Introducing constrained optimization through two simple examples

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2021-06-05 19:00:18

Optimization is all around us. Various physical entities are constantly solving some form of optimization problem: water running downhill (finding state of least energy), light rays trying traveling between two points (finding route of shortest time), or electric currents finding paths of least resistance. Optimization is also used extensively in industry. In explicit forms, it is used to maximize or minimize quantities of interest (portfolio returns, supply chains). In implicit forms, optimization algorithms are the backbone of machine learning and curve-fitting algorithms.

One great way to build intuition for optimization algorithms is to pick a simple problem and solve it numerically using the algorithm of choice. A ‘simple problem’ can be any problem whose solution is known or can be readily guessed. Since we don’t have to think about the answer, we can use such problems to understand how the algorithm arrives at the answer, building valuable intuition in the process.

Given their omnipresence, there is a large pool of simple optimization problems for us to choose for our exploration. We want a problem whose answer we can guess but which also includes all essential elements of a real optimization problem. One such simple-but-rich problem for exploring constrained optimization is the isoperimetric problem:

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