Optimal Routing across Constant Function Market Makers with Gas Fees

Battle Count: 77

Highly relevant for DeFi quantitative trading and algorithmic execution. The paper provides rigorous mathematical foundations for optimal trade routing across fragmented liquidity pools, explicitly modeling gas fees as fixed execution costs. The no-trade conditions and KKT systems directly inform when to execute or refrain from trading. The approximation bounds provide practical guidance for relaxation-based routing algorithms. However, it is primarily theoretical and does not address real-time execution, slippage modeling, or adversarial environments typical in production trading systems.

The mathematical framework is sophisticated, requiring knowledge of nonlinear optimization, KKT conditions, generalized convexity theory, and mixed-integer programming. The numerical implementation involves solving constrained optimization problems with equality constraints (CFMM invariants) and inequality constraints (activation bounds). The KKT system derivation is complex with multiple cases depending on activation status and trade saturation. Practical implementation would require careful handling of the relaxation and verification of constraint qualifications.

The paper provides detailed mathematical formulations, explicit KKT systems, and numerical experiments using SciPy. However, no code repository is mentioned. The numerical examples use specific parameter values and market functions (geometric mean, weighted quasi-arithmetic mean) that are fully specified. Reproduction would require implementing the optimization framework and solving the KKT systems numerically.