忦審偑壽偝傟偰偄傑偡丏 偙傟傪庢傝彍偔偵偼丆free 愰尵傪梡偄傑偡丏 乽栤戣傪僜儖僶乕偵擖椡偡傞偨傔偵偼乿傑偨偼 嶲峫暥專 9 傪嶲徠偟偰偔偩偝偄丏

  4. 夝偗傞偵偼夝偗偨偑丆堄恾偟偨栤戣偲偟偰撉傒崬傑傟偰偄側偄傛偆偩

    丂堄恾偟偨捠傝偺栤戣偵側偭偰偄側偄偲巚傢傟傞応崌偼丆僜儖僶乕偵撉傒崬傑偣偨 LP 僼傽僀儖傪堦搙彂偒弌偝偣傞偲丆栤戣偑惓偟偄偐偳偆偐妋偐傔傜傟傑偡丏 埲壓丆LP 僼傽僀儖偺柤慜傪壖偵 test1.lp 偲偟傑偡丏

  5. 帺暘偱 LP 僼傽僀儖傪惗惉偟偨偑丆嫋梕夝偑偁傞偼偢偺栤戣側偺偵晄擻 (infeasible) 偵側傞

    丂嵟嬤偺僜儖僶乕偵偼丆屳偄偵憡斀偡傞嵟彫偺惂栺廤崌傪弌椡偡傞婡擻偑偮偄偰偄傞偙偲偑偁傝傑偡丏 偙偺婡擻偼丆嵟揔壔栤戣偑晄擻乮幚峴晄擻丆infeasible乯側帪偵嶔傪専摙偡傞偩偗偱側偔丆LP 僼傽僀儖惗惉偺僶僌傪庢傞偺偵傕偲偰傕曋棙偱偡丏 偨偩偟丆栤戣僒僀僘傪彫偝偔偟側偄偲寁嶼帪娫偑偐偐傞偨傔丆側傞傋偔婯柾偺彫偝側栤戣偐傜僥僗僩偡傞偺偑傛偄偱偟傚偆丏

  6. 暿偺僜儖僶乕偱夝偔偲嵟揔夝偑堘偆

    丂嵟揔夝偑暋悢懚嵼偡傞応崌丆夝偑堎側傞偙偲偼偁傝偊傑偡丏 嵟揔抣偦偺傕偺偑堘偭偰偄傞応崌偼丆偍偦傜偔暘巬尷掕朄傪廔椆偝偣傞敾掕忦審偺栤戣偱偡丏 丂側偍丆崿崌惍悢擇師悕嵟揔壔栤戣 (MISOCO) 側偳丆惂栺幃偵擇師崁傪娷傓惍悢嵟揔壔栤戣偼悢抣岆嵎偵庛偔丆僜儖僶乕偑娫堘偭偨夝傪嵟揔夝偲偟偰曉偟偰偔傞偙偲偑偁傝傑偡丏 偙傟偵偼寛掕揑側懳嶔偼側偄偺偱偡偑丆壜擻側傜偽暋悢偺僜儖僶乕側偳偱僠僃僢僋偡傞偲僄儔乕傪尭傜偡偙偲偑偱偒傑偡丏

  7. 惍悢曄悢側偺偵丆0.000001 傑偨偼 0.999999 偺傛偆側夝偑弌傞

    丂傑偢偼丆摼傜傟偨夝傪惍悢偵娵傔偰傒偰丆嫋梕夝偑偒偪傫偲摼傜傟偰偄傞偐偳偆偐傪妋擣偟傑偟傚偆丏 師偵丆big-M 朄偱嫄戝側掕悢傪巊偭偰偄側偄偐妋擣偟偰偔偩偝偄丏 Big-M 朄偱昁梫埲忋偵戝偒側掕悢傪巊偆偙偲偼丆悢抣埨掕惈偍傛傃寁嶼懍搙傪懝偹傑偡丏 側偍丆寁嶼偵偼岆嵎偑娷傑傟傑偡偺偱摼傜傟偨寢壥傪張棟偡傞嵺偵偼拲堄偑昁梫偱偡丏 椺偊偽丆0-1 曄悢偲偟偰偄偰傕 0.000001 偺傛偆側抣偑摼傜傟傞偙偲偑偁傞偨傔丆乽int 宆偺 0 偲斾妑偡傞乿偲偄偆傛偆側張棟偼岆偭偨寢壥傪彽偔偙偲偑偁傝傑偡丏

  8. 暘巬尷掕朄傪恑傔偨傜丆嵟揔惈僊儍僢僾偑憹偊偨

    丂捠忢丆暘巬尷掕朄偵偍偗傞憡懳揑側嵟揔惈僊儍僢僾乮忋奅偲壓奅偺妱崌乯偼扨挷偵尭彮偟傑偡偑丆忋奅偲壓奅偺惓晧偑堎側傞抜奒偱偼丆忋壓奅偑夵慞偝傟偨偲偒偵堦帪揑偵嵟揔惈僊儍僢僾偺抣偑憹壛偟偨傛偆偵尒偊傞偙偲偑偁傝傑偡丏 偙傟偼丆僜儖僶乕偵傛傞嵟揔惈僊儍僢僾偺寁嶼曽朄偵婲場偡傞傕偺偱偡丏 偙偺傛偆側抜奒偵偍偄偰偼丆憡懳揑側嵟揔惈僊儍僢僾偺抣偼暘巬尷掕朄偺恑峴忬嫷偺栚埨偲偟偰偁傑傝嶲峫偵側傝傑偣傫丏 愨懳揑側嵟揔惈僊儍僢僾乮忋奅偲壓奅偺嵎乯傪娤嶡偟偰偔偩偝偄丏
    丂傑偨丆嵟揔抣偑 0 偱偁傞傛偆側栤戣偺応崌丆憡懳揑側嵟揔惈僊儍僢僾偼暘巬尷掕朄偺恑峴忬嫷偺栚埨偲偟偰傎偲傫偳栶偵棫偪傑偣傫丏 偙偺応崌偵傕丆愨懳揑側嵟揔惈僊儍僢僾傪嶲峫偵偟偰偔偩偝偄丏

