DEX Arbitrage, Mathematical Optimizations & Me

If you're a scholar of MEV (Maximal Extractable Value), understanding how to maximize arbitrage profit is essential. Today, we'll explore how mathematical optimizations—specifically convex optimizations—can help solve the arbitrage problem efficiently in decentralized exchanges (DEXs) like Uniswap, Sushiswap, and Balancer.

Understanding Arbitrage in Decentralized Exchanges

Arbitrage is the largest form of MEV on-chain. The concept is simple:

• Identify price differences between exchanges for a given token.
• Execute trades to profit from these discrepancies.
• In the process, align token prices across exchanges.

While the idea is straightforward, finding the highest-yielding arbitrage across multiple exchanges and tokens is complex due to:

• Blockchain constraints (limited block time).
• Dynamic market conditions (changing mempool transactions, fluctuating centralized exchange prices).

Fortunately, research shows that arbitrage in Constant Function Market Makers (CFMMs) is a convex optimization problem—meaning we can efficiently compute the globally optimal arbitrage strategy.

Mathematical Optimization: A Primer

Mathematical optimization involves selecting the best solution from a set of alternatives based on a given criterion.

Key Components:

1. Objective Function: What we aim to maximize/minimize (e.g., arbitrage profit).
2. Constraints: Conditions that must be met (e.g., trading fees, liquidity limits).

Example:

Given a budget, maximize fruit purchases where:

• Objective: Buy the most fruits.

• Constraints:

  • No more than 5 bananas (b ≤ 5).
  • Must buy exactly one orange (o = 1).

A more complex optimization problem:

Maximize:

f(x, y) = x² * y  

Subject to:

x² + y² = 1  

This can be visualized as a 3D graph, where the optimal solution is the highest point along the constraint curve.

Convex Optimization in CFMM Arbitrage

A convex optimization problem must satisfy:

1. Convex objective function.
2. Convex inequality constraints.
3. Linear equality constraints (convertible to convex inequalities).

Why CFMM Arbitrage is Convex:

• Uniswap’s trading function (x * y = k) is convex.
• Balancer’s geometric mean function is convex.
• Constraints (trading fees, reserves) maintain convexity.

Implementing CFMM Arbitrage Optimization

Step 1: Define the Search Space

• Tokens: Labeled 0 to n-1 (e.g., TOKEN-0 = ETH, TOKEN-1 = DAI).
• CFMM Pools: Represented as local_indices (tokens per pool).

Step 2: Model Trades

• Delta (Δ): Tokens tendered to the pool.
• Lambda (Λ): Tokens received from the pool.
• Trading Function (φ): Ensures trade validity (e.g., Uniswap’s x * y ≥ k).

Step 3: Compute Net Network Trade

• Use matrix transformations (A_i) to convert local token indices to global indices.
• Psi (Ψ): Sum of all trades (Σ(Λ_i - Δ_i)) in global token terms.

Step 4: Formulate the Optimization Problem

• Objective: Maximize market_value @ Psi (total USD value of arbitrage).

• Constraints:

  • Trading functions for each CFMM (φ(R + γΔ - Λ) ≥ φ(R)).
  • Arbitrage constraint (Ψ ≥ 0).

Step 5: Solve & Execute Trades

• Use CVXPY (a convex solver) to find optimal trades.
• Brute-force trade ordering to minimize required capital.

👉 Learn more about convex optimization in DeFi

Results: Optimal Arbitrage Strategy

After solving, we obtain:

• Required Capital: Minimal tokens needed to kickstart arbitrage.
• Profit: Net tokens received (converted to USD value).

Example Output:

RECEIVED 1.175 TOKEN-1 = $11.75  
RECEIVED 0.018 TOKEN-2 = $0.036  
RECEIVED 3.25 TOKEN-3 = $9.75  
TOTAL PROFIT: $21.536  

FAQs

1. What is MEV in DeFi?

MEV (Maximal Extractable Value) refers to profits extracted by reordering, inserting, or censoring transactions in a block. Arbitrage is the most common form.

2. Why is convex optimization useful in arbitrage?

Convex optimization ensures global optimality, meaning we can efficiently find the most profitable arbitrage path across multiple pools.

3. How do trading fees impact arbitrage?

Fees (γ) reduce profits by taking a percentage of each trade. Solvers must account for them in constraints.

4. Can this method be applied to flash loans?

Yes! Flash loans allow zero upfront capital, making arbitrage more accessible.

5. What are the risks in DEX arbitrage?

• Slippage: Large trades impact prices.
• Front-running: Competitors may exploit your strategy.

👉 Explore advanced arbitrage strategies

Conclusion

By framing CFMM arbitrage as a convex optimization problem, we unlock an efficient, mathematically sound method to maximize profits. Whether you're a researcher, trader, or developer, mastering these techniques enhances your DeFi strategy toolkit.

Follow me on Twitter @noxx3xxon for more insights!

Further Reading: