Lecture 11: Augmented Lagrangian relaxation

Course: Advanced Optimization and Game Theory for Energy Systems
This is an online intense PhD-level course that took place during the first 3 weeks of January 2021 with 120 students from all over the world.
Lecturer: Jalal Kazempour (Technical University of Denmark, DTU)
Slides are available at: www.jalalkazem...
Course description: kurser.dtu.dk/...
Course content (all video lectures are available in this channel):
Course introduction
Lecture 1: Market clearing as an optimization problem
Lecture 2: Market clearing as an equilibrium problem
Lecture 3: Desirable properties of market-clearing mechanisms
Lecture 4: Market clearing using a cooperative game approach
Lecture 5: Stochastic market clearing
Lecture 6: Robust approaches for market clearing
Lecture 7: Bilevel programming in energy systems
Lecture 8: Optimization problems with decomposable structure
Lecture 9: Benders’ decomposition: Theory
Lecture 10a: Benders’ decomposition: Applications
Lecture 10b: Nested Benders’ decomposition
Lecture 11: Augmented Lagrangian relaxation
Lecture 12: Variants of ADMM and applications

[@amirzare6172]

Thanks a lot, professor, that's very practical.

[@lahlouaziz695]

I have a question : what if we have to optimize a cost (with a monetary unit) as an objective function and the constraint has a different unit, how to deal with this ? I mean the relaxed problem in its objective function will have a term with a monetary unit while the other term with the dual variable will have a different unit. Please correct me if I am wrong

[@yingkaisong4258]

I do not quite understand why the standard LR needs the quadratic assumption. From what was illustrated in the slides, it is not obvious why a linear obj cannot work. Also, I am confused a bit about the logic here: the standard LR needs contiuously differentiable --> thus, the obj needs to be quadratic --> thus linear obj cannot work. Firstly, not only quadratic functions are continuously differentiable; secondly, linear functions are surely continuously differentiable.

[@yingkaisong4258]

I realized that later on, by linear, you actually meant piecewise-linear, but any piecewise-linear obj can be reformulated in a standard way to eliminate such nonsmoothness in the obj

[@jalalkazempour1429]

Hi Yingkai. Thanks for this good question. I think in the case of linear objective function, the first derivative is constant while a gradient-based approach like LR explores the optimal point over iterations based on the change of derivates across the feasible space. This is not an issue in the quadratic objective function (as far as we keep convexity). I suppose if the underlying problem is convex and the objective function has a non-constant first derivate, the standard LR should work. Hope this helps.

[@yingkaisong4258]

@@jalalkazempour1429 It is more clear to me now. Thanks, Prof.!