Mean-Variance Optimization with a Linear Portfolio Constraint

Summary

The document poses a mean-variance portfolio optimization problem with expected return as the reward and a quadratic penalty for risk. It adds a linear inequality requiring the portfolio’s exposure to a specified vector to meet a threshold. The covariance matrix is assumed symmetric and positive definite, and a zero-rate risk-free asset is assumed, so the formulation omits a budget constraint that would otherwise require weights to sum to one.

The question asks whether the optimal weight vector can be written in closed form using the threshold, expected returns, and covariance matrix, and presents a Lagrangian with a multiplier for the inequality. The document contains no answer, derivation, or numerical evidence, so it does not establish the solution or explain when the constraint binds. It serves as a setup for analyzing constrained mean-variance optimization rather than a complete optimization method.

Key ideas

• The objective balances expected portfolio return against a quadratic variance penalty.
• A linear inequality constrains portfolio exposure to a specified vector.
• The assumed positive definite covariance matrix supports a strictly concave objective when risk aversion is positive.
• The Lagrangian introduces a multiplier for the exposure constraint.
• The document asks about a closed-form solution but provides no derivation or answer.

Tags

# Closed form solution for Mean-Variance optimization under constraint


# Closed form solution for Mean-Variance optimization under constraint












Is there a closed form solution for the vector weight $w$ for the following mean-variance optimization problem?

$\max_w w'\mu - \frac{\gamma}{2}w'\Sigma w $

s.t.

$w'z\geq \bar{z}$

where $w, z$ are N x 1 vectors, $\bar{z}$ is a constant, $\Sigma$ is a variance-covariance matrix, that is symmetric and positive definite. Note I assumed the existence of risk-free asset with $r_f=0$, that's why I don't have the usual $w'\boldsymbol{1}=1$ constraint.

If I write the following Lagrangian:

$L(w,\lambda) = w'\mu - \frac{\gamma}{2}w'\Sigma w +\lambda (w'z -\bar{z}) $

Is there a way to have a closed-form solution for $w$ (i.e. as a function of only $\bar{z},\mu,\Sigma$)?

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