海森矩阵

ChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists经济数学MathematicsforEconomists最优化理论OptimumChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists专题一:最优化问题Topic1:OptimumProblemChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists基本概念•设函数f：RnR,称向量xRn为工具向量•SRn为可行的备择集合，称S为机会集合•函数f称为目标函数。•一般形式：s.t.xSRn•瓦尔拉定理:紧集上的连续函数有最大值和最小值)(maxxfxChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists两个变量的极值问题ChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists•如果函数z=f(x,y)二次连续可导，则函数取得极值的一阶必要条件是：•二阶充分条件是：，取得极大值，取得极小值0yxff2,0,0xyyyxxyyxxfffff2,0,0xyyyxxyyxxfffff两个变量的极值问题ChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists•一个企业生产两种产品，收入函数•假设企业的成本函数：•求该企业利润最大时两种产量的产量2211QPQPR两个变量的极值问题：练习22212122QQQQCChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists222121221122πQQQQQPQPCR两个变量的极值问题：练习04ππ21111QQPQ04ππ21222QQPQ154154122211PPQPPQ4π,1π1π,4π22211211可以验证达到效用最大ChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists•如果函数z=f(x1,x2,x3)二次连续可导，则函数取得极值的一阶必要条件是：•二阶充分条件是：海森矩阵负定，取得极大值海森矩阵正定，取得极小值0321fff三个变量的极值问题ChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists333231232221131211fffffffffH111fH222112112ffffHHH3海森矩阵负定，取得极大值:海森矩阵正定，取得极小值0,0,0321HHH0,0,0321HHH三个变量的极值问题ChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists•求函数的极值•一阶条件：•唯一解：2422331222121xxxxxxxf例题：02)(08)(04)(3132123211xxfxxfxxxf0321xxxChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists201081114333231232221131211fffffffffH04111fH031222112112ffffH543HH海森矩阵正定，取得极小值ChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists•求函数的极值232223131323xxxxxxf练习：ChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists•求函数的极值•一阶条件：•解：232223131323xxxxxxf练习：063)(22)(033)(313223211xxfxfxxf)41,1,21()0,1,0(),,(321xxx极大值不是极值ChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists局部最大值LocalMaximum•如果函数为f(x)二次连续可微，则：•如果在x*取得极大值，则•令h=x，x为任意小的非零向量，则hxHfhhxfxfhxfT)(21)(grad)()(0)(21)(grad)()(hxHfhhxfxfhxfT－0)(21)(grad)()(2xxHfxλλxxfxfλxxfT－ChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists局部最大值LocalMaximum•令h=x，x为任意小的非零向量，则•如果0,上式两边都除以并令0，则：gradf(x*)x0•如果0,gradf(x*)x0•所以gradf(x*)x＝0，gradf(x*)＝0•所以，即海森矩阵半负定。0)(21)(grad)()(2xxHfxλλxxfxfλxxfT－0)(xxHfxTChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists局部最小值LocalMinimum•如果在x*取得极小值，gradf(x*)＝0即海森矩阵半正定。极大值和极小值的充分条件•极大值的充分条件是海森矩阵负定•极小值的充分条件是海森矩阵正定0)(xxHfxTChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists专题二:经典规划Topic2：ClassicalProgrammingChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists经典规划•设函数f：RnR,经典规划的一般形式s.t.g(x)=c,或者其中xRng:RnR•方法：建立拉格朗日函数L(x,y)=f(x)+(c-g(x))得到gradf(x)-gradg(x)＝0g(x)＝c),,,()(max21nxxxxfxfcxxxgxgn),,,()(21ChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists经典规划•要判断最大值还是最小值，还要验证拉格朗日函数的海森矩阵。•拉格朗日函数的海森矩阵称为加边海森矩阵。nnnnnnnnLLLgLLLgLLLgggg21222212112111210HChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists充分条件•最大值:•最小值:0)1(,,0,0032221212111212nnLLgLLgggHHH0,,,32nHHHChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists经典规划•设函数f：RnR,经典规划的一般形式s.t.g(x)=c,其中xRng:RnRm•方法：建立拉格朗日函数L(x,y)=f(x)+(c-g(x))得到gradf(x)－gradg(x)＝0g(x)＝c)(maxxfxChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomists经典规划•要判断最大值还是最小值，还要验证拉格朗日函数的海森矩阵。•拉格朗日函数的海森矩阵称为加边海森矩阵。222222),(xLxLxLLyxHL22220xgxfxgxgTChinaAgri.Univ.CollegeofEconomicsandMangementMathematicsforEconomistsnnnnmnnmnmmmmLLLgggLLLgggLLLggggggggggggLnnnnn212122221211121121222111222111212121000000000H