Linearize an L1 Trading Penalty

# 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? ### What a Strong Answer Covers - One nonnegative auxiliary variable per absolute-value term - The pair of linear inequalities enforcing an upper bound on magnitude - Correct objective sign and proof that nonnegative cost makes the bound tight - Split-variable alternatives, asymmetric costs, and complexity ### Follow-up Questions 1. How would different buy and sell costs change the variables? 2. Why does a fixed fee require integer decisions? 3. What does adding a quadratic covariance penalty do to the problem class? ```hint 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. ```