Modeling Absolute Position Bounds in Portfolio Optimization

Summary

The document considers a mean-variance portfolio optimization with short positions allowed, linear equality constraints, and a bound on the absolute size of each position. It asks whether an auxiliary variable and a large penalty are needed to express the absolute-value constraint in a quadratic program. The accepted response shows that the bound can be written directly as two linear inequalities for each weight: an upper bound on the weight and an upper bound on its negation. Together these enforce that each position lies between the negative and positive limits.

Because the objective is unchanged by this transformation, there is no need to introduce a variable intended to equal the absolute weight or to penalize it. The response also notes that portfolio weights may need a separate budget constraint so they sum to one. The discussion addresses componentwise position limits only; it does not cover aggregate gross exposure constraints, solver-specific formulations, or other portfolio feasibility requirements.

Key ideas

• An absolute position bound can be expressed as an upper bound on a weight and on its negation.
• The two inequalities directly constrain each position to lie within the permitted range.
• An auxiliary variable is unnecessary for this componentwise bound in the stated formulation.
• A portfolio may also require a separate constraint that weights sum to one.
• The response does not address aggregate exposure limits or broader portfolio constraints.

Tags

# optimization with absolute constraints


# optimization with absolute constraints












Suppose I have an optimization where I need to impose ADV-like constraint (for a case where Shorting is allowed):

$\max \mu'w - \lambda w'\Sigma w$

$ |w| \le V $

$ Aw = 0$

and I want to use a Quadratic Programming formulation. I read somewhere that I can replace $|w| = z$ by two inequalities. Which one of the two is valid:

Case 1

$\max \mu'w - \lambda w'\Sigma w - M z$

$ z \le V $

$ w \le z$

$ -w \le z$

$ Aw = 0$

where $M$ is a very large constraint, which I think will force $|w| = z$

Case 2

$\max \mu'w - \lambda w'\Sigma w$

$ z \le V $

$ w \le z$

$ -w \le z$

$ Aw = 0$

Case 2 above is what I saw a few places on the net, but it got me thinking that this constraint is equivalent to $|w| \le z$ and I need to find another way to force the equality.

Is either Case 1 or Case 2 or both a correct way to handle the $|w| \le V$ constraint?

## Answer by jaamor (score 3, accepted)

https://quant.stackexchange.com/a/19017

Why not just do:

$$ max \,\, \mu ^T w - \lambda w^T \Sigma w $$ s.t.: $$ w \leq V $$ $$ -w \leq V $$ $$ A w = 0 $$

Google for LP absolute value constraint transformations. Here is a helpful online tutorial.

And if these are portfolio weights, don't forget that they should add up to 1.

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