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,第03章線性規(guī)劃：對(duì)偶LinearProgramming:duality,對(duì)偶理論－對(duì)偶問(wèn)題－對(duì)偶定理－與單純形法的關(guān)系互補(bǔ)松弛－KKT條件基于對(duì)偶的方法－對(duì)偶單純形法(概念、步驟、收斂性),對(duì)偶理論,◎食譜問(wèn)題：確定食品數(shù)量，滿足營(yíng)養(yǎng)需求，花費(fèi)最??？,對(duì)偶問(wèn)題：舉例,n種食品，m種營(yíng)養(yǎng)成份；－第j種食品的單價(jià),－每單位第j種食品所含第i種營(yíng)養(yǎng)的數(shù)量,－為了健康，每天必須食用第i種營(yíng)養(yǎng)的數(shù)量,◎模型：,對(duì)偶問(wèn)題：經(jīng)濟(jì)解釋,◎保健品公司：藥劑師、營(yíng)養(yǎng)丸、定價(jià)問(wèn)題,◎?qū)ε紗?wèn)題,對(duì)偶問(wèn)題：對(duì)稱(chēng)形式的對(duì)偶對(duì),注：對(duì)偶是相互的，即對(duì)偶問(wèn)題的對(duì)偶是原問(wèn)題,◎原始問(wèn)題(primal)：,給定數(shù)據(jù),－原問(wèn)題的變量,對(duì)偶問(wèn)題：非對(duì)稱(chēng)形式的對(duì)偶對(duì),注：為了確定任一線性規(guī)劃問(wèn)題的對(duì)偶，可以利用對(duì)稱(chēng)形式或非對(duì)稱(chēng)形式的對(duì)偶對(duì)！,◎原始問(wèn)題(primal)：,給定數(shù)據(jù),－原問(wèn)題的變量,對(duì)偶問(wèn)題：一般問(wèn)題的對(duì)偶,給定數(shù)據(jù)c,A,b；記A的第j行為aj，A的第i列為ai,◎原問(wèn)題(primal)：,◎?qū)ε紗?wèn)題(dual)：,對(duì)偶問(wèn)題：例子,對(duì)偶定理：弱對(duì)偶定理,弱對(duì)偶定理.設(shè)和分別是原始問(wèn)題和對(duì)偶問(wèn)題的可行解，則,推論2.如果原始問(wèn)題與對(duì)偶問(wèn)題之一無(wú)界，則另一個(gè)問(wèn)題沒(méi)有可行解.,,對(duì)偶定理：強(qiáng)對(duì)偶定理,對(duì)于一般形式的線性規(guī)劃－－利用凸集分離定理證明！,強(qiáng)對(duì)偶定理.如果原始問(wèn)題和對(duì)偶問(wèn)題之一有解，則另一個(gè)問(wèn)題也有解，且最優(yōu)值相等.,與單純形法的關(guān)系：定理,如何由原始問(wèn)題的解得到對(duì)偶問(wèn)題的解？,與單純形法的關(guān)系：例子,考慮問(wèn)題,引入松弛變量→標(biāo)準(zhǔn)形→利用單純形法求解,對(duì)偶問(wèn)題,與單純形法的關(guān)系：例子(續(xù)),原問(wèn)題最優(yōu)解,對(duì)偶問(wèn)題最優(yōu)解,與單純形法的關(guān)系：?jiǎn)渭冃纬俗?◎與基B對(duì)應(yīng)的單純形乘子(simplexmultiplier),◎經(jīng)濟(jì)解釋記A的列向量為aj，對(duì)應(yīng)費(fèi)用為cj，j=1,…,n,－－－－－－－－解釋為單位向量ei的合成價(jià)格！,－－解釋為aj的相對(duì)費(fèi)用系數(shù),◎最優(yōu)性：對(duì)所有的i有,與單純形法的關(guān)系：?jiǎn)渭冃纬俗?續(xù)),靈敏度(sensitivity，工程上),假設(shè)該問(wèn)題的最優(yōu)基是B.則,假設(shè)非退化！,問(wèn)題：當(dāng)向量b變化時(shí)，最優(yōu)值如何變化？,與單純形法的關(guān)系：?jiǎn)渭冃纬俗?續(xù)),影子價(jià)格(shadowprice，經(jīng)濟(jì)上),稱(chēng)為分量所對(duì)應(yīng)資源的邊際價(jià)格(marginalprice)或者影子價(jià)格(shadowprice),互補(bǔ)松弛ComplementarySlackness,互補(bǔ)松弛定理,定理.設(shè)和分別是對(duì)稱(chēng)形式原始問(wèn)題和對(duì)偶問(wèn)題的可行解.則它們是各自最優(yōu)解的充要條件是：對(duì)所有的i和j有,對(duì)偶單純形法DualSimplexMethod,對(duì)偶單純形法：概述,◎適用問(wèn)題：標(biāo)準(zhǔn)形問(wèn)題有一個(gè)不可行的基本解，但對(duì)應(yīng)單純形乘子是對(duì)偶問(wèn)題的可行解,◎單純形表中的表現(xiàn)：,⊙第一張單純形表:相對(duì)費(fèi)用系數(shù)非負(fù)，但有基變量取負(fù)值！