Portfolio weighting optimization is a comprehensive and complicated topic. However, this discussion will primarily concentrate on the implementation of constraints such as maximum and minimum weights, group weighting constraints, and others.

A basic weight capping can be achieved by redistributing the excess weight from larger components proportionally across the rest of the portfolio. This straightforward heuristic method effectively meets the goal while preserving the original weight of each constituent to the greatest extent possible.

Nevertheless, when multiple constraints are required—such as a maximum weight less than or equal to a certain value, a minimum weight greater than or equal to a certain value, and the total weight of the top five securities not exceeding 65%—the “proportional distribution” approach becomes challenging if not impossible to implement. But this problem can be reframed as the minimization of a quadratic function problem.

We can define a vector composed of each constituent’s weight and then calculate the Euclidean distance between this “optimal” vector and the “original” vector, aiming to minimize this distance. Simultaneously, we need to ensure that the total weight equals 100%, the maximum weight is less than or equal to a certain value, the minimum weight is greater than or equal to a certain value, and the combined weight of the top five securities does not exceed 65%.

It’s improbable to solve such a complex problem analytically, but it can be handled through a numerical optimization algorithm like the Sequential Least Squares Programming (SLSQP) method. SLSQP method is essentially an iterative search algorithm. At each step, it computes the gradient of the objective function and uses this information to take a small step towards the optimal solution. This process is repeated, with the algorithm continually making gradient-guided adjustments, until it converges to the solution, or until a specified stopping criterion is met.

Thankfully, the scipy.optimize module offers an algorithm that we can utilize for this purpose.