Lagrangean duality - Revision history

St849: /* Conclusion */

2021-12-16T03:01:13Z

Conclusion

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	==Conclusion==		==Conclusion==
	Overall, ~~Lagrangian~~ duality is a very useful way to solve complex optimization problems, since it is often an effective strategy for reducing the difficulty of finding the solution to the original problem. The process to find the ~~Lagrangian~~ dual is simple and can be implemented in a wide range of problems. ~~Lagrangian~~ duality is used in a variety of applications, including the traveling salesman problem, deep learning models, and computer vision.		Overall, Lagrangean duality is a very useful way to solve complex optimization problems, since it is often an effective strategy for reducing the difficulty of finding the solution to the original problem. The process to find the Lagrangean dual is simple and can be implemented in a wide range of problems. Lagrangean duality is used in a variety of applications, including the traveling salesman problem, deep learning models, and computer vision.

	==References==		==References==

	<references />		<references />

St849: /* Applications */

2021-12-16T03:00:31Z

Applications

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	==Applications==		==Applications==
	The optimal solution to a dual problem is a vector of Karush-Kuhn-Tucker (KKT) multipliers (also known as Lagrange Multipliers or Dual Multipliers), thus the multipliers can be used for nonlinear programming problems to ensure the solution is indeed optimal. Since the Lagrange Multipliers can be used to ensure the optimal solution, ~~Lagrangian~~ duals can be applied to achieve many practical outcomes in optimization, such as determining the lower bounds for non-convex problems, simplifying the solutions of some convex problems meeting the conditions for strong duality, and determining the feasibility of a system of equations or inequalities<ref name=":2" />. The ~~Lagrangian~~ dual is used to compute the Support Vector Classifier (SVC) and effectively provides a lower bound on the objective function, which helps to characterize its solution.<ref>T. Hastie, R. Tibshirani, J. Friedman. [http://statweb.stanford.edu/~tibs/ElemStatLearn/ The Elements of Statistical Learning], Springer, 2009</ref>		The optimal solution to a dual problem is a vector of Karush-Kuhn-Tucker (KKT) multipliers (also known as Lagrange Multipliers or Dual Multipliers), thus the multipliers can be used for nonlinear programming problems to ensure the solution is indeed optimal. Since the Lagrange Multipliers can be used to ensure the optimal solution, Lagrangean duals can be applied to achieve many practical outcomes in optimization, such as determining the lower bounds for non-convex problems, simplifying the solutions of some convex problems meeting the conditions for strong duality, and determining the feasibility of a system of equations or inequalities<ref name=":2" />. The Lagrangean dual is used to compute the Support Vector Classifier (SVC) and effectively provides a lower bound on the objective function, which helps to characterize its solution.<ref>T. Hastie, R. Tibshirani, J. Friedman. [http://statweb.stanford.edu/~tibs/ElemStatLearn/ The Elements of Statistical Learning], Springer, 2009</ref>

	One application where the ~~Lagrangian~~ dual may be applied to find bounds for a complex problem is the symmetric [[Traveling salesman problem\|Traveling Salesman Problem]]. This problem is analytically difficult, but its ~~Lagrangian~~ dual is a convex problem whose optimal solution provides a lower bound on solutions to the primal problem. While the solution to the dual problem may not be feasible in the primal problem, it is possible to use heuristic methods to find a primal solution based on the optimal dual solution<ref>S. Timsj, [https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.31.843&rep=rep1&type=pdf An Application of Lagrangian Relaxation to the Traveling Salesman Problem]</ref>.		One application where the Lagrangean dual may be applied to find bounds for a complex problem is the symmetric [[Traveling salesman problem\|Traveling Salesman Problem]]. This problem is analytically difficult, but its Lagrangean dual is a convex problem whose optimal solution provides a lower bound on solutions to the primal problem. While the solution to the dual problem may not be feasible in the primal problem, it is possible to use heuristic methods to find a primal solution based on the optimal dual solution<ref>S. Timsj, [https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.31.843&rep=rep1&type=pdf An Application of Lagrangian Relaxation to the Traveling Salesman Problem]</ref>.

