基于MATLAB的条件极值研究
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摘要
求解条件极值的基本方法为拉格朗日乘数法,直接从隐函数组出发推导拉格朗日乘数法既繁琐又不易让人理解,但注意到在一定的条件下,二元函数f (x,y)的极值点必定出现在该函数的等高线与约束曲线g (x,y)=0的切点上,由于等高线和约束曲线上任意一点处的梯度向量?f,?g分别垂直于该等高线和约束曲线,可以得到?f,?g在极值点处相互平行的几何结论,从而容易得出拉格朗日乘数法。
Lagrange multipliers is the basic method to solve the problem of conditional extremum in actual teaching. Deriving Lagrange multipliers directly from the group of implicit function is not only cumbersome but also difficult to understand. It should, however, be noted that in certain conditions, the extreme points of binary function f( x, y) will appear in the points of tangency between the contours of the curve and constraint curve g( x, y) = 0,since the gradient vector of the contour curve?f, ?g is perpendicular to this contour curve itself, so does the constraint curve. We can get the geometric conclusion that the?f and ?g are parallel to each other at extreme points, so Lagrange multipliers is very easy to get. In this paper, we use the powerful mapping functions of MATLAB to reproduce the vivid image of this process so as to help the students understand the geometry and mathematical thinking of Lagrange multipliers.
Lagrange multipliers is the basic method to solve the problem of conditional extremum in actual teaching. Deriving Lagrange multipliers directly from the group of implicit function is not only cumbersome but also difficult to understand. It should, however, be noted that in certain conditions, the extreme points of binary function f( x, y) will appear in the points of tangency between the contours of the curve and constraint curve g( x, y) = 0,since the gradient vector of the contour curve?f, ?g is perpendicular to this contour curve itself, so does the constraint curve. We can get the geometric conclusion that the?f and ?g are parallel to each other at extreme points, so Lagrange multipliers is very easy to get. In this paper, we use the powerful mapping functions of MATLAB to reproduce the vivid image of this process so as to help the students understand the geometry and mathematical thinking of Lagrange multipliers.
引文
[1]华东师范大学数学系.数学分析(下册)[M].北京:高等教育出版社,2001,6:164-169.Department of Mathematics,East China Normal University.Mathematical Analysis(Volume 2)[M].Beijing:Higher Education Press,2001,6:164-169.
[2]刘玉琏,傅沛仁.数学分析讲义(下册)[M].北京:高等教育出版社,1992,7:227-238.LIU Y L,FU P R.Notes on Mathematical Analysis(Volume 2)[M].Beijing:Higher Education Press,1992,7:227-238.
[3]DALE V,EDWIN J P,STEVEN E R.Calculus[M].Beijing:China Machine Press,2003,4:656-680.
[4]BRIEF A,DON D.Brief Calculus[M].Beijing:Higher Education Press,2004,11:649-658.
[5]WEIR,HASS,GIORDANO.Thomas’Calculus[M].Beijing:Higher Education Press,2016,6:1045-1120.
[6]JAMES S.Calculus[M].Beijing:Higher Education Press,2016,6:1045-1120.
[7]DUANE H M.Bruce Littlefield.Proficient in Matlab 7[M].BeiJing:Tsinghua University Press,2006,5:271-317.
[8]陈建发.关于拉格朗日乘数法的几何意义[J].高等数学研究,2016,19(2):35-36.CHEN J F.Geometric significance of Lagrange multiplier method[J].Studies in College Mathematics,2016,19(2):35-36.
[9]罗棋,朱珊珊.拉格朗日乘数法求解条件极值问题[J].商丘职业技术学院学报,2018,17(4):74-76,92.LUO Q,ZHU S S.Discussion and application of the condition eextreme value problem[J].Journal of Shangqiu Vocational and Technical College,2008,17(4):74-76,92.
[10]刘剑锋,李宏伟,刘智慧.拉格朗日乘数法求解极值点的讨论[J].高等数学研究,2017,20(2):13-14.LIU J F,LI H W,LIU Z H.Discussion on Lagrange multiplier method in solving of extreme value point problems[J].Studies in College Mathematics,2017,20(2):13-14.
