Optimization Theory & Applications – Rishabh Kumar

November 22, 2021

Feedback by Rishabh Kumar, 5th year UG, Mathematics and Computing

“Theory of optimization plays an important role in Engineering management and mathematics and is closely related to several other fields in decision science. The objective of this micro specialization framework is to provide a solid foundation of various optimization techniques and their applications.”

Source: Brief Description of the micro specialization curricula

Q.1. The general structure of all micro specialization curricula is composed of 3-components where the 1st one is the mandatory Foundation Course which is supposed to provide a firm base for the advanced courses. In your case, it was the Operations Research theory and lab. What do you have to say about this course regarding and additionally how important was it to catch up with the advanced courses you took later on?

Response: These courses are mandatory courses in the MnC curriculum and are also the prerequisites to all other courses of this micro-specialization. The formalism of the entire course is based on Linear Programming. The biggest takeaway is Duality Theory which is a recurring theme in all the other courses in rest of optimization theory. Although duality theory is formally developed over Linear Algebra(which is also extensively used along with Multivariable Calculus in Nonlinear Programming) and we had a formal course in our 5th sem, but still students can build-up a threshold understanding based on whatever they have studied in Mathematics-2 course, even if they didn’t have any other formal course on LA. Also, students advancing in AI have to eventually learn LA and sort of the same concepts will be useful in optimization theory.

Recently a subject on application of Linear Algebra for AI(AI61003) has been introduced, which would potentially clear up the basics needed for learning optimization as well.

Q.2 In the 2nd component you had to complete any 1 subject from the 2 options you had. Which subject did you choose and what was the thought procedure of choosing it? Also, tell us a bit about each of the subjects.

Response: In the semester I planned to complete this component, the other option Optimization Through Vector Spaces was not offered.

In Nonlinear Programming, I was mostly interested in learning the techniques used in convex optimization, which covers most of the course, because it is useful in many aspects like ML, statistics, finance and other engineering disciplines too like signal processing and control theory. Specifically, if we talk about ML then most regression problems, with or without constraints, could be modelled as a convex optimization problem. Also most of the optimizers that are commonly used in MATLAB, Python etc have the building blocks as convex optimizers.

Since most of the convex optimization problems could be solved with utmost relative ease using code rather than solving manually, we were also given coding assignments for such problems in this course, covering a good part of the spectrum.

Q.3 Most other micros have project/term paper as component-3 or a laboratory/advanced theory course as an alternative option in some cases. You had Optimization Methods in Finance, Multi-Objective Programming, Numerical Optimization where supposedly each of them focuses on separate broad areas of optimization theory. But if we think from the application perspective then any project may predominantly require more than one of those methods. Like many engineering problems do require constrained multi-objective optimization to be solved numerically. Same goes for finance. So how does a person choose just one of them and tell us about your choice?

Also, do you feel a project would have been a better Component-3 for this micro and these subjects could have been accommodated in component-2?

Response: I will be choosing Optimization Methods in Finance this semester mostly because I am interested in learning to model optimization problems from other diverse domains like finance. I could have chosen Multi-Objective Programming in the previous sem but prioritized Computational Linear Algebra over it.

And regarding the confusion that both theory and numerical techniques are essential, then how to choose only 1 from Multi-Objective Programming and Numerical Optimization; it is a misleading impression. The course Multi-Objective Programming is self-complete and the required numerical methods will be taught there because without them a person learns only to model problems but not how to actually solve them, since most of them won’t have analytical solutions and have to be solved computationally anyhow.

Regarding the possibility that component-3 could be filled with projects/term paper rather than more math courses this is not feasible due to the following reasons:

1. The advanced techniques taught in component-3 are actually essential from the real-world applications’ perspective and can’t be overlooked. Even if the courses were accommodated in component-2, then there was a possibility of not taking any of the 3 courses which won’t be good.

2. All the three courses are theoretically quite advanced and the techniques would be used somehow in any project on optimization theory that is done. It requires proper training through a formal course to get acquainted with them, otherwise, it would take a lot of time and effort to be self-learned while doing a project parallelly.

Q.4 Clearly the micro-specialization does promise to deliver what it is meant for. There is a possibility that any student, even if majoring in MnC, may have been underexposed to these very crucial techniques taught in this micro-curriculum. Personally, how much and how exactly will the courses be impactful in whatever field you’ve aimed to expertise in or in any other engineering or management discipline?

Response: I believe that the basics of optimization theory like Operations Research and it’s related computational methods should be taught to students from all the engineering disciplines. With the increasing complexity of any system to be engineered either from design or control perspective, optimization theory becomes more and more crucial and unavoidable. Also in the management domains, there are many fields like supply chain management, route optimization techniques etc where these techniques are extensively used. Specifically, in finance, portfolio optimization problem is inherently a convex optimization problem.

Talking about the field I aim to expertise in, i.e Probability Theory and Dynamics, there is a very advanced subfield known as Theory of Optimal Transport which has given many Fields Medals in this century. In the big picture, if probability theory lets you mathematically model a very abstract physical scenario, then at the same time optimization theory lets you compute initial conditions that would maximise or minimize the probability of certain outcomes. For example, my current project with the University of Ottawa is, in simple terms, based on finding conditions when two coupled stochastic processes can meet “optimally”, so here optimization theory is extensively deployed in maximizing the probability of the same.