  9. [CPLEX]丂ILOG CPLEX Optimization Studio 偱 LP 僼傽僀儖傪惗惉偟偨偑丆揧帤偑偢傟傞

    丂CPLEX 偵晅懏偟偰偄傞 CPLEX Optimization Studio 偵偍偄偰 OPL 尵岅偱栤戣傪婰弎偟丆幚峴偡傞偲 LP 僼傽僀儖傪惗惉偡傞傛偆偵愝掕偟偰偄傞偲偒丆攝楍曄悢偺揧帤傪 1 偐傜巒傔傞傛偆偵偟偰偄偰傕丆LP 僼傽僀儖偱偼 0 偐傜巒傑傞傛偆偵側偭偰偄傞応崌偑偁傝傑偡乮偦偆側傜側偄帪傕偁傝傑偡乯丏 LP 僼傽僀儖偵傕栤戣帺懱偼惓偟偔婰弎偝傟偰偄傞偺偱偡偑丆偙偺尰徾偑婲偒傞偲揧帤偑偢傟偰偄傞偺偱恖娫偑夝傪夝庍偡傞嵺偵娫堘偭偰偟傑偆壜擻惈偑偁傝傑偡丏 偙偺栤戣偑敪惗偟偨応崌偼丆OPL 尵岅忋偱攝楍曄悢偺揧帤傪 0 偐傜巒傔傞傛偆偵偟丆揧帤 0 偵懳墳偡傞曄悢偵偼掕悢傪梌偊偰偟傑偊偽揧帤偑偢傟傞偙偲偼偲傝偁偊偢側偔側傝傑偡丏
栚師傊栠傞

  1. LP 僼傽僀儖偵偮偄偰

丂LP 僼傽僀儖偵偮偄偰偺丆傛傝嵶偐偄忣曬偱偡丏

  1. LP 僼傽僀儖偺徻嵶側暥朄偑抦傝偨偄

    丂LP 僼傽僀儖偵偼丆摑堦偝傟偨僼僅乕儅僢僩偑懚嵼偣偢丆僜儖僶乕偛偲偵撈帺偺奼挘偑偝傟偰偄傑偡丏 杮儁乕僕偍傛傃 嶲峫暥專 9 偱偼丆懡偔偺娐嫬偱埖偊傞偲巚傢傟傞丆CPLEX 偱梡偄傜傟偰偄傞 LP 僼傽僀儖宍幃偺堦晹暘傪傕偲偵婰弎偟偰偄傑偡丏 僜儖僶乕偛偲偵掕傔傜傟偰偄傞 LP 僼傽僀儖偺徻嵶側暥朄偼丆埲壓傪嶲徠偟偰偔偩偝偄丏

  2. 栚揑娭悢偑柍偄栤戣傪埖偄偨偄

    丂minimize乮偁傞偄偼 maximize乯偺師偺峴偵 subject to 偲偟偰惂栺幃傪彂偒巒傔偰偔偩偝偄丏 偨偩偟丆栚揑娭悢偑柍偄栤戣偱傕丆揔摉側栚揑娭悢傪愝掕偟偰傗傞偙偲偵傛傝寁嶼偑崅懍偵側傞偙偲偑偁傝傑偡乮抶偔側傞偙偲傕偁傝傑偡乯丏 椺偊偽丆曄悢偺弌尰偟偨弴偵廳傒偺學悢傪 1, 2, ... 偲偟偰栚揑娭悢偵擖傟偰傗傞摍偱偡丏

  3. LP 僼傽僀儖偱僔僌儅婰崋傗攝楍偵憡摉偡傞傕偺傪梡偄偨偄

    丂LP 僼傽僀儖偵偦偺傛偆側婡擻偼側偄偨傔丆1000 屄偺曄悢偺榓偼乮揔媂夵峴偟側偑傜乯儀僞偵彂偔偙偲偵側傝傑偡丏 傑偨 x(1) 偲彂偄偰傕丆x(2) 偲偼摿偵娭學偑偁傝傑偣傫偟丆x2)))) 偲偄偆傛偆側曄悢傕壜擻偱偡丏 傕偪傠傫丆恖娫偑曄悢偺堄枴傪棟夝偟傗偡偄傛偆偵丆x(1), y(3,4), z_7_5_6 側偳偲攝楍晽偵彂偔偙偲偼悇彠偝傟傑偡丏 乮偨偩偟丆[ ] 偼 LP 僼傽僀儖偱擇師幃傪昞偡嵺偺婰崋偺偨傔梡偄傞偙偲偑偱偒傑偣傫丏乯

