Discrete Optimization is something all of us use in our daily activities when say, we order at a restaurant, decide which subject to study, take up a new activity… or look for a change.

It comprises of choosing between alternatives that best suit some objective we have in mind. When such things are formalized, i.e. the objective and the ability of each choice to fulfill that objective are quantified, we get a mathematical expression of the problem we would optimize.

The classical or statistical method of enumerating all solutions and choosing the best out of them is the ideal way of solving any optimization problem, and will always lead to the global optimal solution— however complex be the discrete optimization (or Combinatorics) problem.

But such a brute force solution is only feasible for some smaller problems involving a handful of features. As soon as the dimension of the problem starts growing, brute force fails, sheerly from time considerations. We then have to think of better ways to solve… and come across methods or heuristics such as a greedy algorithm, which chooses the most beneficial solution step at each iteration. Such a procedure gives an acceptable solution fast enough, but not always able to find the shortest route (our original objective). This results in a compromise or trade-off between accuracy and speed, without which most practical problems would never be solved.

The major treatise of optimization is considered equivalent to finding the shortest route through a series of cities. This comprises the generic Travelling Salesman Problem (TSP), generic in the sense that most discrete optimization problems can be reduced to the TSP very easily. Different algorithms can be employed to solve this problem. The solution methods in this discrete optimization course are practically illustrated with different instances of the TSP (and a knapsack problem) as examples.

Nature-inspired metaheuristics give us some excellent ways to solve a discrete optimization problem in an elegant way. Ant Colony Optimization (ACO) is one such algorithm proposed by Marco Dorigo in the 1990’s, and is considered a state-of-the-art method to solve the TSP.

The course progressively relates live real-world experiences to optimization problems and casts them in the language of mathematics. The methods to solve the TSP is introduced lucidly, and with care. Three example problems of increasing difficulty are solved through different methods introduced in the course, and their individual results compared.