DEX Arbitrage, Mathematical Optimisations & Me

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In the world of decentralized finance (DeFi), arbitrage is one of the most prevalent and profitable forms of Miner Extractable Value (MEV). If you're interested in maximizing returns from on-chain trading opportunities, understanding how to efficiently identify and execute DEX arbitrage using mathematical optimization is essential.

At its core, arbitrage involves spotting price discrepancies for the same token across different decentralized exchanges (DEXs), executing a sequence of trades, and profiting while helping align prices across markets. While this concept seems straightforward, identifying the optimal arbitrage path across multiple pools and tokens—especially under tight block time constraints—is a complex computational challenge.

Fortunately, research has shown that arbitrage involving Constant Function Market Makers (CFMMs)—such as Uniswap, SushiSwap, and Balancer—can be modeled as a convex optimization problem. This means we can solve it efficiently to find the globally optimal trade set: the combination of swaps that yields the highest possible profit.

Let’s explore how this works, from foundational math to practical implementation.

Understanding Mathematical Optimization

Before diving into convex optimization, it's important to understand the broader concept of mathematical optimization:

Mathematical optimization is the selection of the best element, with respect to some criterion, from a set of available alternatives.

In practice, this often involves solving min/max problems, where you aim to either minimize or maximize an objective function subject to certain constraints.

Objective Functions and Constraints

Constraints can be:

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For example, imagine buying fruit with a fixed budget. You want to maximize quantity, but you’re constrained by prices and limits like "no more than 5 bananas" or "must buy exactly one orange." This is a classic optimization scenario.

Now consider this more advanced example:

def objective_function(x, y):
    return x**2 * y

# Subject to constraint: x² + y² = 1

Here, you’re trying to maximize f(x, y) = x² * y, but only for values of x and y that lie on the unit circle (x² + y² = 1). This can be visualized in 3D: walk around the circle on the ground and measure how high the function value rises above each point. The goal? Find the highest point.

This framing—maximizing an objective under constraints—is exactly how we model arbitrage.

Why Convex Optimization Matters for DEX Arbitrage

A problem qualifies as convex optimization if:

  1. The objective function is convex.
  2. The inequality constraints are convex.
  3. The equality constraints are linear (which can be converted into two convex inequalities).

What Makes a Function Convex?

A real-valued function is convex if the line segment between any two points on its graph lies above or on the graph.

Visually, convex functions curve upward—like a bowl. Non-convex functions have dips or curves that make global optimization difficult.

CFMMs like Uniswap use invariant functions such as x * y = k, which are convex when plotted. This geometric property allows us to apply powerful convex solvers to find optimal arbitrage paths quickly and reliably.

Modeling Arbitrage as a Convex Problem

We now shift from theory to structure: how do we frame DEX arbitrage as a convex optimization task?

Key Components

1. Tokens and Pools

For instance:

This structure enables us to map local token indices within pools to global token identities.

2. Trade Variables: Δ (Delta) and Λ (Lambda)

Each trade involves:

These values must satisfy the pool’s trading function, adjusted for fees (γ = 1 - fee). For example, Uniswap V2 requires:

(x + γΔ)(y - Λ) ≥ xy

This ensures protocol reserves grow after each trade.

3. Net Network Trade (Ψ)

After all trades, we calculate the net gain/loss per token across the entire system:

Ψ = Σ(Aᵢ(Λᵢ - Δᵢ))

Where Aᵢ is a matrix mapping local pool indices to global token indices.

This net result tells us our final position: which tokens we gained or lost.

Building the Optimization Model in Code

Using Python and the cvxpy library, we can implement this model programmatically.

Step-by-Step Implementation

  1. Define Pool Data

    • Reserves: current token balances in each pool.
    • Fees: γ values (e.g., 0.997 for 0.3% fee).
    • Market values: external USD prices for normalization.

  2. Set Up Variables

    deltas = [cp.Variable(len(l), nonneg=True) for l in local_indices]
lambdas = [cp.Variable(len(l), nonneg=True) for l in local_indices]

  3. Compute Net Trade (Ψ)

    psi = cp.sum([A_i @ (L - D) for A_i, D, L in zip(A, deltas, lambdas)])

  4. Objective Function
    Maximize total USD value of net gains:

    obj = cp.Maximize(market_value @ psi)

  5. Apply Constraints

    • Trading function constraints for each pool type:

      • Balancer: geometric mean with weights.
      • Uniswap V2: geometric mean.
      • Constant Sum: sum-based invariant.
    • Arbitrage constraint: psi >= 0 — no net outflow of any token.

  6. Solve

    prob = cp.Problem(obj, cons)
prob.solve()

The solver returns the optimal Δ and Λ values—the exact trades to execute.

Determining Execution Order

Once we know which trades to make, we need to determine in what order to execute them. Since arbitrage typically starts with initial capital (or a flash loan), our goal is to minimize the upfront capital required.

With 5 trades, there are 5! = 120 possible execution orders. We use brute-force search over permutations to find the sequence that minimizes required starting funds.

For each permutation:

The optimal order reduces risk and capital efficiency barriers.

FAQ: Common Questions About DEX Arbitrage Optimization

Q: What are CFMMs?

A: Constant Function Market Makers (CFMMs) are automated market makers that use mathematical formulas (like x * y = k) to determine asset prices based on reserve ratios. Examples include Uniswap and Balancer.

Q: Why is convex optimization ideal for arbitrage?

A: Because CFMM trading functions are convex, the overall arbitrage problem becomes convex—ensuring fast convergence to the global optimum without getting stuck in local maxima.

Q: Can this method work across multiple blockchains?

A: In principle, yes—but latency and cross-chain communication introduce complexity. Most implementations focus on single-chain arbitrage due to speed requirements.

Q: How are real-time prices incorporated?

A: External market data (e.g., from centralized exchanges) provides reference prices used in the objective function to normalize token values in USD terms.

Q: Is gas cost considered in this model?

A: Not in this basic version—but advanced models integrate gas as a cost term in the objective function to ensure profitability after transaction fees.

Q: Can I customize the objective to target specific tokens?

A: Absolutely. By setting market values of unwanted tokens to zero, the optimizer will only maximize gains in your desired assets.

Final Output and Profitability

Running the full script yields:

For example:

USD VALUE REQUIRED = $10.50
SUM OF RECEIVED TOKENS USD VALUE = $15.80
PROFIT = $5.30

Even small inefficiencies add up at scale—especially when automated bots scan hundreds of pools per second.

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Conclusion

DEX arbitrage isn’t just about spotting price differences—it’s about solving a high-dimensional optimization problem under real-time constraints. By modeling arbitrage as a convex optimization problem, we unlock powerful tools that guarantee optimal results with computational efficiency.

Whether you're building MEV bots or designing DeFi protocols, understanding this intersection of finance and mathematics gives you a critical edge.

And once you've mastered arbitrage logic, you can expand into more advanced routing problems—like multi-hop trades or portfolio rebalancing—all within the same mathematical framework.

👉 Start applying these models with powerful trading tools today.