Lagrange Multipliers - Revision history

Lila: /* Resources */

2021-11-02T21:52:08Z

Resources

Lila: /* Resources */

2021-10-06T18:22:51Z

Resources

Lila at 18:22, 6 October 2021

2021-10-06T18:22:09Z

Vu.nguyen at 02:44, 26 October 2020

2020-10-26T02:44:52Z

Vu.nguyen at 02:43, 26 October 2020

2020-10-26T02:43:04Z

Vu.nguyen at 02:36, 26 October 2020

2020-10-26T02:36:12Z

Vu.nguyen at 02:35, 26 October 2020

2020-10-26T02:35:39Z

Vu.nguyen at 02:33, 26 October 2020

2020-10-26T02:33:31Z

Vu.nguyen: Created page with "Lagrange Multipliers"

2020-10-26T02:26:46Z

Created page with "Lagrange Multipliers"

New page

Lagrange Multipliers

← Older revision		Revision as of 21:52, 2 November 2021
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	*[https://www.youtube.com/watch?v=nDuS5uQ7-lo Lagrange multipliers (3 variables) \| MIT 18.02SC Multivariable Calculus, Fall 2010] Video by MIT OpenCourseWare		*[https://www.youtube.com/watch?v=nDuS5uQ7-lo Lagrange multipliers (3 variables) \| MIT 18.02SC Multivariable Calculus, Fall 2010] Video by MIT OpenCourseWare
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus_Optimization_Methods/Lagrange_Multipliers Lagrange Multipliers, WikiBooks: Calculus Optimization Methods] under a CC BY-SA license

← Older revision		Revision as of 18:22, 6 October 2021
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	==Resources==		==Resources==
		+	* [https://en.wikibooks.org/wiki/Calculus_Optimization_Methods/Lagrange_Multipliers Lagrange Multipliers], WikiBooks: Calculus Optimization Methods

	===Videos===		===Videos===

← Older revision		Revision as of 18:22, 6 October 2021
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−		+	The method of Lagrange multipliers solves the constrained optimization problem by transforming it into a non-constrained optimization problem of the form:
		+	* <math>\operatorname{\mathcal{L}}(x_1,x_2,\ldots, x_n,\lambda)= \operatorname{f}(x_1,x_2,\ldots, x_n)+\operatorname{\lambda}(k-g(x_1,x_2,\ldots, x_n))</math>
		+	Then finding the gradient and Hessian as was done above will determine any optimum values of <math>\operatorname{\mathcal{L}}(x_1,x_2,\ldots, x_n,\lambda)</math>.
		+
		+	Suppose we now want to find optimum values for <math>f(x,y)=2x^2+y^2</math> subject to <math>x+y=1</math> from [2].
		+
		+	Then the Lagrangian method will result in a non-constrained function.
		+	* <math>\operatorname{\mathcal{L}}(x,y,\lambda)= 2x^2+y^2+\lambda (1-x-y)</math>
		+
		+	The gradient for this new function is
		+	* <math>\frac{\partial \mathcal{L}}{\partial x}(x,y,\lambda)= 4x+\lambda (-1)=0</math>
		+	* <math>\frac{\partial \mathcal{L}}{\partial y}(x,y,\lambda)= 2y+\lambda (-1)=0</math>
		+	* <math>\frac{\partial \mathcal{L}}{\partial \lambda}(x,y,\lambda)=1-x-y=0</math>
		+
		+	Finding the stationary points of the above equations can be obtained from their matrix from.
		+
		+	: <math> \begin{bmatrix}
		+	4 & 0 & -1 \\
		+	0& 2 & -1 \\
		+	-1 & -1 & 0
		+	\end{bmatrix} \begin{bmatrix}
		+	x\\
		+	y \\
		+	\lambda \end{bmatrix}= \begin{bmatrix}
		+	0\\
		+	0\\
		+	-1
		+	\end{bmatrix}
		+	</math>
		+
		+	This results in <math>x=1/3, y=2/3, \lambda=4/3</math>.
		+
		+	Next we can use the Hessian as before to determine the type of this stationary point.
		+
		+	: <math> H(\mathcal{L})=
		+	\begin{bmatrix}
		+	4 & 0 & -1 \\
		+	0& 2 & -1 \\
		+	-1&-1&0
		+	\end{bmatrix}
		+	</math>
		+
		+	Since <math> H(\mathcal{L}) >0 </math> then the solution <math>(1/3,2/3,4/3)</math> minimizes <math>f(x,y)=2x^2+y^2</math> subject to <math>x+y=1</math> with <math>f(x,y)=2/3</math>.
		+
		+
		+	==Resources==
		+
		+	===Videos===
	*[https://www.youtube.com/watch?v=ry9cgNx1QV8&list=RDCMUCFe6jenM1Bc54qtBsIJGRZQ&index=5 LaGrange Multipliers - Finding Maximum or Minimum Values ]Video by patrickJMT		*[https://www.youtube.com/watch?v=ry9cgNx1QV8&list=RDCMUCFe6jenM1Bc54qtBsIJGRZQ&index=5 LaGrange Multipliers - Finding Maximum or Minimum Values ]Video by patrickJMT