	~~Lagrangian~~ duals have also proven useful in deep learning problems where the objective is to learn optimization problems constrained by hard physical laws. By incorporating duality during the training cycle, researchers can achieve improved results on a variety of problems, including optimizing power flows; optimizing gas compressor usage in natural gas pipelines; transprecision computing, in which the bit width of variables may be reduced to reduce power usage in embedded systems at the cost of increased error; and devising classifiers which are “fair” with respect to a defined protected characteristic<ref>F. Fioretto, P. Van Hentenryck, T. W. K. Mak, C. Tran, F. Baldo, M. Lombardi, [https://web.ecs.syr.edu/~ffiorett/files/papers/ecml2020_arxiv.pdf Lagrangian Duality for Constrained Deep Learning]</ref>.		Lagrangean duals have also proven useful in deep learning problems where the objective is to learn optimization problems constrained by hard physical laws. By incorporating duality during the training cycle, researchers can achieve improved results on a variety of problems, including optimizing power flows; optimizing gas compressor usage in natural gas pipelines; transprecision computing, in which the bit width of variables may be reduced to reduce power usage in embedded systems at the cost of increased error; and devising classifiers which are “fair” with respect to a defined protected characteristic<ref>F. Fioretto, P. Van Hentenryck, T. W. K. Mak, C. Tran, F. Baldo, M. Lombardi, [https://web.ecs.syr.edu/~ffiorett/files/papers/ecml2020_arxiv.pdf Lagrangian Duality for Constrained Deep Learning]</ref>.

	The convex relaxation provided by the ~~Lagrangian~~ dual is also useful in the domains of machine learning<ref>J. Hui, [https://jonathan-hui.medium.com/machine-learning-lagrange-multiplier-dual-decomposition-4afe66158c9 Machine Learning - Lagrange multiplier & Dual decomposition]</ref>; combinatorial optimization, where it generalizes two well-known relaxation techniques in the field, max-stable and max-cut<ref>C. Lemarechal, F. Oustry, [https://hal.inria.fr/inria-00072958/document Semidefinite Relaxations and Lagrangian Duality with Application to Combinatorial Optimization]</ref>; and 3D registration of geometric data in computer vision, where primal techniques often fail to escape local optima<ref>J. Briales, J. Gonzalez-Jimenez, [https://openaccess.thecvf.com/content_cvpr_2017/papers/Briales_Convex_Global_3D_CVPR_2017_paper.pdf Convex Global 3D Registration with Lagrangian Duality]</ref>. ~~Lagrangian~~ relaxation is a standard and efficient way to determine the optimal solution of NP-hard combinatorial problems.<ref>A. Ben-Tal, L.E. Ghaoui, A. Nemirovski. [http://www2.isye.gatech.edu/~nemirovs/FullBookDec11.pdf Robust optimization]. Princeton University Press, 2009</ref>		The convex relaxation provided by the Lagrangean dual is also useful in the domains of machine learning<ref>J. Hui, [https://jonathan-hui.medium.com/machine-learning-lagrange-multiplier-dual-decomposition-4afe66158c9 Machine Learning - Lagrange multiplier & Dual decomposition]</ref>; combinatorial optimization, where it generalizes two well-known relaxation techniques in the field, max-stable and max-cut<ref>C. Lemarechal, F. Oustry, [https://hal.inria.fr/inria-00072958/document Semidefinite Relaxations and Lagrangian Duality with Application to Combinatorial Optimization]</ref>; and 3D registration of geometric data in computer vision, where primal techniques often fail to escape local optima<ref>J. Briales, J. Gonzalez-Jimenez, [https://openaccess.thecvf.com/content_cvpr_2017/papers/Briales_Convex_Global_3D_CVPR_2017_paper.pdf Convex Global 3D Registration with Lagrangian Duality]</ref>. Lagrangean relaxation is a standard and efficient way to determine the optimal solution of NP-hard combinatorial problems.<ref>A. Ben-Tal, L.E. Ghaoui, A. Nemirovski. [http://www2.isye.gatech.edu/~nemirovs/FullBookDec11.pdf Robust optimization]. Princeton University Press, 2009</ref>

	In some real-world applications, the dual formulation of a problem has a specific real-world meaning. The following are examples of real-world applications for ~~Lagrangian~~ Duality:		In some real-world applications, the dual formulation of a problem has a specific real-world meaning. The following are examples of real-world applications for Lagrangean Duality:

