Lecture 14 | Lagrange Dual Function | Convex Optimization by Dr. Ahmad Bazzi
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5 سال پیش
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In Lecture 14 of this course on Convex Optimization, we introduce the Lagrangian duality theory. In essence, for each optimization problem (convex or not), we can associate a certain function referred to as the Lagrangian function. This function, in turn, has a dual function (which serves as an infimum over the variable of interest x). It turns out that, for any optimisation problem, the dual function is a lower bound on the optimal value of the optimisation problem in hand. This lecture focuses on many examples that derive the Lagrangian and the associated dual functions. MATLAB implementations are also presented to give useful insights. This lecture is outlined as follows:
00:00 Intro
01:00 Lagrangian function and duality
04:02 Lagrangian dual function
06:46 Lower bound on the optimal value
09:16 MATLAB: Lower bound verification
15:28 Example 1 - Least Squares
17:48 Example 2 - Linear Programming
20:48 Example 3 - Two-way Partitioning
26:04 Relationship between conjugate function and the dual function
31:22 Example 4 - Equality Constrained Norm minimization
33:37 Example 5 - Entropy Maximization
35:44 Outro
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Lecture 1 | Introduction to Convex Optimization: Lecture 1 | Convex Optimization | Int...
Lecture 2 | Convex Sets: Lecture 2 | Convex Sets | Convex Opti...
Lecture 3 | Convex Functions: Lecture 3 | Convex Functions | Convex...
Lecture 4 | Convex Optimization Principles : Lecture 4 | Convex Optimization Princ...
Lecture 5 | Linear Programming & SIMPLEX algorithm w MATLAB: Lecture 5 | Linear Programming & SIMP...
Lecture 6 | Quadratic Programs: Lecture 6 | Quadratic Programs | Conv...
Lecture 7 | Quadratically Constrained Quadratic Programs: Lecture 7 | Quadratically Constrained...
Lecture 8 | Second Order Cone Programming: Lecture 8 | Second Order Cone Program...
Lecture 9 | Geometric Programs: Lecture 9 | Geometric Programs (GP) |...
Lecture 10 | Generalized Geometric Programs: Lecture 10 | Generalized Geometric Pr...
Lecture 11 | SemiDefinite Programming Lecture 11 | Semidefinite Programming...
Lecture 12| Vector and Multicriterion Optimization | Pareto Optimal points and the Pareto Frontier Lecture 12 | Vector and Multicriterio...
Lecture 13 | Optimal Trade-off Analysis Lecture 13 | Optimal Trade-off Analys...
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References:
[1] Boyd, Stephen, and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004.
[2] Nesterov, Yurii. Introductory lectures on convex optimization: A basic course. Vol. 87. Springer Science & Business Media, 2013.
Reference no. 3:
[3] Ben-Tal, Ahron, and Arkadi Nemirovski. Lectures on modern convex optimization: analysis, algorithms, and engineering applications. Vol. 2. Siam, 2001.
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Instructor: Dr. Ahmad Bazzi
IG: Instagram: drahmadbazzi
FB: https://www.facebook.com/profile.php?...
RG: https://www.researchgate.net/profile/...
MSE: https://math.stackexchange.com/users/...
YT: ahmadbazzi
---------------------------------------------------------------------------------------------------------
Credits :
Microsoft OneNote: https://products.office.com/en-gb/one...
#ConvexOptimization #Lagrange #Dual
In Lecture 14 of this course on Convex Optimization, we introduce the Lagrangian duality theory. In essence, for each optimization problem (convex or not), we can associate a certain function referred to as the Lagrangian function. This function, in turn, has a dual function (which serves as an infimum over the variable of interest x). It turns out that, for any optimisation problem, the dual function is a lower bound on the optimal value of the optimisation problem in hand. This lecture focuses on many examples that derive the Lagrangian and the associated dual functions. MATLAB implementations are also presented to give useful insights. This lecture is outlined as follows:
00:00 Intro
01:00 Lagrangian function and duality
04:02 Lagrangian dual function
06:46 Lower bound on the optimal value
09:16 MATLAB: Lower bound verification
15:28 Example 1 - Least Squares
17:48 Example 2 - Linear Programming
20:48 Example 3 - Two-way Partitioning
26:04 Relationship between conjugate function and the dual function
31:22 Example 4 - Equality Constrained Norm minimization
33:37 Example 5 - Entropy Maximization
35:44 Outro
---------------------------------------------------------------------------------------------------------
Lecture 1 | Introduction to Convex Optimization: Lecture 1 | Convex Optimization | Int...
Lecture 2 | Convex Sets: Lecture 2 | Convex Sets | Convex Opti...
Lecture 3 | Convex Functions: Lecture 3 | Convex Functions | Convex...
Lecture 4 | Convex Optimization Principles : Lecture 4 | Convex Optimization Princ...
Lecture 5 | Linear Programming & SIMPLEX algorithm w MATLAB: Lecture 5 | Linear Programming & SIMP...
Lecture 6 | Quadratic Programs: Lecture 6 | Quadratic Programs | Conv...
Lecture 7 | Quadratically Constrained Quadratic Programs: Lecture 7 | Quadratically Constrained...
Lecture 8 | Second Order Cone Programming: Lecture 8 | Second Order Cone Program...
Lecture 9 | Geometric Programs: Lecture 9 | Geometric Programs (GP) |...
Lecture 10 | Generalized Geometric Programs: Lecture 10 | Generalized Geometric Pr...
Lecture 11 | SemiDefinite Programming Lecture 11 | Semidefinite Programming...
Lecture 12| Vector and Multicriterion Optimization | Pareto Optimal points and the Pareto Frontier Lecture 12 | Vector and Multicriterio...
Lecture 13 | Optimal Trade-off Analysis Lecture 13 | Optimal Trade-off Analys...
---------------------------------------------------------------------------------------------------------
References:
[1] Boyd, Stephen, and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004.
[2] Nesterov, Yurii. Introductory lectures on convex optimization: A basic course. Vol. 87. Springer Science & Business Media, 2013.
Reference no. 3:
[3] Ben-Tal, Ahron, and Arkadi Nemirovski. Lectures on modern convex optimization: analysis, algorithms, and engineering applications. Vol. 2. Siam, 2001.
---------------------------------------------------------------------------------------------------------
Instructor: Dr. Ahmad Bazzi
IG: Instagram: drahmadbazzi
FB: https://www.facebook.com/profile.php?...
RG: https://www.researchgate.net/profile/...
MSE: https://math.stackexchange.com/users/...
YT: ahmadbazzi
---------------------------------------------------------------------------------------------------------
Credits :
Microsoft OneNote: https://products.office.com/en-gb/one...
#ConvexOptimization #Lagrange #Dual
5 سال پیش
در تاریخ 1398/08/12 منتشر شده
است.
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