# TLDR

We propose a convex optimization approach for determining the optimal liquidation order in cross-margined AMM-based exchanges, minimizing the negative impact on exchange solvency.

# Motivation

Liquidation is crucial in maintaining an exchange’s solvency. In cross-margined exchanges, determining the optimal combination of liquidation orders can be complex. Current permissionless liquidation mechanisms in a competitive setting may not be ideal for AMM-based protocols.

# Problem Formalization

We simplify the problem with these assumptions:- Liquidations in the context of a risk-indifferent AMM
- Liquidations are atomic operations executing at uniform prices within a block
- No liquidation penalties or fees

We define an optimal liquidation order as the minimizer of the total notional amount liquidated while satisfying the maintenance margin and the risk-indifference invariant. We represent the liquidation of a portfolio as a convex optimization problem:

See the full write-up here for more details on the notation.

In words, the problem solves for an optimal partial liquidation for each position such that the portfolio’s margin post-liquidation (RHS of the second constraint) is at least the maintenance margin determined by total notional size post-liquidation (LHS of the second constraint). Note that if the problem is infeasible, the portfolio will be fully liquidated.

# Simulations

We use entropic value-at-risk with a confidence level of 0.99 and assume the price vector follows a normal distribution. We use GO-GARCH to forecast the covariance matrix. See our simulation code here.

# Future Research

We can extend the framework to include liquidation penalties, although it may lead to more infeasible solutions and full liquidations.

# Questions

- What alternative objectives could better reflect the needs for specific protocols?
- Are there reward mechanisms that incentivize optimal liquidations in a decentralized setting (e.g. liquidation reward is maximized as the distance between the liquidation strategy and an optimal liquidation strategy is minimized)?