  4. 嫄戝側 LP 僼傽僀儖傪偳偆傗偭偰惗惉偡傟偽傛偄偐

    丂戝偒偔暘偗偰埲壓偺 3 偮偺曽朄偑偁傝傑偡丗 乽僾儘僌儔儈儞僌尵岅偱 LP 僼傽僀儖傪惗惉偡傞乿 乽悢棟嵟揔壔儌僨儕儞僌僣乕儖傪巊偆乿 乽僜儖僶乕偵弨旛偝傟偰偄傞 API 側偳偱丆乮LP 僼傽僀儖傪夘偝偢乯僟僀儗僋僩偵栤戣惗惉偍傛傃媮夝傪峴偆乿丏 偦傟偧傟偺棙揰偲寚揰偼埲壓偺捠傝偱偡丏
    丂乽僾儘僌儔儈儞僌尵岅偱 LP 僼傽僀儖傪惗惉偡傞乿丗 巊偄姷傟偰偄傞僾儘僌儔儈儞僌尵岅偱僥僉僗僩傪弌椡偝偣傟偽傛偄偺偱丆堦斣偍庤寉偱偡丏 杮儁乕僕偱傕丆LP 僼傽僀儖亄奺僜儖僶乕偺 CLI 傪梡偄偨曽朄傪憐掕偟偰徯夘偟偰偄傑偡丏 傑偨丆栤戣傪 LP 僼傽僀儖傪夘偟偰埖偆偨傔丆偁傞僜儖僶乕偐傜懠偺僜儖僶乕偵忔傝姺偊傞偙偲傕梕堈側偺傕儊儕僢僩偱偡丏 偨偩偟偁偔傑偱傕 LP 僼傽僀儖儀乕僗偺偨傔丆懠偺曽朄偲斾妑偡傞偲僜儖僶乕偺暋嶨側惂屼偑擄偟偄偲偄偆僨儊儕僢僩傕偁傝傑偡偑丆栤戣傪撈棫偵夝偔偩偗側傜偽偙偺曽朄偱傕傎偲傫偳晄曋偼側偄偲巚傢傟傑偡丏
    丂乽悢棟嵟揔壔儌僨儕儞僌僣乕儖傪巊偆乿丗 儊儕僢僩偼丆悢幃偱昞尰偝傟偨儌僨儖偐傜僾儘僌儔儉偑嶌傝傗偡偄偙偲偱偡丏 椺偊偽丆偁傞儌僨儕儞僌尵岅偱偼乽x1 偐傜 x100 傑偱偺榓偑 3 埲壓乿傪昞尰偡傞偺偵 sum(i in 1..100)(x[i]) <= 3 偲彂偔偙偲偑偱偒傞側偳丆悢幃偐傜僾儘僌儔儉傊捈姶揑偵彂偒姺偊偑壜擻偲側偭偰偄傑偡丏 懠偵傕丆傛偔巊偆僞僀僾偺惂栺幃偵懳墳偡傞梊栺岅偑弨旛偝傟偰偄傞側偳丆悢妛揑掕幃壔偑姰惉偟偰偐傜偺曄姺嶌嬈偑旕忢偵崅懍偵峴偊傞偙偲偑棙揰偱偡丏 偨偩偟丆悢棟嵟揔壔儌僨儕儞僌僣乕儖傪峸擖偟丆偦偙偱巊梡偝傟偰偄傞儌僨儕儞僌尵岅傪廗摼偡傞庤娫偑偐偐傝傑偡丏 斈梡偺僾儘僌儔儈儞僌尵岅傪妎偊傞傛傝偼梕堈偱偡偑丆偁傞掱搙偺帪娫揑丒嬥慘揑側弶婜搳帒偑昁梫偵側傝傑偡丏 僜儖僶乕偵傛偭偰偼丆悢棟嵟揔壔儌僨儕儞僌僣乕儖偑摨崼偝傟偰偄傞傕偺傕偁傝傑偡丏
    丂乽僜儖僶乕偵弨旛偝傟偰偄傞 API 側偳偱丆乮LP 僼傽僀儖傪夘偝偢乯僟僀儗僋僩偵栤戣惗惉偍傛傃媮夝傪峴偆乿丗 栤戣僼傽僀儖偺惗惉偵偲偳傑傜偢丆暘巬弴彉偺憖嶌側偳暘巬尷掕朄偺崅搙側惂屼偑壜擻偵側傞偺偑堦斣偺儊儕僢僩偲側傝傑偡丏 楍惗惉朄側偳丆暋悢偺栤戣娫偱忣曬傪傗傝偲傝偡傞昁梫偑偁傞応崌偵偼丆偙偺曽朄傪棙梡偡傞偺偑尰幚揑偱偟傚偆丏 偨偩偟 API 偺彂幃側偳偼僜儖僶乕偛偲偵堎側傞偨傔丆暿僜儖僶乕傊偺棳梡偼擄偟偄偺傕妋偐偱偡丏

  5. LP 僼傽僀儖傪帺摦揑偵惍宍偟偨偄

    丂LP 僼傽僀儖傪堦搙僜儖僶乕偵撉傒崬傑偣偰丆偦傟偐傜彂偒弌偡偲偒傟偄偵惍宍偟偰偔傟傑偡丏 埲壓丆惍宍偟偨偄 LP 僼傽僀儖偺柤慜傪 test1.lp 偲偟傑偡丏

  6. MPS 僼傽僀儖偲偼壗偐丏MPS 僼傽僀儖偲 LP 僼傽僀儖傪曄姺偟偨偄

    丂MPS 僼傽僀儖偲偼丆悢棟嵟揔壔栤戣傪昞偡僼傽僀儖宍幃偺堦庬偱偡丏 僥僉僗僩僼傽僀儖側偺偱偡偑丆恖娫偺栚偱偼偁傑傝撉傒傗偡偔偼偁傝傑偣傫丏 儀儞僠儅乕僋栤戣側偳偼 MPS 宍幃偱傾僢僾儘乕僪偝傟偰偄傞偙偲傕懡偄偱偡丏

  7. LP 僼傽僀儖傪 MPS 僼傽僀儖偵曄姺偟偨傜丆栚揑娭悢抣偑儅僀僫僗 1 攞偵側偭偨

    丂偄偔偮偐偺僜儖僶乕偱偼丆嵟戝壔栤戣偺 LP 僼傽僀儖傪 MPS 僼傽僀儖偵曄姺偡傞偲丆學悢傪儅僀僫僗 1 攞偟偨嵟彫壔栤戣偵抲偒姺偊傞傛偆偱偡丏

  8. LP 僼傽僀儖偱擇師偺栚揑娭悢傪昞偟偨偄

    丂嵟彫壔栤戣偺応崌偵撌擇師栚揑娭悢丆嵟戝壔栤戣偺応崌偵墯擇師栚揑娭悢偑捈愙埖偊傞僜儖僶乕偑偁傝傑偡丏
    丂栚揑娭悢偵偍偄偰丆慄宍崁傪愭偵彂偒丆偦傟偵懕偄偰擇師崁傪 + [ 2 x^2 - 4 x * y + 9 y^2 ] /2 偺傛偆偵彂偒傑偡丏 曄悢偺 2 忔偼 ^2 傪丆僋儘僗崁偵偼 * 傪巊偄傑偡丏 傑偨丆學悢偼 2 攞偟偰彂偒丆擇師崁偺慡懱傪 [ ] /2 偱偔偔傝傑偡乮[ ] 偲偦傟傛傝撪懁偼僗儁乕僗偱嬫愗傞乯丏

