New insights and augmented Lagrangian algorithm for optimal portfolio liquidation with market impact

In financial markets, investors may be forced to unwind their portfolios to meet a determined leverage ratio within regulatory policy or risk management requirements. This paper studies the optimal portfolio liquidation problem with market impact. Some new insights are given into this problem, and trading orders are discussed for various financial parameters. Specifically, we establish the equivalence between equity and liability maximization. This means if one wants to maximize the equity, it is to maximize the liability and vice versa. The computational complexity of the problem is examined to be NP-hard. We expose the hidden convexity through monotonicity analysis and linearization techniques. Although good properties are established for the Lagrangian algorithm, a counter-example is constructed to show one deficiency of this algorithm. Therefore, we propose an augmented Lagrangian algorithm for solving the problem. The inverse Hessian of the augmented Lagrangian function is explicitly calculated for the subproblem solved by the projected Newton method. Meanwhile, we consider how to choose a good initial point, which is essential for seeking a high-quality solution. Some numerical results are presented, which validate the usefulness of the augmented Lagrangian algorithm.