	* In an electrical network, if the primal problem describes current flows, the dual problem describes voltage differences.<ref name=":0">R. M. Freund, [https://ocw.mit.edu/courses/sloan-school-of-management/15-094j-systems-optimization-models-and-computation-sma-5223-spring-2004/lecture-notes/duality_article.pdf Applied Lagrange Duality for Constrained Optimization], Massachusetts Institute of Technology, 2004</ref>		* In an electrical network, if the primal problem describes current flows, the dual problem describes voltage differences.<ref name=":0">R. M. Freund, [https://ocw.mit.edu/courses/sloan-school-of-management/15-094j-systems-optimization-models-and-computation-sma-5223-spring-2004/lecture-notes/duality_article.pdf Applied Lagrange Duality for Constrained Optimization], Massachusetts Institute of Technology, 2004</ref>

St849: /* Numerical Example */

2021-12-16T02:57:43Z

Numerical Example

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	In this section, we will apply the process to solve an example problem.		In this section, we will apply the process to solve an example problem.

	Construct the ~~Lagrangian~~ dual for the following optimization problem:		Construct the Lagrangean dual for the following optimization problem:

	<blockquote><math>\min</math> <math>x_1+2x_2+5x_3</math>		<blockquote><math>\min</math> <math>x_1+2x_2+5x_3</math>
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	<math>x_3^2-x_2-x_1=0</math>		<math>x_3^2-x_2-x_1=0</math>

	</blockquote>Step 1: Construct the ~~Lagrangian~~.		</blockquote>Step 1: Construct the Lagrangean.

	<blockquote><math>L(x,\lambda,\nu) = x_1 + 2x_2 + 5x_3 + \sum_{i=1}^1\lambda_i * x_1x_2 + \sum_{i=1}^1\nu_i * (x_3^2-x_2-x_1)</math>		<blockquote><math>L(x,\lambda,\nu) = x_1 + 2x_2 + 5x_3 + \sum_{i=1}^1\lambda_i * x_1x_2 + \sum_{i=1}^1\nu_i * (x_3^2-x_2-x_1)</math>
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	<math>\nu_1\geq0</math>		<math>\nu_1\geq0</math>

	</blockquote></blockquote>Step 2: Take the partial derivatives of the ~~Lagrangian~~ with respect to <math>x</math>, set each equal to zero, and solve for <math>x^*</math>.		</blockquote></blockquote>Step 2: Take the partial derivatives of the Lagrangean with respect to <math>x</math>, set each equal to zero, and solve for <math>x^*</math>.

	<blockquote><math>{dL(x,\lambda,\nu) \over dx_1}=1+\lambda_1x_2-\nu_1=0</math>		<blockquote><math>{dL(x,\lambda,\nu) \over dx_1}=1+\lambda_1x_2-\nu_1=0</math>
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	<math>x_3^* = \frac{-5}{2\nu_1} </math>		<math>x_3^* = \frac{-5}{2\nu_1} </math>

	</blockquote>Step 3: Substitute <math>x^*</math> terms into the ~~Lagrangian~~ to create the dual function (<math>\phi</math>), which is only a function of the dual variables (<math>\lambda, \nu</math>).		</blockquote>Step 3: Substitute <math>x^*</math> terms into the Lagrangean to create the dual function (<math>\phi</math>), which is only a function of the dual variables (<math>\lambda, \nu</math>).

	<blockquote><math>L(x^,\lambda,\nu) = x_1^ + 2x_2^* + 5x_3^* + \lambda_1x_1^x_2^ + \nu_1(x_3^)^2-\nu_1x_2^-\nu_1x_1^*</math>		<blockquote><math>L(x^,\lambda,\nu) = x_1^ + 2x_2^* + 5x_3^* + \lambda_1x_1^x_2^ + \nu_1(x_3^)^2-\nu_1x_2^-\nu_1x_1^*</math>

St849: /* Process */

2021-12-16T02:56:44Z

Process

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	==Process==		==Process==

	Constructing the ~~Lagrangian~~ dual can be done in four easy steps:		Constructing the Lagrangean dual can be done in four easy steps:

	<blockquote>Step 1: Construct the ~~Lagrangian~~. The dual variables are non-negative to ensure strong duality.		<blockquote>Step 1: Construct the Lagrangean. The dual variables are non-negative to ensure strong duality.