  9. LP 僼傽僀儖偱擇師偺惂栺幃傪昞偟偨偄

    丂撌擇師惂栺丆擇師悕惂栺偑捈愙埖偊傞僜儖僶乕偑偁傝傑偡丏
    丂惂栺幃偵偍偗傞擇師崁偼 [ x^2 - 2 x * y + 7 y^2 ] <= 58 傗 [ x(1)^2 + 3 x(2)^2 - y * z ] <= 0 偺傛偆偵彂偒傑偡乮[ ] 偲偦傟傛傝撪懁偼僗儁乕僗偱嬫愗傞乯丏 曄悢偺 2 忔偼 ^2 傪丆僋儘僗崁偵偼 * 傪巊偄傑偡丏 栚揑娭悢偵偍偗傞擇師崁偲偺堘偄偼丆學悢傪 2 攞偟偰 /2 偡傞昁梫偑側偄偲偄偆偙偲偱偡丏

  10. LP 僼傽僀儖偱栚揑娭悢偵掕悢崁傪娷傔偨偄

    丂LP 僼傽僀儖偵偼摑堦偝傟偨僼僅乕儅僢僩偑懚嵼偣偢丆僜儖僶乕偛偲偵撈帺偺奼挘偑偝傟偰偄傑偡丏 栚揑娭悢偵掕悢崁傪娷傔偨 LP 僼傽僀儖傪捈愙埖偊傞僜儖僶乕傕偁傞偺偱偡偑丆僄儔乕偲側傞僜儖僶乕傕偁傝丆尰忬偱偼栚揑娭悢偵掕悢崁傪娷傔偨 LP 僼傽僀儖傪彂偔偙偲偼偁傑傝偍偡偡傔偱偒傑偣傫丏 栚揑娭悢抣傪掕悢崁傪娷傔偨抣偲偟偰昞帵偝偣偨偄応崌偼丆椺偊偽栚揑娭悢偵掕悢崁偵懳墳偡傞曄悢傪壛偊丆惂栺幃偱偦偺曄悢偺抣傪屌掕偟偰偟傑偊偽摨摍偺偙偲偑偱偒傑偡丏

  11. LP 僼傽僀儖偵彂偔曄悢傗惂栺幃偺弴斣傪曄偊偨傜寁嶼帪娫偑曄傢偭偨

    丂僜儖僶乕偼暘巬尷掕朄偺拞偱暘巬弴彉傪條乆側忣曬偐傜寛掕偟偰偄傑偡偑丆 嵟廔揑偵桪愭搙偑摍偟偄応崌偵偼壗傜偐偺弴彉傪偮偗傞昁梫偑偁傞偨傔丆暘巬弴彉偍傛傃寁嶼帪娫偑曄悢傗惂栺幃傪婰弎偟偨弴斣偵埶懚偡傞偙偲偑偁傝傑偡丏 偙偺偁偨傝偺忣曬偼 嶲峫暥專 4 偵徻偟偄偱偡丏

  12. Lazy Constraints 傪巊偄偨偄

    丂僜儖僶乕偵傛偭偰偼 Lazy Constraints 偑巊偊傑偡丏 Lazy Constraints 偲掕媊偝傟偨惂栺幃偼丆嵟弶偼栤戣偐傜庢傝彍偐傟偰偍傝丆嫋梕夝偑尒偮偐傞偨傃偵栤戣偵捛壛偡傞傋偒偐偳偆偐僠僃僢僋偝傟傑偡丏 惂栺幃偺杮悢偑朿戝偵側偭偰偟傑偆偑丆幚嵺偵偼偦偺偆偪偺偛偔堦晹偟偐堄枴偑側偄傛偆側栤戣偱岠壥傪敪婗偟傑偡丏
    丂埲壓偱偼丆LP 僼傽僀儖偱 Lazy Constraints 傪埖偆曽朄傪愢柧偟傑偡丏 偦傕偦傕惂栺幃慡懱傪 LP 僼傽僀儖偵彂偒偒傞偺偑擄偟偄応崌傗丆摦揑偵 Lazy Constraints 傪捛壛偟偨偄傛偆側応崌偼丆奺僜儖僶乕偺 API 傪棙梡偡傞偺偑曋棙偱偡丏 徻嵶偼儅僯儏傾儖傪嶲徠偟偰偔偩偝偄丏
栚師傊栠傞

  1. 僜儖僶乕偺挷惍偲寁嶼懍搙偵偮偄偰

丂寁嶼懍搙偵娭學偡傞晹暘偱偺丆僜儖僶乕偺挷惍偵偮偄偰偱偡丏

  1. 嵟彫壔栤戣偱側偐側偐壓奅偑忋偑傜側偄丏嵟戝壔栤戣偱側偐側偐忋奅偑壓偑傜側偄

    丂掕幃壔偑椙偔側偄壜擻惈傕偁傝傑偡偑丆偲傝偁偊偢僜儖僶乕偺僷儔儊乕僞傪挷惍偟偰傒傞壙抣偼偁傞偲巚傢傟傑偡丏

  2. 嵟揔夝偵偙偩傢傜側偄偺偱丆側傞傋偔崅懍偵幙偺椙偄夝偑傎偟偄

    丂僜儖僶乕偺僷儔儊乕僞傪挷惍偡傞偺偑庤偭庢傝憗偄偲巚傢傟傑偡丏

  3. 寁嶼帪娫傪惂尷偟偰丆暘巬尷掕朄傪帺摦揑偵掆巭偡傞傛偆偵偟偨偄

    丂埲壓偺僐儅儞僪偱丆暘巬尷掕朄傪 10000 昩偱廔椆偝偣傜傟傑偡丏 悢帤傪曄偊傞偙偲偵傛傝昩悢傪帺桼偵愝掕偱偒傑偡丏

  4. 掕傔偨鑷抣偲摨偠偐傛傝椙偄栚揑娭悢抣偺嫋梕夝偑弌偨傜丆暘巬尷掕朄傪帺摦揑偵掆巭偡傞傛偆偵偟偨偄

    丂梡搑偵傛偭偰偼丆儐乕僓乕偑掕傔偨鑷抣偲摨偠偐傛傝椙偄栚揑娭悢抣乮嵟彫壔栤戣偱偼鑷抣埲壓丆嵟戝壔栤戣偱偼鑷抣埲忋乯傪帩偮嫋梕夝偑摼傜傟傟偽廫暘側応崌偑偁傝傑偡丏 埲壓偺僐儅儞僪偱丆掕傔偨鑷抣偲摨偠偐傛傝椙偄栚揑娭悢抣偺嫋梕夝偑弌偨傜丆暘巬尷掕朄傪帺摦揑偵掆巭偡傞傛偆偵偱偒傑偡丏 埲壓偱偼鑷抣傪 200 偲偟偰愢柧偟傑偡偑丆悢帤傪曄偊傞偙偲偵傛傝鑷抣偼帺桼偵愝掕偱偒傑偡丏