[11]赵宇,黄金莹,王希圆.数学分析中的条件极值问题[J].大学数学,2013,29(2):102-106.ZHAO Y,HUANG J Y,WANG X Y.Extremum problem with conditional of mathematical analysis[J].College Mathematics,2013,29(2):102-106.
[12]颜超,陈涛.拉格朗日乘数法的几何理解[J].高等数学研究,2017,20(2):15-16,19.YAN C,CHEN T.The geometric interpretation of Lagrange multiplier method[J].Studies in College Mathematics,2017,20(2):15-16.
[13]刘三明,李修勇.Lagrange乘数法的几何直观推导[J].河南科技大学学报(自然科学版),2004(6):22.LIU S M,LI X Y.Lagrange multiplier method introduced from view geometry[J].Journal of Henan University of Science and Technology(Natural Science),2004(6):22.
[14]DOLGOPOLIK M V.Existence of augmented Lagrange multipliers:reduction to exact penalty functions and localization principle[J].Mathematical Programming,2017,166(1-2):297-326.
[15]ZHOU Y Y,ZHOU J C,YANG X Q.Existence of augmented Lagrange multipliers for cone constrained optimization problems[J].Journal of Global Optimization,2014,58(2):243-260.
[2]刘玉琏,傅沛仁.数学分析讲义(下册)[M].北京:高等教育出版社,1992,7:227-238.LIU Y L,FU P R.Notes on Mathematical Analysis(Volume 2)[M].Beijing:Higher Education Press,1992,7:227-238.
[3]DALE V,EDWIN J P,STEVEN E R.Calculus[M].Beijing:China Machine Press,2003,4:656-680.
[4]BRIEF A,DON D.Brief Calculus[M].Beijing:Higher Education Press,2004,11:649-658.
[5]WEIR,HASS,GIORDANO.Thomas’Calculus[M].Beijing:Higher Education Press,2016,6:1045-1120.
[6]JAMES S.Calculus[M].Beijing:Higher Education Press,2016,6:1045-1120.
[7]DUANE H M.Bruce Littlefield.Proficient in Matlab 7[M].BeiJing:Tsinghua University Press,2006,5:271-317.
[8]陈建发.关于拉格朗日乘数法的几何意义[J].高等数学研究,2016,19(2):35-36.CHEN J F.Geometric significance of Lagrange multiplier method[J].Studies in College Mathematics,2016,19(2):35-36.
[9]罗棋,朱珊珊.拉格朗日乘数法求解条件极值问题[J].商丘职业技术学院学报,2018,17(4):74-76,92.LUO Q,ZHU S S.Discussion and application of the condition eextreme value problem[J].Journal of Shangqiu Vocational and Technical College,2008,17(4):74-76,92.
[10]刘剑锋,李宏伟,刘智慧.拉格朗日乘数法求解极值点的讨论[J].高等数学研究,2017,20(2):13-14.LIU J F,LI H W,LIU Z H.Discussion on Lagrange multiplier method in solving of extreme value point problems[J].Studies in College Mathematics,2017,20(2):13-14.
[11]赵宇,黄金莹,王希圆.数学分析中的条件极值问题[J].大学数学,2013,29(2):102-106.ZHAO Y,HUANG J Y,WANG X Y.Extremum problem with conditional of mathematical analysis[J].College Mathematics,2013,29(2):102-106.
[12]颜超,陈涛.拉格朗日乘数法的几何理解[J].高等数学研究,2017,20(2):15-16,19.YAN C,CHEN T.The geometric interpretation of Lagrange multiplier method[J].Studies in College Mathematics,2017,20(2):15-16.
[13]刘三明,李修勇.Lagrange乘数法的几何直观推导[J].河南科技大学学报(自然科学版),2004(6):22.LIU S M,LI X Y.Lagrange multiplier method introduced from view geometry[J].Journal of Henan University of Science and Technology(Natural Science),2004(6):22.
[14]DOLGOPOLIK M V.Existence of augmented Lagrange multipliers:reduction to exact penalty functions and localization principle[J].Mathematical Programming,2017,166(1-2):297-326.
[15]ZHOU Y Y,ZHOU J C,YANG X Q.Existence of augmented Lagrange multipliers for cone constrained optimization problems[J].Journal of Global Optimization,2014,58(2):243-260.