	Step 2: Take the partial derivatives (gradient) of the ~~Lagrangian~~ with respect to <math>x</math>, set each equal to zero, and solve for <math>x^*</math>.		Step 2: Take the partial derivatives (gradient) of the Lagrangean with respect to <math>x</math>, set each equal to zero, and solve for <math>x^*</math>.

	Step 3: Substitute <math>x^*</math> terms into the ~~Lagrangian~~ to create the dual function (<math>\phi</math>), which is only a function of the dual variables (<math>\lambda, \nu</math>).		Step 3: Substitute <math>x^*</math> terms into the Lagrangean to create the dual function (<math>\phi</math>), which is only a function of the dual variables (<math>\lambda, \nu</math>).

	Step 4: Construct the dual with the dual function (<math>\phi</math>) and the new constraints. The dual variables are non-negative.		Step 4: Construct the dual with the dual function (<math>\phi</math>) and the new constraints. The dual variables are non-negative.

St849: /* Complementary Slackness */

2021-12-16T02:55:35Z

Complementary Slackness

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	<blockquote><math>\leq (\ f_0(x^) + \sum_{i=1}^m(\lambda_i^\ f_i(x^)) + \sum_{i=1}^p(\nu_i^\ h_i(x^*)) </math>		<blockquote><math>\leq (\ f_0(x^) + \sum_{i=1}^m(\lambda_i^\ f_i(x^)) + \sum_{i=1}^p(\nu_i^\ h_i(x^*)) </math>

	</blockquote>The inequality shows the <math>x^* </math> minimizes <math>L(x, \lambda, \nu)</math> over <math>x</math>. It is possible to have other minimizers however in this instance <math>x^ </math> is the minimizer. This inequality follows as the ~~Lagrangian~~ infimum over <math>x</math> is less than or equal to <math>x = x*</math>.		</blockquote>The inequality shows the <math>x^* </math> minimizes <math>L(x, \lambda, \nu)</math> over <math>x</math>. It is possible to have other minimizers however in this instance <math>x^ </math> is the minimizer. This inequality follows as the Lagrangean infimum over <math>x</math> is less than or equal to <math>x = x*</math>.

	<blockquote><math>\leq \ f_0(x^*) </math>		<blockquote><math>\leq \ f_0(x^*) </math>

St849: /* Algorithm */

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Algorithm

St849: /* Introduction */

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Introduction

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	==Introduction==		==Introduction==
	~~Lagrangian~~ duality is a specific form of a broader concept known as [[Duality]]. The theory of duality originated as part of an intellectual debate and observation amongst mathematicians and colleagues John von Neumann and George Dantzig. During World War II, a discussion occurred where Dantzig shared his theory on Linear Programming and his simplex method. Von Neumann quickly surmised that Dantzig's Linear Programming shared an equivalence with his Game Theory and Expanding Economy model. Von Neumann then conceptualized the theory of using an alternate perspective, referred to as the dual problem, so that either the primal or dual problem could be solved to yield an optimal solution. In 1947, von Neumann formally published his theory of Duality. This principle would then see several formulations and implementations such as Fenchel, Wolfe, and ~~Lagrangian~~.<ref>A Bachem, Marting Grötschel, B H Korte, Mathematical Programming: The State of the Art. Springer-Verlag, 1983</ref>		Lagrangean duality is a specific form of a broader concept known as [[Duality]]. The theory of duality originated as part of an intellectual debate and observation amongst mathematicians and colleagues John von Neumann and George Dantzig. During World War II, a discussion occurred where Dantzig shared his theory on Linear Programming and his simplex method. Von Neumann quickly surmised that Dantzig's Linear Programming shared an equivalence with his Game Theory and Expanding Economy model. Von Neumann then conceptualized the theory of using an alternate perspective, referred to as the dual problem, so that either the primal or dual problem could be solved to yield an optimal solution. In 1947, von Neumann formally published his theory of Duality. This principle would then see several formulations and implementations such as Fenchel, Wolfe, and Lagrangean.<ref>A Bachem, Marting Grötschel, B H Korte, Mathematical Programming: The State of the Art. Springer-Verlag, 1983</ref>

