Optimization Interactive
Quadratic Programming
Optimize with quadratic objectives and linear constraints. Powers portfolio optimization, hedging, and risk management.
๐ QP Standard Form
Objective
min ยฝx'Qx + c'x
Minimize risk + cost
Constraints
Ax โค b
Inequality constraints
Equality
Ex = f
Equality constraints
Preferences
4 12
0.5 5
๐ Portfolio Metrics
Expected Return 9.0%
Portfolio Risk (ฯ) 14.4%
Sharpe Ratio 0.49
Optimal Allocation
Large Cap
16.8%
Small Cap
30.4%
Bonds
21.0%
International
31.8%
Return: 8%
Risk: 15%
Return: 12%
Risk: 25%
Return: 4%
Risk: 5%
Return: 10%
Risk: 20%
Efficient Frontier
Risk (ฯ) โ
Return โ
Individual assets Optimal portfolio Efficient frontier
๐ฐ Betting Applications
Portfolio Allocation
Optimal bet sizing across markets
Hedge Optimization
Minimize cost to offset liability
Parlay Construction
Maximize EV subject to correlation
Risk Parity
Equal risk contribution per bet
Python / CVXPY
# Mean-variance portfolio optimization
import cvxpy as cp
import numpy as np
# Data
returns = np.array([0.08, 0.12, 0.04, 0.10])
cov_matrix = np.array([...]) # 4x4 covariance matrix
# Decision variable
weights = cp.Variable(4)
# Mean-variance objective
expected_return = returns @ weights
portfolio_variance = cp.quad_form(weights, cov_matrix)
# Risk aversion parameter
lambda_risk = 2
# Objective: maximize return - lambda * variance
objective = cp.Maximize(expected_return - lambda_risk * portfolio_variance)
# Constraints
constraints = [
cp.sum(weights) == 1, # Fully invested
weights >= 0, # No short selling
expected_return >= 0.1 # Minimum return
]
# Solve
problem = cp.Problem(objective, constraints)
problem.solve()
print(f"Optimal weights: {weights.value}")
print(f"Expected return: {expected_return.value * 100:.1f}}%")โ Key Takeaways
- โข QP: quadratic objective, linear constraints
- โข Powers mean-variance optimization
- โข Convex QP โ global optimum guaranteed
- โข Use CVXPY or quadprog to solve
- โข Risk aversion ฮป controls risk-return tradeoff
- โข Foundation of portfolio theory