Solving Mean-Variance Portfolios with Weight Bounds

Summary

The document addresses a Markowitz portfolio optimization problem with lower and upper bounds on each asset weight. It explains that ordinary componentwise upper and lower limits are linear constraints, so they can be included through Kuhn–Tucker conditions or handled by numerical solvers designed for linear constraints. The covariance matrix and expected-return vector remain the usual ingredients of the mean-variance setup.

A distinct modeling choice arises when an asset must either receive zero weight or, if selected, receive at least a positive minimum allocation. That on-or-off rule cannot be represented by simple continuous bounds alone; it introduces discrete selection variables and turns the problem into a mixed-integer optimization. The answer notes that this class is substantially harder and mentions specialized solvers. It does not derive the optimizer step by step, provide a numerical example, or discuss how to estimate inputs, so the main lesson is the difference between continuous bounds and minimum-position rules.

Key ideas

• Ordinary lower and upper limits on portfolio weights are linear constraints.
• Kuhn–Tucker conditions or numerical optimization solvers can handle these continuous bounds.
• A rule requiring either zero weight or a minimum positive allocation adds a discrete choice.
• The zero-or-minimum allocation formulation becomes a mixed-integer optimization problem.
• Mixed-integer portfolio problems are harder to solve than continuous constrained Markowitz problems.

Tags

# Solving a Markowitz problem with restrictions (lower and upper bound) to the weights vector


# Solving a Markowitz problem with restrictions (lower and upper bound) to the weights vector












I would like to find a step by step solutionfor the following Markowitx problem. It is a standard markowitz problem. The unique detail (wich is why I am posting this question here) is that there is a upper bound and a lowerbound for the weights vector.

The problem I want to solve:

The detail:

Definitions:

$w$ is the weight vector, $\Sigma$ is the covariance matrix, and $\mu$ is the returns vector.

## Answer by Richi Wa (score 2, accepted)

https://quant.stackexchange.com/a/33167

The constraints $$ w \le b_u $$ and $$ b_l \le w \Leftrightarrow-w \ge - b_l $$ can all be handled using the Kuhn–Tucker conditions. Numerical solvers exist for these linear constraints too (e.g. this is in R). See als this.

However, with the lower bound you often want the optomizer to choose some assets to be zero and if greater zero then greater than some lower bound. In this case you get a mixed-integer problem which are much harder to handle (I have sucessfully applied LINDO to this class of problems).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.