	The ~~Lagrangian~~ dual has become a common approach to solving optimization problems. Many problems can be efficiently solved by constructing the ~~Lagrangian~~ function of the problem and solving the dual problem instead of the primal problem. Optimization problems can be reformulated by taking the ~~Lagrangian~~ and its corresponding gradient to find the optimal value of the optimization problem’s variables. These values are then substituted back into the ~~Lagrangian~~ and the dual problem is constructed by reformulating the optimization problem in terms of a different set of variables (dual variables).		The Lagrangean dual has become a common approach to solving optimization problems. Many problems can be efficiently solved by constructing the Lagrangean function of the problem and solving the dual problem instead of the primal problem. Optimization problems can be reformulated by taking the Lagrangean and its corresponding gradient to find the optimal value of the optimization problem’s variables. These values are then substituted back into the Lagrangean and the dual problem is constructed by reformulating the optimization problem in terms of a different set of variables (dual variables).

	Even when a dual problem cannot be proven to have the same optimal solution as the primal problem, ~~Lagrangian~~ relaxation still provides benefits. This is when an approximation can be determined by using ~~Lagrangian~~ multipliers to add a penalty to constraint violations. The advantage of performing ~~Lagrangian~~ relaxation is that in many situations it is easier to solve the dual than the primal problem.<ref>Claude Lemaréchal, Computational combinatorial optimization: Lagrangian Relaxation, Springer-Verlag, 2001</ref> For example, non-convex problems are generally more difficult to solve than convex problems. By reformulating the primal optimization problem into the ~~Lagrangian~~ dual problem, a difficult non-convex problem can oftentimes be turned into an easier convex problem.<ref name=":1">S. Boyd, L. Vandenberghe, [https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf Convex Optimization], Cambridge University Press, 2009</ref>		Even when a dual problem cannot be proven to have the same optimal solution as the primal problem, Lagrangean relaxation still provides benefits. This is when an approximation can be determined by using Lagrangean multipliers to add a penalty to constraint violations. The advantage of performing Lagrangean relaxation is that in many situations it is easier to solve the dual than the primal problem.<ref>Claude Lemaréchal, Computational combinatorial optimization: Lagrangian Relaxation, Springer-Verlag, 2001</ref> For example, non-convex problems are generally more difficult to solve than convex problems. By reformulating the primal optimization problem into the Lagrangean dual problem, a difficult non-convex problem can oftentimes be turned into an easier convex problem.<ref name=":1">S. Boyd, L. Vandenberghe, [https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf Convex Optimization], Cambridge University Press, 2009</ref>

	==Algorithm==		==Algorithm==

DanielYeung at 22:52, 15 December 2021

2021-12-15T22:52:19Z

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	==Weak and Strong Duality==		==Weak and Strong Duality==
	[[File:Weak and Strong Dualtiy.png\|link=https://optimization.cbe.cornell.edu/File:Weak%20and%20Strong%20Dualtiy.png\|thumb\|Illustration of weak and strong duality<ref>B. Ghojogh, A. Ghodsi, F. Karray, M. Crowley, [https://arxiv.org/pdf/2110.01858.pdf KKT Conditions, First-Order and Second-Order Optimization, and Distributed Optimization: Tutorial and Survey], University of Waterloo</ref>.\|alt=\|~~left~~]] Strong duality is a concept in mathematical optimization that states the primal optimal objective and the dual optimal objective value are equal under certain conditions. Whereas, in the weak duality, the optimal value from the primal objective is greater than or equal to the dual objective. Thus, in the weak duality, the duality gap is greater than or equal to zero<ref>R.J. Vanderbei, Linear Programming: Foundations and Extensions. Springer, 2008, http://link.springer.com/book/10.1007/978-0-387-74388-2</ref>.		[[File:Weak and Strong Dualtiy.png\|link=https://optimization.cbe.cornell.edu/File:Weak%20and%20Strong%20Dualtiy.png\|thumb\|Illustration of weak and strong duality<ref>B. Ghojogh, A. Ghodsi, F. Karray, M. Crowley, [https://arxiv.org/pdf/2110.01858.pdf KKT Conditions, First-Order and Second-Order Optimization, and Distributed Optimization: Tutorial and Survey], University of Waterloo</ref>.\|alt=\|right]] Strong duality is a concept in mathematical optimization that states the primal optimal objective and the dual optimal objective value are equal under certain conditions. Whereas, in the weak duality, the optimal value from the primal objective is greater than or equal to the dual objective. Thus, in the weak duality, the duality gap is greater than or equal to zero<ref>R.J. Vanderbei, Linear Programming: Foundations and Extensions. Springer, 2008, http://link.springer.com/book/10.1007/978-0-387-74388-2</ref>.