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-•[https://www.youtube.com/watch?v=ry9cgNx1QV8&list=RDCMUCFe6jenM1Bc54qtBsIJGRZQ&index=5 LaGrange Multipliers - Finding Maximum or Minimum Values ]Video by patrickJMT
+*[https://www.youtube.com/watch?v=ry9cgNx1QV8&list=RDCMUCFe6jenM1Bc54qtBsIJGRZQ&index=5 LaGrange Multipliers - Finding Maximum or Minimum Values ]Video by patrickJMT
-•[https://www.youtube.com/watch?v=x6j6yFzTUgU Lagrange Multipliers Practice Problems] Video by ames Hamblin 2017
-•[https://www.youtube.com/watch?v=HyqBcD_e_Uw Lagrange multipliers | MIT 18.02SC Multivariable Calculus, Fall 2010] Video by MIT OpenCourseWare
+*[https://www.youtube.com/watch?v=x6j6yFzTUgU Lagrange Multipliers Practice Problems] Video by ames Hamblin 2017
-•[https://www.youtube.com/watch?v=qXhcpqslNUU Lagrange Multipliers - Two Constraints -patrickJMT 2009 ] Video by patrickJMT 2009
-•[https://www.youtube.com/watch?v=nDuS5uQ7-lo Lagrange multipliers (3 variables) | MIT 18.02SC Multivariable Calculus, Fall 2010] Video by MIT OpenCourseWare
+*[https://www.youtube.com/watch?v=HyqBcD_e_Uw Lagrange multipliers | MIT 18.02SC Multivariable Calculus, Fall 2010] Video by MIT OpenCourseWare

← Older revision		Revision as of 02:43, 26 October 2020
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−	[https://www.youtube.com/watch?v=ry9cgNx1QV8&list=RDCMUCFe6jenM1Bc54qtBsIJGRZQ&index=5 LaGrange Multipliers - Finding Maximum or Minimum Values ]Video by patrickJMT	+	•[https://www.youtube.com/watch?v=ry9cgNx1QV8&list=RDCMUCFe6jenM1Bc54qtBsIJGRZQ&index=5 LaGrange Multipliers - Finding Maximum or Minimum Values ]Video by patrickJMT
		+	•[https://www.youtube.com/watch?v=x6j6yFzTUgU Lagrange Multipliers Practice Problems] Video by ames Hamblin 2017
		+	•[https://www.youtube.com/watch?v=HyqBcD_e_Uw Lagrange multipliers \| MIT 18.02SC Multivariable Calculus, Fall 2010] Video by MIT OpenCourseWare
		+	•[https://www.youtube.com/watch?v=qXhcpqslNUU Lagrange Multipliers - Two Constraints -patrickJMT 2009 ] Video by patrickJMT 2009
		+	•[https://www.youtube.com/watch?v=nDuS5uQ7-lo Lagrange multipliers (3 variables) \| MIT 18.02SC Multivariable Calculus, Fall 2010] Video by MIT OpenCourseWare

← Older revision		Revision as of 02:36, 26 October 2020
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−	[https://www.youtube.com/watch?v=ry9cgNx1QV8&list=RDCMUCFe6jenM1Bc54qtBsIJGRZQ&index=5 LaGrange Multipliers - Finding Maximum or Minimum Values ]	+	[https://www.youtube.com/watch?v=ry9cgNx1QV8&list=RDCMUCFe6jenM1Bc54qtBsIJGRZQ&index=5 LaGrange Multipliers - Finding Maximum or Minimum Values ]Video by patrickJMT

← Older revision		Revision as of 02:35, 26 October 2020
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−	[~~ttps~~://www.youtube.com/watch?v=ry9cgNx1QV8&list=RDCMUCFe6jenM1Bc54qtBsIJGRZQ&index=5 LaGrange Multipliers - Finding Maximum or Minimum Values] ~~-patrickJMT~~	+	[https://www.youtube.com/watch?v=ry9cgNx1QV8&list=RDCMUCFe6jenM1Bc54qtBsIJGRZQ&index=5 LaGrange Multipliers - Finding Maximum or Minimum Values ]

← Older revision		Revision as of 02:33, 26 October 2020
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−	~~Lagrange~~ Multipliers	+
		+	[ttps://www.youtube.com/watch?v=ry9cgNx1QV8&list=RDCMUCFe6jenM1Bc54qtBsIJGRZQ&index=5 LaGrange Multipliers - Finding Maximum or Minimum Values] -patrickJMT