  5. 嫋梕夝傪梌偊偨忬懺偱暘巬尷掕朄傪僗僞乕僩偟偨偄

    丂乮幙偺椙偄乯嫋梕夝傪梌偊傞偙偲偵傛傝丆暘巬尷掕朄偺崅懍壔傪偹傜偄傑偡丏 摿偵丆嫋梕夝偑尒偮偐傝偵偔偄栤戣偱偼寁嶼帪娫偺抁弅偑婜懸偱偒傑偡丏

  6. 僜儖僶乕傪僸儏乕儕僗僥傿僋僗偲偟偰巊偄偨偄

    丂忋婰偺乽嵟揔夝偵偙偩傢傜側偄偺偱丆側傞傋偔崅懍偵幙偺椙偄夝偑傎偟偄乿乽寁嶼帪娫傪惂尷偟偰丆暘巬尷掕朄傪帺摦揑偵掆巭偡傞傛偆偵偟偨偄乿乽掕傔偨鑷抣偲摨偠偐傛傝椙偄栚揑娭悢抣偺嫋梕夝偑弌偨傜丆暘巬尷掕朄傪帺摦揑偵掆巭偡傞傛偆偵偟偨偄乿乽嫋梕夝傪梌偊偨忬懺偱暘巬尷掕朄傪僗僞乕僩偟偨偄乿傪慻傒崌傢偣偰巊偆偲椙偄偲巚傢傟傑偡丏

  7. 暘巬弴彉傪庤摦偱巜掕偟偨偄

    丂嵟嬤偺僜儖僶乕偼偐側傝尗偔側偭偰偄傞偺偱丆僨僼僅儖僩偺暘巬弴彉偼偐側傝崅懍偱偡丏 庤嶌嬈偵傛傞暘巬弴彉偺挷惍偵懡偔偺帪娫傪妡偗傞偙偲偼偁傑傝偍偡偡傔偟傑偣傫丏 偨偩偟丆摿掕偺曄悢傪屌掕偡傞偲栤戣偑挊偟偔彫偝偔側傞側偳丆栤戣峔憿傪徻偟偔抦偭偰偄傞応崌偵偼暘巬弴彉偺庤摦巜掕偑桳岠側応崌傕偁傝傑偡丏

  8. 暲楍暘巬尷掕朄偱僗儗僢僪悢傪巜掕偟偨偄

    丂暲楍暘巬尷掕朄偑幚憰偝傟偰偄傞僜儖僶乕偱偼丆儅儖僠僐傾 CPU 偺棙揰傪惗偐偡偙偲偑偱偒傑偡丏 偁傞偄偼丆懠偺僞僗僋偺幾杺傪偟側偄傛偆偵僗儗僢僪悢傪尭傜偡偙偲傕壜擻偱偡丏

  9. 暲楍暘巬尷掕朄偵偍偄偰丆僗儗僢僪悢傪憹傗偟偨傜抶偔側偭偰偟傑偭偨

    丂僗儗僢僪悢傪憹傗偟偰傕抶偔側傞応崌偼偁傝傑偡丏 嶲峫暥專 4 偺恾 4 偑徻偟偄偱偡丏 偱偡偑丆椺偊偽 1 僗儗僢僪偲 16 僗儗僢僪偔傜偄偩偲偦傟側傝偵堘偄偑弌偰偔傞傛偆偱偡丏 乽暲楍暘巬尷掕朄偱僗儗僢僪悢傪巜掕偟偨偄乿偺崁傕嶲徠偟偰偔偩偝偄丏

  10. 僜儖僶乕偺僆僾僔儑儞 A 偺傒傪僆儞偵偟偨傜寁嶼偑懍偔側傝丆僆僾僔儑儞 B 偺傒傪僆儞偵偟偨傜寁嶼偑懍偔側偭偨偺偱丆僆僾僔儑儞 A 偲 B 傪椉曽僆儞偵偟偨偲偙傠寁嶼偑抶偔側偭偰偟傑偭偨

    丂偦偆偄偆偙偲偼偟偽偟偽偁傝傑偡丏

  11. 寁嶼傪堦搙拞抐偟偰偐傜嵞奐偟偨傜丆拞抐偟側偄応崌偲寁嶼夁掱偑堎側偭偨

    丂僜儖僶乕偵傛偭偰偼丆偦偆偄偆偙偲偑偁傝傑偡丏

  12. 寁嶼帪娫傪惂尷偟偨傜丆惂尷偟側偄応崌偲寁嶼夁掱偑堎側偭偨

    丂僜儖僶乕偵傛偭偰偼丆寁嶼帪娫偺忋尷傪巜掕偡傞偲丆偦偺忋尷偵嬤偯偄偰傕嵟揔惈僊儍僢僾偑戝偒偄偲偒乮寁嶼偑嵟屻傑偱廔椆偟側偦偆側偲偒乯偵嫋梕夝偺敪尒傪桪愭偡傞儌乕僪偵愗傝懼傢傞傕偺偑偁傞傛偆偱偡丏

  13. 慄宍嵟揔壔栤戣傪暋悢夞夝偄偰傒偨傜丆嵟揔夝偑堎側偭偨

    丂僜儖僶乕偵傛偭偰偼丆儅儖僠僗儗僢僪偱慄宍嵟揔壔栤戣傪夝偔偲丆暋悢偺傾儖僑儕僘儉乮扨懱朄傗撪揰朄乯傪梡偄偰寁嶼偟堦斣懍偔廔椆偟偨傕偺傪弌椡偟傑偡丏 偙偺偨傔丆婬側働乕僗偱偡偑丆傾儖僑儕僘儉娫偺寁嶼帪娫偺嵎堎偑旝彫側偲偒偼帋峴偛偲偵堎側傞嵟揔夝偑弌傞壜擻惈偑偁傞傛偆偱偡丏
栚師傊栠傞