	The verification of gaps is a convenient tool to check the optimality of solutions. As shown in the illustration, left, weak duality creates an optimality gap, while strong duality does not. Thus, the strong duality only holds true if the duality gap is equal to 0.<br /><br />		The verification of gaps is a convenient tool to check the optimality of solutions. As shown in the illustration, left, weak duality creates an optimality gap, while strong duality does not. Thus, the strong duality only holds true if the duality gap is equal to 0.<br /><br />

Msk328: /* Introduction */

2021-12-15T22:27:34Z

Introduction

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	==Introduction==		==Introduction==
	Lagrangian duality is a specific form of a broader concept known as [[Duality]]. The theory of duality originated as part of an intellectual debate and observation amongst mathematicians and colleagues John von Neumann and George Dantzig. During World War II, a discussion occurred where Dantzig shared his theory on Linear Programming and his simplex method. Von Neumann quickly surmised that Dantzig's Linear Programming shared an equivalence with his Game Theory and Expanding Economy model. Von Neumann then conceptualized the theory of using an alternate perspective, referred to as the dual problem, so that either the primal or dual problem could be solved to yield an optimal solution. In 1947, von Neumann formally published his theory of Duality. This principle would then see several formulations and implementations such as Fenchel, Wolfe, and Lagrangian<ref>A Bachem, Marting Grötschel, B H Korte, Mathematical Programming: The State of the Art. Springer-Verlag, 1983</ref>.		Lagrangian duality is a specific form of a broader concept known as [[Duality]]. The theory of duality originated as part of an intellectual debate and observation amongst mathematicians and colleagues John von Neumann and George Dantzig. During World War II, a discussion occurred where Dantzig shared his theory on Linear Programming and his simplex method. Von Neumann quickly surmised that Dantzig's Linear Programming shared an equivalence with his Game Theory and Expanding Economy model. Von Neumann then conceptualized the theory of using an alternate perspective, referred to as the dual problem, so that either the primal or dual problem could be solved to yield an optimal solution. In 1947, von Neumann formally published his theory of Duality. This principle would then see several formulations and implementations such as Fenchel, Wolfe, and Lagrangian.<ref>A Bachem, Marting Grötschel, B H Korte, Mathematical Programming: The State of the Art. Springer-Verlag, 1983</ref>

	The Lagrangian dual has become a common approach to solving optimization problems. Many problems can be efficiently solved by constructing the Lagrangian function of the problem and solving the dual problem instead of the primal problem. Optimization problems can be reformulated by taking the Lagrangian and its corresponding gradient to find the optimal value of the optimization problem’s variables. These values are then substituted back into the Lagrangian and the dual problem is constructed by reformulating the optimization problem in terms of a different set of variables (dual variables).		The Lagrangian dual has become a common approach to solving optimization problems. Many problems can be efficiently solved by constructing the Lagrangian function of the problem and solving the dual problem instead of the primal problem. Optimization problems can be reformulated by taking the Lagrangian and its corresponding gradient to find the optimal value of the optimization problem’s variables. These values are then substituted back into the Lagrangian and the dual problem is constructed by reformulating the optimization problem in terms of a different set of variables (dual variables).

