Constrained Mean-Variance Portfolio Optimization

Worked Solution

How to Think About It: This is the bread-and-butter optimization problem in quantitative portfolio management. The unconstrained Markowitz problem has a clean closed-form solution via Lagrange multipliers, but the moment you add long-only constraints ($w_i \geq 0$), the closed form goes away and you need numerical methods. The real-world twist is part (iii): estimated $\mu$ values are notoriously noisy, and optimizers love to load up on whichever assets have the largest estimation errors. This is why naive mean-variance optimization produces garbage portfolios in practice.

Key Insight: The QP formulation is standard, but the practical value is understanding why the optimizer "breaks" with noisy inputs and how regularization fixes it.

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Part (i): QP Formulation and KKT Conditions

The quadratic program is:

$\min_w \frac{1}{2} w^T \Sigma w$

subject to: - $\mathbf{1}^T w = 1$ (budget constraint) - $\mu^T w \geq r_0$ (return constraint) - $w_i \geq 0$ for $i = 1, \ldots, d$ (long-only)

The Lagrangian is:

$\mathcal{L}(w, \lambda, \gamma, \nu) = \frac{1}{2} w^T \Sigma w - \lambda (\mathbf{1}^T w - 1) - \gamma (\mu^T w - r_0) - \nu^T w$

where $\lambda \in \mathbb{R}$ is the multiplier for the equality constraint, $\gamma \geq 0$ for the return constraint, and $\nu \in \mathbb{R}^d$ with $\nu_i \geq 0$ for the long-only constraints.

The KKT conditions are:

1. Stationarity: $\Sigma w - \lambda \mathbf{1} - \gamma \mu - \nu = 0$
2. Primal feasibility: $\mathbf{1}^T w = 1$, $\mu^T w \geq r_0$, $w_i \geq 0$
3. Dual feasibility: $\gamma \geq 0$, $\nu_i \geq 0$
4. Complementary slackness: $\gamma (\mu^T w - r_0) = 0$ and $\nu_i w_i = 0$ for all $i$

Since $\Sigma$ is positive semidefinite, the problem is convex. If $\Sigma$ is positive definite and a feasible point exists, the KKT conditions are necessary and sufficient for a global optimum.

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Part (ii): Numerical Method

*Interior-point method:*

An interior-point (barrier) method handles the inequality constraints by adding logarithmic barrier terms. Replace the long-only constraints and return constraint with barrier functions:

$\min_w \frac{1}{2} w^T \Sigma w - t^{-1} \left( \sum_{i=1}^d \ln w_i + \ln(\mu^T w - r_0) \right)$

subject to $\mathbf{1}^T w = 1$, where $t > 0$ is a barrier parameter that increases over iterations.

Each iteration requires solving a linear system of the form $(\Sigma + D)\Delta w = b$, where $D$ is a diagonal-plus-rank-one correction from the barrier Hessian. This involves:

• Forming the Hessian: $O(d^2)$
• Solving the $d \times d$ linear system: $O(d^3)$ in general (via Cholesky factorization)

Per-iteration complexity: $O(d^3)$.

With an iterative solver (conjugate gradient) and sparse structure, this can be reduced to roughly $O(d^2)$ per iteration, but $O(d^3)$ is the standard dense case.

*Projected gradient alternative:* Each iteration computes a gradient step $w \leftarrow w - \eta \Sigma w$ (cost $O(d^2)$ for the matrix-vector product), then projects onto the constraint set $\{w : \mathbf{1}^T w = 1, w \geq 0, \mu^T w \geq r_0\}$. The projection onto this simplex-like set costs $O(d \log d)$ via sorting. So projected gradient runs at $O(d^2)$ per iteration but converges more slowly than interior-point.

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Part (iii): Estimation Error and Robust Alternative

The expected return vector $\mu$ is estimated from historical data, typically as a sample mean $\hat{\mu}$. The estimation error in $\hat{\mu}$ is on the order of $\sigma / \sqrt{T}$ per asset, where $T$ is the sample size. This is large -- for daily returns with $T = 250$ and $\sigma \approx 0.20$ annually, the standard error of each mean return is about $0.20/\sqrt{250} \approx 1.3\%$, which is comparable to the mean itself.

The optimizer treats $\hat{\mu}$ as exact and tilts the portfolio aggressively toward assets whose returns were overestimated by chance. The constraint $\hat{\mu}^T w \geq r_0$ effectively becomes a bet on estimation noise rather than true expected returns. The result is a concentrated, unstable portfolio that performs poorly out of sample.

Robust alternative -- ridge penalty on weights:

$\min_w \frac{1}{2} w^T \Sigma w + \frac{\delta}{2} \|w\|_2^2$

subject to $\mathbf{1}^T w = 1$, $\mu^T w \geq r_0$, $w_i \geq 0$.

This is equivalent to replacing $\Sigma$ with $\Sigma + \delta I$, which shrinks the portfolio toward equal weights. The parameter $\delta > 0$ controls the strength of regularization. This has two effects: - It penalizes concentration, so the optimizer cannot load up on a single asset even if its estimated $\hat{\mu}$ is large. - It makes the effective covariance matrix better conditioned, reducing sensitivity to estimation error in both $\Sigma$ and $\mu$.

Shrinkage alternative: Replace $\hat{\mu}$ with a shrinkage estimator $\tilde{\mu} = (1 - \alpha) \hat{\mu} + \alpha \bar{\mu} \mathbf{1}$, where $\bar{\mu}$ is the grand mean of all asset returns and $\alpha \in [0, 1]$ controls shrinkage intensity. The modified problem uses $\tilde{\mu}$ in the return constraint: $\tilde{\mu}^T w \geq r_0$. This pulls extreme return estimates toward the cross-sectional average, reducing the influence of estimation noise.

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Answer: The long-only mean-variance problem is a convex QP with $d$ nonnegativity constraints, one equality constraint, and one inequality constraint. The KKT conditions require stationarity ($\Sigma w = \lambda \mathbf{1} + \gamma \mu + \nu$), feasibility, and complementary slackness on the active constraints. An interior-point method solves it in $O(d^3)$ per iteration. To handle estimation error in $\mu$, add a ridge penalty $\frac{\delta}{2}\|w\|_2^2$ (equivalently, replace $\Sigma$ with $\Sigma + \delta I$) or shrink $\hat{\mu}$ toward a common mean.