,⊙轉(zhuǎn)軸過(guò)程中：保持相對(duì)費(fèi)用系數(shù)非負(fù)，直到基變量全部取非負(fù)值！,則稱(chēng)x是標(biāo)準(zhǔn)形問(wèn)題的對(duì)偶可行基本解.,定義.假設(shè)是Ax=b的基本解.如果,基本解＋可行＋對(duì)偶可行＝最優(yōu)解,對(duì)偶單純形法：對(duì)偶可行基本解,是對(duì)偶問(wèn)題的可行解，即,目的：找新的使前m個(gè)等式中的某個(gè)與后n-m個(gè)不等式中的某個(gè)角色互換，同時(shí)使對(duì)偶問(wèn)題的目標(biāo)函數(shù)值增大！,對(duì)偶單純形法：推導(dǎo)I,對(duì)偶單純形法：推導(dǎo)II,令，其中ui是B-1的第i行，則,出基變量：取負(fù)值的基變量(*****),進(jìn)基變量：取到最小正比值的非基變量(*****),步0給定對(duì)偶可行基本解對(duì)應(yīng)的單純形表.,步1若對(duì)每個(gè)i都有，停；當(dāng)前DFBS是最優(yōu)的.,步2選取i滿足yi0<0，這時(shí)，第i個(gè)基變量出基.,步4以yiq為轉(zhuǎn)軸元進(jìn)行轉(zhuǎn)軸，更新單純形表，返步1.,對(duì)偶單純形法：計(jì)算步驟,步3若，停，問(wèn)題無(wú)可行解；否則，選q滿足,引入盈余變量；并給等式兩邊同乘-1；得初始表格,對(duì)偶單純形法：例子,對(duì)偶單純形法：例子(續(xù)),最優(yōu)解：,結(jié)論：由對(duì)偶可行基本解確定的單純形乘子集合與對(duì)偶問(wèn)題可行集的極點(diǎn)是完全相同的。,對(duì)偶單純形法：進(jìn)一步的理解,原問(wèn)題,,,,,對(duì)偶單純形法：收斂性,定理.如果標(biāo)準(zhǔn)形線性規(guī)劃問(wèn)題的任一對(duì)偶可行基本解所對(duì)應(yīng)的非基變量的相對(duì)費(fèi)用系數(shù)大于零，則對(duì)偶單純形法在有限步內(nèi)終止.,◎如果線性規(guī)劃問(wèn)題可以用對(duì)偶單純形法求解，則必有界！其計(jì)算結(jié)果只能是不可行或者有解！,◎如果線性規(guī)劃問(wèn)題可以用單純形法求解，則其無(wú)界或有解！◎兩階段法可以求解任一線性規(guī)劃問(wèn)題；第I階段的結(jié)果分有可行解或者無(wú)可行解兩種；對(duì)有可行解的，在第II階段可得問(wèn)題無(wú)界或有解！,◎典型情況(有顯然的對(duì)偶可行基本解),◎一般情況,⊙已有一個(gè)標(biāo)準(zhǔn)形問(wèn)題的最優(yōu)解和最優(yōu)基,◇添加一個(gè)“不等式約束”后的新問(wèn)題,對(duì)偶單純形法：?jiǎn)?dòng),⊙“不等式約束”＋,后的新問(wèn)題,線性規(guī)劃的應(yīng)用：博弈論(GameTheory),石頭－布－剪刀(Rock-Paper-Scissors),二人博弈,支付矩陣：行palyer給列player的支付(payoff),注：某個(gè)player利用任一確定性(純)策略，均可被另一個(gè)player擊敗.,二人零和博弈(Two-PersonZero-sumGames),給定mn矩陣A,注：A的行代表rowboy的確定性策略，而A的列代表colgirl的確定性策略.,行player(rowboy)選擇策略列player(colgirl)選擇策略rowboy支付給colgirlaij美元,確定性策略可能會(huì)很差！,隨機(jī)化策略(RandomizedStrategies),假設(shè)rowboy選擇策略i的概率是yi假設(shè)colgirl選擇策略j的概率是xj,如果使用隨機(jī)(混合)策略y，colgirl使用隨機(jī)策略x，則rowboy向colgirl的期望支付是,Colgirl的分析,假設(shè)colgirl準(zhǔn)備采取策略x,則rowboy最好的防衛(wèi)是利用y,其求解問(wèn)題,因此colgirl應(yīng)該選取x,其極大化這種可能性,求解Max-Min問(wèn)題(轉(zhuǎn)化成線性規(guī)劃問(wèn)題),內(nèi)層優(yōu)化很容易：,引入標(biāo)量變量v表示內(nèi)層極小化的值：,因此colgirl的問(wèn)題是,Rowboy的問(wèn)題,類(lèi)似地，rowboy選擇y,該問(wèn)題等價(jià)于：,注：colgirl的問(wèn)題是rowboy的問(wèn)題的對(duì)偶,MinMax定理,設(shè)x*表示colgirl的max-min問(wèn)題的解；設(shè)y*是rowboy的min-max問(wèn)題的解.則,證明.由強(qiáng)對(duì)偶定理，我們有,且,