	Even when a dual problem cannot be proven to have the same optimal solution as the primal problem, Lagrangian relaxation still provides benefits. This is when an approximation can be determined by using Lagrangian multipliers to add a penalty to constraint violations. The advantage of performing Lagrangian relaxation is that in many situations it is easier to solve the dual than the primal problem.<ref>Claude Lemaréchal, Computational combinatorial optimization: Lagrangian Relaxation, Springer-Verlag, 2001</ref>. For example, non-convex problems are generally more difficult to solve than convex problems. By reformulating the primal optimization problem into the Lagrangian dual problem, a difficult non-convex problem can oftentimes be turned into an easier convex problem.<ref name=":1">S. Boyd, L. Vandenberghe, [https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf Convex Optimization], Cambridge University Press, 2009</ref>		Even when a dual problem cannot be proven to have the same optimal solution as the primal problem, Lagrangian relaxation still provides benefits. This is when an approximation can be determined by using Lagrangian multipliers to add a penalty to constraint violations. The advantage of performing Lagrangian relaxation is that in many situations it is easier to solve the dual than the primal problem.<ref>Claude Lemaréchal, Computational combinatorial optimization: Lagrangian Relaxation, Springer-Verlag, 2001</ref> For example, non-convex problems are generally more difficult to solve than convex problems. By reformulating the primal optimization problem into the Lagrangian dual problem, a difficult non-convex problem can oftentimes be turned into an easier convex problem.<ref name=":1">S. Boyd, L. Vandenberghe, [https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf Convex Optimization], Cambridge University Press, 2009</ref>

	==Algorithm==		==Algorithm==

Msk328: /* Introduction */

2021-12-15T22:26:47Z

Introduction

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	The Lagrangian dual has become a common approach to solving optimization problems. Many problems can be efficiently solved by constructing the Lagrangian function of the problem and solving the dual problem instead of the primal problem. Optimization problems can be reformulated by taking the Lagrangian and its corresponding gradient to find the optimal value of the optimization problem’s variables. These values are then substituted back into the Lagrangian and the dual problem is constructed by reformulating the optimization problem in terms of a different set of variables (dual variables).		The Lagrangian dual has become a common approach to solving optimization problems. Many problems can be efficiently solved by constructing the Lagrangian function of the problem and solving the dual problem instead of the primal problem. Optimization problems can be reformulated by taking the Lagrangian and its corresponding gradient to find the optimal value of the optimization problem’s variables. These values are then substituted back into the Lagrangian and the dual problem is constructed by reformulating the optimization problem in terms of a different set of variables (dual variables).

	Even when a dual problem cannot be proven to have the same optimal solution as the primal problem, Lagrangian relaxation still provides benefits. This is when an approximation can be determined by using Lagrangian multipliers to add a penalty to constraint violations. The advantage of performing Lagrangian relaxation is that in many situations it is easier to solve the dual than the primal problem.<ref>Claude Lemaréchal, Computational combinatorial optimization: Lagrangian Relaxation, Springer-Verlag, 2001.</ref> For example, non-convex problems are generally more difficult to solve than convex problems. By reformulating the primal optimization problem into the Lagrangian dual problem, a difficult non-convex problem can oftentimes be turned into an easier convex problem.<ref name=":1">S. Boyd, L. Vandenberghe, [https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf Convex Optimization], Cambridge University Press, 2009</ref>		Even when a dual problem cannot be proven to have the same optimal solution as the primal problem, Lagrangian relaxation still provides benefits. This is when an approximation can be determined by using Lagrangian multipliers to add a penalty to constraint violations. The advantage of performing Lagrangian relaxation is that in many situations it is easier to solve the dual than the primal problem.<ref>Claude Lemaréchal, Computational combinatorial optimization: Lagrangian Relaxation, Springer-Verlag, 2001</ref>. For example, non-convex problems are generally more difficult to solve than convex problems. By reformulating the primal optimization problem into the Lagrangian dual problem, a difficult non-convex problem can oftentimes be turned into an easier convex problem.<ref name=":1">S. Boyd, L. Vandenberghe, [https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf Convex Optimization], Cambridge University Press, 2009</ref>

	==Algorithm==		==Algorithm==

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	<math>h_i(x)=0, i = 1, ..., p</math>		<math>h_i(x)=0, i = 1, ..., p</math>

	</blockquote>The ~~Lagrangian~~ for the optimization problem above takes the form:		</blockquote>The Lagrangean for the optimization problem above takes the form:

	<blockquote><math>L(x,\lambda,\nu) = f_0(x) + \sum_{i=1}^m\lambda_i * f_i(x) + \sum_{i=1}^p\nu_i * h_i(x)</math>		<blockquote><math>L(x,\lambda,\nu) = f_0(x) + \sum_{i=1}^m\lambda_i * f_i(x) + \sum_{i=1}^p\nu_i * h_i(x)</math>
Line 29:		Line 29:
	<math>\lambda_i\geq0</math>		<math>\lambda_i\geq0</math>

	<math>\nu_i\geq0</math></blockquote>The Gradient of the ~~Lagrangian~~ is set equal to zero, to find the optimal values of <math>x</math> (<math>x^*</math>).		<math>\nu_i\geq0</math></blockquote>The Gradient of the Lagrangean is set equal to zero, to find the optimal values of <math>x</math> (<math>x^*</math>).

	<blockquote><math>\nabla L(x^,\lambda^,\nu^) = \nabla f_0(x^) + \sum_{i=1}^m\lambda_i^\nabla f_i(x^) + \sum_{i=1}^p\nu_i^\nabla h_i(x^)=0</math>		<blockquote><math>\nabla L(x^,\lambda^,\nu^) = \nabla f_0(x^) + \sum_{i=1}^m\lambda_i^\nabla f_i(x^) + \sum_{i=1}^p\nu_i^\nabla h_i(x^)=0</math>

	</blockquote>The ~~Lagrangian~~ dual function (<math>\phi</math>) equals the ~~Lagrangian~~ with <math>x^*</math> values substituted out.		</blockquote>The Lagrangean dual function (<math>\phi</math>) equals the Lagrangean with <math>x^*</math> values substituted out.

	<blockquote><math>\phi(\lambda,\nu)=L(x^,\lambda,\nu) = f_0(x^) + \sum_{i=1}^m\lambda_i * f_i(x^) + \sum_{i=1}^p\nu_i h_i(x^*)</math>		<blockquote><math>\phi(\lambda,\nu)=L(x^,\lambda,\nu) = f_0(x^) + \sum_{i=1}^m\lambda_i * f_i(x^) + \sum_{i=1}^p\nu_i h_i(x^*)</math>

	</blockquote>The ~~Lagrangian~~ dual optimization problem can be formulated using the following ~~Lagrangian~~ dual function:		</blockquote>The Lagrangean dual optimization problem can be formulated using the following Lagrangean dual function:

	<blockquote><math>\min</math> <math>\phi(\lambda, \nu)</math>		<blockquote><math>\min</math> <math>\phi(\lambda, \nu)</math>
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	<math>s. t.</math> <math>\lambda_i\geq0</math>		<math>s. t.</math> <math>\lambda_i\geq0</math>

	<math>\nu_i\geq0</math></blockquote>At the optimal values of variables <math>x^</math>, there exist scalar ~~Lagrangian~~ multiplier values <math>\lambda^</math> where the gradient of the ~~Lagrangian~~ is zero<ref>J. Nocedal, S. J. Wright, [http://link.springer.com/book/10.1007/b98874 Numerical Optimization]. Springer, 1999</ref>. Hence, solving the dual problem, which is a function of the ~~Lagrangian~~ multipliers (<math>\lambda^</math>) yields the same solution as the primal problem, which is a function of the original variables (<math>x^</math>). For the dual solution and the primal solutions to be equal the condition has to be a strong duality. The ~~Lagrangian~~ takes all of the constraints and combines it with the objective function into one larger formula.		<math>\nu_i\geq0</math></blockquote>At the optimal values of variables <math>x^</math>, there exist scalar Lagrangean multiplier values <math>\lambda^</math> where the gradient of the Lagrangean is zero<ref>J. Nocedal, S. J. Wright, [http://link.springer.com/book/10.1007/b98874 Numerical Optimization]. Springer, 1999</ref>. Hence, solving the dual problem, which is a function of the Lagrangean multipliers (<math>\lambda^</math>) yields the same solution as the primal problem, which is a function of the original variables (<math>x^</math>). For the dual solution and the primal solutions to be equal the condition has to be a strong duality. The Lagrangean takes all of the constraints and combines it with the objective function into one larger formula.

	<blockquote>		<blockquote>