  1. 僜儖僶乕傪傕偭偲曋棙偵巊偄偨偄

丂僜儖僶乕偺曋棙側婡擻偺徯夘傪偟傑偡丏

  1. 僜儖僶乕偑慜張棟傪偟偨 LP 僼傽僀儖傪尒偨偄

    丂偄偔偮偐偺僜儖僶乕偱偼丆僜儖僶乕偑慜張棟偟偨屻偺 LP 僼傽僀儖傪尒傞偙偲偑偱偒傑偡丏

  2. 偳偺傛偆偵暘巬尷掕朄偑恑傫偱偄傞偐傪徻嵶偵娤嶡偟偨偄

    丂暘巬尷掕朄偺壜帇壔傪峴偆嵺側偳偼丆僲乕僪娫偺娭學偑昁梫偵側傝傑偡丏

  3. 惍悢曄悢偺抣偑堎側傞嵟揔夝偺屄悢傪悢偊偨偄

    丂尨棟揑偵偼壜擻偱偡丏 偱偡偑丆嵟揔夝偺屄悢偼栤戣偵傛偭偰偼旕忢偵懡偔側傝偊傑偡丏 摿偵丆曄悢娫偵懳徧惈偑偁傞応崌偵偼梕堈偵旕尰幚揑側悢偵側傝傑偡乮帠幚忋楍嫇偟偒傟側偄乯丏 偟偨偑偭偰丆彫偝側栤戣偐傜帋偡偺偑摼嶔偱偡丏

  4. 儘僌僼傽僀儖偺弌椡愭傪曄峏偟偨偄

    丂暋悢偺栤戣傪夝偔応崌偵偼丆儘僌僼傽僀儖傪暘偗偰偍偔偲曋棙偱偡丏

  5. 暋悢偺栤戣傪帺摦揑偵夝偐偣偨偄

    丂偄偔偮偐偺曽朄偑偁傝傑偡偑丆偲傝偁偊偢 OS 偺僐儅儞僪儔僀儞乮Windows 偺応崌偼僐儅儞僪僾儘儞僾僩乯傪梡偄偰惂屼偡傞曽朄傪徯夘偟傑偡丏

  6. 暋悢偺栤戣娫偱忣曬傪傗傝偲傝偟偨偄丏懠偺僾儘僌儔儉偲楢実偟偨偄

    丂LP 僼傽僀儖傪僐儅儞僪儔僀儞丒僀儞僞乕僼僃乕僗偱夝偔曽朄偼丆嵟傕扨弮側曽朄偱偡丏 楍惗惉朄側偳丆暋悢偺栤戣偺娫偱忣曬傪傗傝偲傝偡傞昁梫偑偁傞応崌偵偼丆僜儖僶乕偺 API 側偳傪梡偄傞偺偑摼嶔偱偡丏 奺僜儖僶乕偺儅僯儏傾儖偍傛傃僒儞僾儖僼傽僀儖傪嶲徠偟偰偔偩偝偄丏

  7. 寁嶼傪拞抐偡傞偙偲側偔丆巄掕夝偦偺傕偺傪帺摦揑偵婰榐偟偨偄

    丂梡搑偵傛偭偰偼丆嵟揔夝埲奜偺夝傕尒偨偄偙偲偑偁傝傑偡丏

  8. 曄峏偟偨僷儔儊乕僞愝掕傪昞帵丒曐懚丒暅尦偟偨偄

    丂僜儖僶乕偺曄峏偟偨僷儔儊乕僞愝掕傪昞帵丒曐懚偡傞曽朄偱偡丏

  9. 栤戣偺摑寁忣曬偑尒偨偄

    丂撉傒崬傫偩栤戣偵偮偄偰丆曄悢偺屄悢傗惂栺幃偺杮悢傪昞帵偱偒傑偡丏

  10. 梌偊偨惍悢嵟揔壔栤戣偺楢懕娚榓栤戣傪摼偨偄

    丂撉傒崬傫偩惍悢嵟揔壔栤戣偵偮偄偰丆偦偺楢懕娚榓栤戣乮惍悢惂栺傪庢傝彍偄偨栤戣乯傪弌椡偝偣傞偙偲偑偱偒傑偡丏

  11. 梌偊偨慄宍嵟揔壔栤戣偺憃懳栤戣傪摼偨偄

    丂撉傒崬傫偩慄宍嵟揔壔栤戣偵偮偄偰丆偦偺憃懳栤戣傪弌椡偝偣傞偙偲偑偱偒傑偡丏

  12. [CPLEX]丂巄掕夝偑摼傜傟偨偲偒偺宱夁帪娫傪帺摦揑偵婰榐偟偨偄

    丂CPLEX 12.5 偐傜娙扨偵側偭偰偄傑偡丏 Interactive Optimizer 偱偼丆乽set mip display 3乿乮偁傞偄偼乽set mip display 4乿乯偲偟偰暘巬尷掕朄傪幚峴偟偰偔偩偝偄丏

  13. [CPLEX]丂Benders 暘夝傪峴偄偨偄

    丂崿崌惍悢嵟揔壔栤戣傪夝偔嵺偵丆Benders 暘夝朄傪梡偄傞偙偲偑偱偒傑偡丏 堦晹偺栤戣偱偼岠壥偑婜懸偱偒傑偡丏 Interactive Optimizer 偱偼丆乽set benders strategy 3乿偲偟偰栤戣傪夝偔偲丆慡偰偺惍悢曄悢偼儅僗僞乕栤戣傊丆慡偰偺楢懕曄悢偼僒僽栤戣傊偲暘妱偝傟偰 Benders 暘夝朄偑幚峴偝傟傑偡丏 傑偨丆儐乕僓乕帺恎偱栤戣偺暘妱傪巜掕偡傞偙偲傕偱偒傑偡丏 徻偟偔偼儅僯儏傾儖傪嶲徠偟偰偔偩偝偄丏
栚師傊栠傞

