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Linearize an L1 Trading Penalty

Point72Quantitative Researcher · Online Assessment
MediumQuantitative ResearcherOnline AssessmentStatistics & Math

Last updated

Linearize weighted absolute trading costs with auxiliary variables, complete LP constraints, the correct objective sign, and an equivalence proof.

Linearize an L1 Trading Penalty

Convert a portfolio problem into a standard linear program. The objective contains a linear expected-return term and a nonnegative weighted absolute-value penalty on trades from current holdings. The model also has linear equality constraints and box bounds on target holdings. Introduce auxiliary variables, write the complete transformed objective and constraints, and prove equivalence.

Constraints & Assumptions

  • h is the target vector, h0 is fixed, and c_i >= 0 is the cost per unit of |h_i - h0_i|.
  • Expected-profit coefficients are alpha.
  • Existing constraints are A h = b and lower <= h <= upper.
  • There is no quadratic risk term in this formulation.

Clarifying Questions to Ask

  • Is the original problem written as maximization or minimization?
  • Are buy and sell costs symmetric?
  • Do any fixed fees, minimum lots, or integer holdings apply?
  • Are holding and trade units consistent with the coefficients?

Hints

Hint 1

Bound both signs of each trade: An auxiliary magnitude must be at least the trade and at least its negative; a positive objective cost then drives it to equality.

Follow-up questions

  • How would different buy and sell costs change the variables?
  • Why does a fixed fee require integer decisions?
  • What does adding a quadratic covariance penalty do to the problem class?