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In Lecture 23 of this course on Convex Optimization, we focus on algorithms that solve unconstrained minimization type problems. The lecture evolves around unconstrained minimization problems that might or might not enjoy closed form solutions. Descent methods are discussed along with exact line search and backtracking. MATLAB implementations are given along the way.

This lecture is outlined as follows:

00:00:00 Introduction
00:01:06 Unconstrained Minimization
00:01:36 Iterative Algorithm Assumptions
00:04:28 Gradient Equivalence
00:09:04 Unconstrained Least Squares
00:20:13 Unconstrained Geometric Program
00:28:10 Initial Subset Assumption
00:35:16 Intuitive Solution of Logarithmic Barrier Minimization
00:40:42 Generalization of Logarithmic Barriers
00:42:57 Descent Methods
00:50:42 Gradient Descent
00:52:59 Exact Line Search
00:56:23 Backtracking
01:00:25 MATLAB: Gradient Descent with Exact Line Search
01:17:35 MATLAB: Gradient Descent with Backtracking
01:20:12 MATLAB: Gradient Descent with Explicit Step Size Update
01:28:07 Summary
01:30:59 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...  
Lecture 14 | Lagrange Dual Function    • Lecture 14 | Lagrange Dual Function |...  
Lecture 15 | Lagrange Dual Problem    • Lecture 15 | Lagrange Dual Problem | ...  
Lecture 16 | Certificate of Suboptimality    • Lecture 16 | Certificate of Suboptima...  
Lecture 17 | Complementary Slackness    • Lecture 17 | Complementary Slackness ...  
Lecture 18 | KKT Conditions    • Lecture 18 | KKT Conditions | Convex ...  
Lecture 19 | Perturbation and Sensitivity Analysis    • Lecture 19 | Perturbation and Sensiti...  
Lecture 20 | Equivalent Reformulations    • Lecture 20 | Equivalent Reformulation...  
Lecture 21 | Weak Alternatives    • Lecture 21 | Weak Alternatives | Conv...  
Lecture 22 | Strong Alternatives    • Lecture 22 | Strong Alternatives | Co...  
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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
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Credits :
Microsoft OneNote: https://products.office.com/en-gb/one...

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