L1 Norm Equality Constraints in Portfolio Optimization

Summary

The document asks whether a portfolio optimization problem can use an equality constraint on the sum of absolute position weights instead of the usual equality constraint on their signed sum. The distinction is that the L1 constraint fixes total absolute exposure, while the signed-sum constraint fixes net exposure. The note’s brief summary characterizes the L1 equality as conservative and says it backs up short positions; it also describes it as encompassing long-only and short-only equal-weight cases.

The stated tradeoff is that this constraint prevents exact-form solutions. No derivation, optimization example, or comparison of outcomes is provided, so the claimed implications are not demonstrated in detail. In practice, interpretation depends on whether weights may be long or short and on the rest of the objective and constraints. The note raises a useful modeling choice about gross versus net exposure, but leaves questions about feasibility, solution methods, and when the equality is appropriate unanswered.

Key ideas

• A signed-sum equality constrains net portfolio exposure, whereas an L1 equality constrains total absolute exposure.
• An L1 equality includes absolute short positions in the exposure budget.
• The note characterizes this constraint as conservative but offers no worked example.
• It states that exact-form solutions are prevented, without detailing the resulting optimization method.

Tags

# L1 norm equality constraints in portfolio optimization, pros and cons


# L1 norm equality constraints in portfolio optimization, pros and cons












Can I use L1 norm equality constraints $\sum_{i}{|w_i|}=c,~c>0$ (or $\|\mathbf{w}\|_1=c$) in portfolio optimization, instead of the $\sum_{i}{w_i}=c$ (or $\mathbf{1}^T\mathbf{w}=c$) constraint? Any implications, pros, and cons?

Thanks for the comments below. I try to summarize what we have for now:

> L1 norm equality constraint is conservative and backups all short positions. It generalizes both equal-weight long and equal-weight short. However, it prevents exact-form solutions.

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