Formulating a Long-Only Mean-Variance Portfolio Optimization Problem

Summary

The document poses a constrained portfolio optimization problem: minimize variance while requiring portfolio weights to sum to one, achieving a specified target return, and keeping every weight nonnegative. This last condition imposes a long-only portfolio by ruling out short positions. The question focuses on whether the problem has an analytical solution and whether Karush–Kuhn–Tucker conditions can help when a textbook says a direct Lagrange multiplier solution is unavailable.

No solution, derivation, or numerical example is provided, so the document serves as a problem statement rather than a complete method. It highlights the distinction between unconstrained mean-variance optimization and optimization with inequality constraints, where the active set of zero-weight assets may matter. Readers would need additional material to establish when a closed-form answer exists or how to solve the constrained problem in practice.

Key ideas

• The objective is to minimize portfolio variance for a specified target return.
• Weights must sum to one, and nonnegative weights prohibit short selling.
• The question asks whether KKT conditions can support an analytical solution.
• No derivation or answer is included, so the document does not provide a usable solution method.

Tags

# Analytical solution to short-sale constrained portfolio


# Analytical solution to short-sale constrained portfolio












Say that we want to find the efficient mean-variance portfolio (i.e. minimize variance given that weights sum to 1 and given a set target return) and impose a short sale constraint such that $w_i \geq 0$ for each $i$.

My textbook states that Lagrange multipliers cannot be used to analytically solve this problem (which I understand), so my question is if there exists an analytical solution? (Perhaps using KKT-conditions?)

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.