式13: "I" left parenthesis, "u" , right parenthesis equals negative StartFraction, 1 Over "r" , EndFraction natural log left parenthesis, negative "u" , right parenthesisIu=−1rln−u
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Lagrange Multiplier for Inequality Constraints
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式15: lambda left parenthesis, "x" , right parenthesis equals negative StartFraction, 1 Over "r" "x" , EndFractionλx=−1rx
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Lagrangian
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式17: "L" left parenthesis, "x" , "y" , right parenthesis equals "f" left parenthesis, "x" , "y" , right parenthesis plus lambda left parenthesis, "g" Subscript, 1 , Baseline left parenthesis, "x" , "y" , right parenthesis , right parenthesis "I" left parenthesis, "g" Subscript, 1 , Baseline left parenthesis, "x" , "y" , right parenthesis , right parenthesis plus lambda left parenthesis, "g" Subscript, 2 , Baseline left parenthesis, "x" , "y" , right parenthesis , right parenthesis "I" left parenthesis, "g" Subscript, 2 , Baseline left parenthesis, "x" , "y" , right parenthesis , right parenthesisLx,y=fx,y+λg1x,yIg1x,y+λg2x,yIg2x,y
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FOC of Lagrangian
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式19: left bracket, StartFraction, "d" Over "d" "x" Subscript, 0 , Baseline , EndFraction left parenthesis, "L" left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis , right parenthesis , StartFraction, "d" Over "d" "y" Subscript, 0 , Baseline , EndFraction left parenthesis, "L" left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right bracket tilde left bracket, 0 , 0 , right bracketddx0Lx0,y0,ddy0Lx0,y0~0,0
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Drag the r value up to increase precision. If you drag it too fast, then it goes towards another optimum point (the maximizer)
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式21: "r" equals 1r=1
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100100
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式22:
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`x_0= `${x_0}, `y_0=`${y_0} left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesisx0,y0
