README.md

Cardinality-Constrained Portfolio Optimization

This project explores portfolio optimization under cardinality constraints, where the number of assets selected in a portfolio is explicitly limited. Using integer programming, the project demonstrates how adding a cardinality constraint transforms a convex Markowitz optimization problem into a combinatorial, non-convex challenge—motivating the need for advanced solvers and quantum-inspired approaches.

---

🧠 What It Does

Given:

• A covariance matrix of asset returns $$\Sigma$$
• A vector of expected returns $$\mu$$
• A target expected return $$r^*$$
• A cardinality limit $$K$$ (max number of non-zero assets)

The optimizer solves:

$$ \min_w \quad w^T \Sigma w \quad \text{subject to:} $$

• $\sum w_i = 1$ (fully invested)
• $\mu^T w \geq r^*$ (target return)
• $w_i \leq z_i, \quad z_i \in {0, 1}$ (selection constraint)
• $\sum z_i \leq K$ (cardinality constraint)
• $w_i \geq 0$ (long-only)

---

🔍 Features

• ✅ Solves the cardinality-constrained minimum-variance problem
• ✅ Compares with the unconstrained Markowitz solution
• ✅ Uses mip for mixed-integer quadratic programming
• ✅ Visualizes how limiting asset count affects portfolio composition and performance
• ✅ Prepares groundwork for future QUBO or quantum annealing implementations

---

📊 Visualization

• Efficient frontier with vs. without cardinality constraint
• Portfolio weights vs. cardinality limit
• Risk-return tradeoff curves and asset sparsity plots

---

📦 Requirements

• Python 3.x
• NumPy
• Matplotlib
• mip (Mixed Integer Programming)

Install dependencies:

pip install numpy matplotlib mip