Optimization Techniques - W2023 - Lecture 5 (Sensitivity Analysis & Lagrangian Function)

The course "Optimization Techniques" (ENGG*6140, section 2) at the School of Engineering at the University of Guelph. Instructor: Benyamin Ghojogh
Lecture 5 continues the sensitivity analysis in linear programming and then introduces the Lagrangian function, Lagrange multipliers, dual variables, the dual function, primal and dual optimization problems, weak and strong duality, and the Slater's condition.

Chapters:

0:00 - Talking about off-class question-answering
4:58 - Question about if solutions of different methods are the same
9:31 - A clarification on duality in linear programming
11:16 - Discussion on rounding the solution of linear programming for integer programming
17:55 - Questions about some typos in the slides
21:43 - Sensitivity analysis (overview)
22:54 - Sensitivity analysis (overview of case 1-1)
27:34 - Sensitivity analysis (overview of case 1-2)
30:09 - Sensitivity analysis (overview of case 2-1)
32:58 - Sensitivity analysis (overview of case 2-2)
34:54 - Sensitivity analysis (case 3: adding new variable)
52:37 - Sensitivity analysis (case 4: adding new constraint)
58:51 - Sensitivity analysis (case 4-1)
1:00:14 - Sensitivity analysis (case 4-2)
1:05:58 - Sensitivity analysis (case 4-3)
1:09:59 - Discussion on compatibility of units
1:14:28 - Start of the lecture of KKT conditions
1:16:14 - General form of optimization problem
1:21:55 - Lagrangian function, Lagrange multipliers, and dual variables
1:27:45 - Sign of the terms in Lagrangian
1:31:06 - Interpretation of the Lagrangian function
1:41:26 - The Lagrange dual function
1:45:29 - Primal feasibility
1:47:38 - Feasibility in dual function
1:51:08 - Introducing (lemma, theorem, proposition, corollary, ...)
1:54:48 - Dual feasibility
2:02:50 - Dual function as a lower bound
2:02:59 - Sign of dual variable for inequality constraint
2:07:05 - Primal and dual problems
2:11:25 - Weak and strong duality
2:13:14 - Weak and strong duality in iterative optimization
2:20:57 - Slater's condition
2:24:47 - An example for Slater's condition
2:27:14 - Talking about the material of midterm
2:28:55 - Talking about the course project
2:37:25 - A short talk about enlightenment in optimization

[AliArjomandi-fxev]

I think interpretation of $v_i*h_i(x)$ as a penalty is not precise. Because negative $h_i(x)$ does not act as a penalty.