Optimization Interactive

Convex Optimization

When the problem is convex, local optima are global optima. Efficient algorithms guarantee finding the best solution.

📊 What is Convex?

Convex Set

Line between any two points in the set stays inside the set.

✓ Convex

✗ Non-convex

Convex Function

"Bowl-shaped" — line between any two points lies above the curve.

f(λx + (1-λ)y) ≤ λf(x) + (1-λ)f(y)

Gradient Descent

-8 8

0.01 0.5

10 100

📊 Optimization Result

Optimum x* -1

Found x -0.9999

f(x) 0.0000

Error |x - x*| 1.28e-4

✓ Converged to optimum

Gradient Descent Path

Start

Path

End

f(x) = x² + 2x + 1

Why Convexity Matters

Global Minimum

Local minimum = global minimum

→ No "traps"

Guaranteed Convergence

Gradient descent converges

→ Smooth landscape

Efficient Algorithms

Polynomial time solutions

→ Well-studied theory

Duality

Strong duality often holds

→ Bounds on optimum

🎰 Betting Applications

Portfolio Optimization

Obj: Minimize variance

s.t. Target return

Liability Hedging

Obj: Minimize cost

s.t. Risk limits

Odds Calibration

Obj: Minimize Brier score

s.t. Probs sum to 1

Resource Allocation

Obj: Maximize profit

s.t. Budget limits

Python Code

# Convex optimization with CVXPY
import cvxpy as cp
import numpy as np

# Portfolio optimization (minimize variance)
n_assets = 5
expected_returns = np.array([0.12, 0.10, 0.08, 0.15, 0.11])
cov_matrix = np.random.randn(n_assets, n_assets)
cov_matrix = cov_matrix @ cov_matrix.T  # Make positive semidefinite

# Decision variables
weights = cp.Variable(n_assets)

# Objective: minimize portfolio variance
objective = cp.Minimize(cp.quad_form(weights, cov_matrix))

# Constraints
constraints = [
    cp.sum(weights) == 1,          # Weights sum to 1
    weights >= 0,                   # No short selling
    weights @ expected_returns >= 0.10  # Target return
]

# Solve
problem = cp.Problem(objective, constraints)
problem.solve()

print(f"Optimal weights: {weights.value}")
print(f"Portfolio variance: {problem.value}")

✅ Key Takeaways

• Convex = no local minima traps
• Gradient descent guaranteed to converge
• Many betting problems are convex

• Use CVXPY (Python) or cvx (MATLAB)
• Check if problem is convex before solving
• Non-convex: use multiple restarts