  1. 掕幃壔偵偮偄偰

丂掕幃壔偵娭偟偰丆傛偔偁傞幙栤傪婰嵹偟傑偡丏

  1. 幚嵺偺尰応偱偼巊偊側偄傛偆側摎偊偑弌偰偔傞

    丂弌偰偒偨夝偑婰弎偟偨惂栺忦審傪慡偰枮偨偟偰偄傞側傜偽丆僜儖僶乕傗惍悢嵟揔壔帺懱偺栤戣偲偄偆傛傝偼丆尰幚偺惂栺忦審偑慡偰惂栺幃偲偟偰昞尰偝傟偰偄側偄偙偲偵傛傞傕偺偱偡丏 恖娫偑埫栙偺偆偪偵壖掕偟偰偄傞偑丆柧帵揑偵彂偄偰偄側偄惂栺幃偼側偄偐偳偆偐僠僃僢僋偟偰偔偩偝偄丏

  2. 慄宍幃偱彂偗側偄偲巚傢傟傞惂栺忦審傑偨偼栚揑娭悢偑偁傞

    丂愨懳抣傗 max 墘嶼巕側偳丆堦尒偟偰慄宍幃偱昞偣側偝偦偆側応崌偱傕丆曗彆曄悢側偳傪摫擖偟偰掕幃壔偱偒傞偙偲偑偁傝傑偡丏 嶲峫暥專 1 偵丆擔杮岅偱傛偔傑偲傑偭偨忣曬偑偁傝傑偡丏

  3. 0-1 曄悢 x, y 偵懳偟偰丆乽z = x * y乿偵懳墳偡傞 0-1 曄悢 z 傪巊偄偨偄

    丂z >= x + y - 1, z <= x, z <= y 偺 3 杮偺晄摍幃偍傛傃 x, y, z 偵娭偡傞 0-1 惂栺偵傛傝昞尰偱偒傑偡乮懠偺彂偒曽傕偁傝傑偡丏嶲峫暥專 14 側偳傪嶲徠偟偰偔偩偝偄乯丏 偙偺庬偺掕幃壔偺僥僋僯僢僋偼丆嶲峫暥專 1 偵傑偲傔傜傟偰偄傑偡丏
    丂堦曽偱丆0-1 曄悢偺応崌偵偼丆x * y 偲偄偆昞尰傪捈愙庴偗偮偗傞偙偲偺偱偒傞僜儖僶乕傕偁傞傛偆偱偡丏 偟偐偟丆娚榓栤戣偑娚偔側傞傛偆側曄宍傪梡偄偰 x * y 傪埖偊傞傛偆偵偟偰偄傞偙偲偑懡偄偨傔丆僜儖僶乕偑庴偗偮偗偰偄傞偲偟偰傕丆忋婰偺晄摍幃 3 杮偵傛傞昞尰傪帋偟偰傒傞壙抣偼偁傝傑偡丏

  4. 惂栺幃傪懠偺曄悢偵墳偠偰僆儞丒僆僼偟偨偄丏乽傕偟亊亊側傜偽仮仮乿偲偄偆傛偆側惂栺幃偑彂偒偨偄

    丂Big-M 朄偲屇偽傟傞曽朄傪巊偄傑偡丏 椺偊偽丆0-1 曄悢 z 偲旕晧偺楢懕曄悢 x 偵懳偟偰丆z = 0 偺帪偵乽x = 0乿偲偄偆惂栺幃傪擖傟丆z = 1 偺帪偵偦偺惂栺幃傪柍岠偵偟偨偄偲偟傑偡丏 旕晧偺楢懕曄悢 x 偺忋奅偑惓偺乮戝偒側乯掕悢 M 偩偲壗傜偐偺棟桼偱傢偐偭偰偄傞応崌丆x <= M * z 偲偡傞偙偲偱丆忋婰傪払惉偡傞偙偲偑偱偒傑偡丏 側偍丆掕幃壔偺嵺偵丆M 偺抣偼嫋偝傟傞斖埻偱側傞傋偔彫偝偔偟偨曽偑寁嶼偑崅懍偵側傝傑偡丏 徻偟偔偼 嶲峫暥專 1 傪嶲徠偟偰偔偩偝偄丏 偨偩偟丆big-M 朄傪 1 偮偺栤戣偺拞偱懡梡偡傞偙偲偼寁嶼懍搙傪棊偲偡尨場偵側傞偺偱丆big-M 朄傪巊傢側偄偱掕幃壔偱偒傞応崌偼偦偪傜傪偍偡偡傔偟傑偡丏
    丂傑偨丆Special Ordered Set type 1 (SOS type 1) 惂栺偑巊偊傞僜儖僶乕偺応崌丆x <= M * z 偲彂偔偐傢傝偵丆SOS type 1 惂栺偱 x 偲 z' 傪儊儞僶乕偵擖傟傞偙偲偱丆摨條偺惂栺偲側傝傑偡丗偙偙偱 z' 偼 z + z' = 1 偲側傞 0-1 曄悢偱偡丏
    丂偝傜偵丆擇師悕惂栺偑巊偊傞僜儖僶乕偱偼旕晧偺楢懕曄悢 y 偵懳偟 x^2 <= y * z 偲偄偆擇師悕惂栺幃偱 x <= M * z 偲摨條偺惂栺傪昞偡偙偲偑偱偒傑偡丏 偙傟偼 perspective 嵞掕幃壔 (perspective reformulation) 偲屇偽傟傞僥僋僯僢僋偺堦庬偱偡丏 Perspective 嵞掕幃壔偺徻嵶偼丆椺偊偽 嶲峫暥專 20 傪偍撉傒偔偩偝偄丏 丂側偍丆乽A 傑偨偼 B乿偲偄偆傛偆側惂栺幃傪彂偒偨偄応崌偵偼丆disjunctive constraint乮棧愙惂栺乯傪梡偄偰掕幃壔偡傞曽朄傕偁傝傑偡丏 棧愙惂栺傪埖偆傛偆側栤戣偼 disjunctive programming 偲屇偽傟偰偄傑偡丏 徻偟偔偼 嶲峫暥專 16丆嶲峫暥專 21 傪嶲徠偟偰偔偩偝偄丏

  5. 曄悢偺屄悢偑彮側偄栤戣側偺偵夝偗側偄丏曄悢偺屄悢偑彮側偄掕幃壔偵曄偊偨傜抶偔側偭偨

    丂曄悢偑彮側偄乮偁傞偄偼惂栺幃偑彮側偄乯栤戣偁傞偄偼掕幃壔偑丆昁偢偟傕寁嶼帪娫偑抁偄傢偗偱偼偁傝傑偣傫丏 偳偺傛偆側掕幃壔偑椙偄掕幃壔乮佮崅懍偵夝偗傞掕幃壔乯偵側傞偐偼働乕僗僶僀働乕僗偱偡偑丆堦斒揑偵偼乽慄宍娚榓栤戣偺嵟揔抣偲惍悢嵟揔壔栤戣偺嵟揔抣偺槰棧偑彫偝偄乿掕幃壔偑椙偄掕幃壔偵側傞偙偲偑懡偄偱偡丏 偨偩偟儀僗僩傪恠偔偟偰傕丆偁傑傝偵傕曄悢偑懡偄栤戣傗惍悢嵟揔壔偵岦偄偰偄側偄栤戣偼夝偔偺偑擄偟偄偺傕帠幚偱偡丏
    椺丗彫偝偄偺偵擄偟偄栤戣偺 LP 僼傽僀儖 乮MIPLIB 2017 偺 pb-market-split8-70-4乯丏 0-1 曄悢 71 屄丆惂栺幃 17 杮偺彫偝側栤戣偱偡偑乮杮幙揑偵偼 0-1 曄悢 70 屄丆摍幃僫僢僾僒僢僋惂栺 8 杮偺嫋梕惈敾掕栤戣乯丆2023 擭 11 寧傑偱嫋梕夝偑堦偮傕尒偮偐偭偰偍傜偢丆傑偨晄擻偱偁傞偙偲傕徹柧偝傟偰偄傑偣傫偱偟偨丏 尰嵼偼嫋梕夝乮偐偮嵟揔夝乯偑敪尒偝傟偰偄傑偡乮 嶲峫暥專 23 乯丏

  6. 慄宍傗擇師偱側偄栚揑娭悢傪埖偄偨偄

    丂懡崁幃娭悢傪捈愙埖偊側偄僜儖僶乕偱傕丆撌擇師惂栺傪埖偊傞応崌偼埲壓偺傛偆側僩儕僢僋偱師悢傪忋偘傞偙偲偑偱偒傑偡丏 惂栺幃偱 x^2 <= y 偐偮 y^2 <= z 偲偟丆z 傪嵟彫壔偡傟偽 x 偺 4 忔傪嵟彫壔偟偰偄傞偙偲偲摍壙偱偡乮LP 僼傽僀儖偺宍幃偱偼 - y + [ x^2 ] <= 0 偐偮 - z + [ y^2 ] <= 0乯丏 傑偨丆擇師悕惂栺傪埖偊傞僜儖僶乕偱偼丆旕晧曄悢 x, s, t 偵懳偟 x^2 <= s * t 偐偮 s^2 <= x 偲偟丆t 傪嵟彫壔偡傟偽 x 偺 1.5 忔傪嵟彫壔偟偰偄傞偙偲偲摍壙偱偡乮LP 僼傽僀儖偺宍幃偱偼 [ x^2 - s * t ] <= 0 偐偮 - x + [ s^2 ] <= 0乯丏 偙偺庬偺僥僋僯僢僋偼丆嶲峫暥專 22 偵傑偲傔傜傟偰偄傑偡丏
栚師傊栠傞

  1. 偦偺懠偺忣曬

丂偦偺懠丆栶偵棫偮儕儞僋愭側偳偺忣曬傪傑偲傔偰偍偒傑偡丏

  1. 惍悢嵟揔壔栤戣偺儀儞僠儅乕僋栤戣廤偑抦傝偨偄

    丂偄偔偮偐偁傝傑偡偑丆桳柤側偺偼 MIPLIB (Mixed Integer Programming LIBrary) 偱偡丏 尰帪揰偱偼 MIPLIB 2017 ( 嶲峫暥專 19, https://miplib.zib.de/ ) 偑嵟怴僶乕僕儑儞偱偡丏 彮偟慜傑偱偼 MIPLIB 2010 ( 嶲峫暥專 4, https://miplib2010.zib.de/ ) 偑傛偔梡偄傜傟偰偄傑偟偨丏

  2. 彜梡僜儖僶乕偲旕彜梡僜儖僶乕偼寁嶼帪娫偑偳偺掱搙堎側傞偺偐

    丂尰帪揰偱偼丆嵟崅懍偺彜梡僜儖僶乕偲嵟崅懍偺旕彜梡僜儖僶乕偲傪斾妑偡傞偲丆慜幰偺曽偑偐側傝崅懍偱偡丏 Hans Mittelmann 偺儀儞僠儅乕僋僒僀僩乮 https://plato.asu.edu/bench.html 乯偵傛傞偲丆2 帪娫埲撪偱夝偗傞娙扨側栤戣偱丆10 攞嬤偄懍搙嵎偑偁傞傛偆偱偡丏 偝傜偵丆擄偟偄栤戣傎偳僜儖僶乕偺惈擻嵎偑尠挊偵尰傟傑偡乮悢暘偲悢帪娫丆偁傞偄偼悢帪娫偲悢擔偲偄偆傛偆側働乕僗傕婬偱偼偁傝傑偣傫乯丏

  3. 偳偺彜梡僜儖僶乕偑懍偄偺偐

    丂栤戣偵傛偭偰僜儖僶乕 A 偺曽偑懍偐偭偨傝丆僜儖僶乕 B 偺曽偑懍偐偭偨傝偲偄偆偙偲偑偁傝傑偡丏 側偺偱丆偳偺僜儖僶乕偑懍偄偺偐偲偄偆偺偼摑寁揑偵尒傞偟偐側偄偙偲偵偛拲堄偔偩偝偄丏 僜儖僶乕暿偺儀儞僠儅乕僋寢壥偵偮偄偰偼丆Hans Mittelmann 偺儀儞僠儅乕僋僒僀僩乮 https://plato.asu.edu/bench.html 乯偑桳柤偱偡丏 側偍丆尰嵼偼堦晹偺彜梡僜儖僶乕偵偮偄偰丆儀儞僠儅乕僋寢壥偺宖嵹傪掆巭偟偰偄傞傛偆偱偡乮屆偄寢壥偼巆偭偰偄傑偡乯丏

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