Long-Only Global Minimum Variance Portfolio

• Long-only global minimum variance portfolio is a fully invested, no-short-sales quadratic program that minimizes portfolio variance based solely on asset return covariances.
• The method relies on active set selection and factor-model geometry, where assets with lower beta exposures are chosen to form the optimal simplex-constrained allocation.
• Robust covariance estimation techniques, including sparse, latent variable, and decision-focused methods, are pivotal in enhancing the stability and performance of these portfolios.

The long-only global minimum variance portfolio is the fully invested, no-short-sales solution to variance minimization. In its standard form, it solves

$$\min_{w \in \mathbb{R}^p} \; w^\top \Sigma w \quad \text{s.t.} \quad \mathbf{1}^\top w = 1,\; w_i \ge 0,\; i=1,\dots,p,$$

so its defining input is the covariance matrix of asset returns rather than expected returns. Unlike the unconstrained GMV portfolio, which admits the closed-form solution $$\Sigma^{-1}\mathbf{1}/(\mathbf{1}^\top \Sigma^{-1}\mathbf{1})$$, the long-only problem is a convex quadratic program whose behavior depends jointly on covariance estimation, conditioning, and the geometry of the feasible simplex (Oya, 2021, Fonseca, 25 Jun 2026).

1. Definition and optimization structure

A particularly explicit long-only formulation appears in work that defines the feasible set as

$$X = \left\{ \bm{x} = (x_1,\dots,x_p)^\top : \sum_{i=1}^p x_i = 1,\; x_i \ge 0,\; i=1,\dots,p \right\},$$

and then solves

$$\min_{\bm{x}\in X} \; \bm{x}^\top \bm{\Sigma} \bm{x}.$$

This makes clear that the long-only GMV portfolio is a simplex-constrained quadratic minimization problem, with no direct role for $\bm{\mu}$ in the objective (Yadav et al., 28 Jan 2026).

The contrast with the unconstrained GMV is fundamental. If sign constraints are dropped, the solution is

$$w^{\star} = \frac{\widehat{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}^\top \widehat{\Sigma}^{-1}\mathbf{1}},$$

or equivalently $$\Omega \mathbf{1}/(\mathbf{1}^\top \Omega \mathbf{1})$$ when $\Omega=\widehat{\Sigma}^{-1}$. Long-only constraints remove this closed form. The resulting optimization remains convex, but “there is no closed-form solution” and one must solve it numerically as a quadratic program (Bongiorno et al., 2 Jul 2025).

This distinction has two immediate implications. First, long-only GMV inherits all the sensitivity of covariance estimation, because the variance objective is unchanged. Second, the no-short-sales restriction changes the qualitative structure of the solution: inactive assets appear at zero weight, active sets become economically meaningful, and the optimizer can move from a diffuse unconstrained allocation to a corner solution on a much smaller subset of names. Several later developments in the literature can be read as attempts to control precisely this interaction between covariance error and simplex constraints.

2. Active sets and factor-model geometry

Under factor-model structure, the long-only GMV portfolio can be characterized much more explicitly than in the generic quadratic-program view. In the one-factor model

$$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta, \qquad \Delta = \mathrm{diag}(\delta_1^2,\dots,\delta_p^2),$$

the long-only solution $w^L$ is determined by its active set

$$\Sigma^{-1}\mathbf{1}/(\mathbf{1}^\top \Sigma^{-1}\mathbf{1})$$0

Once $$\Sigma^{-1}\mathbf{1}/(\mathbf{1}^\top \Sigma^{-1}\mathbf{1})$$1 is known, the positive weights are exactly the unconstrained GMV weights on the reduced covariance matrix $$\Sigma^{-1}\mathbf{1}/(\mathbf{1}^\top \Sigma^{-1}\mathbf{1})$$2; assets outside $$\Sigma^{-1}\mathbf{1}/(\mathbf{1}^\top \Sigma^{-1}\mathbf{1})$$3 receive weight zero (Kercheval et al., 11 Apr 2026).

When the betas are ordered as $$\Sigma^{-1}\mathbf{1}/(\mathbf{1}^\top \Sigma^{-1}\mathbf{1})$$4, the active set is an initial segment $$\Sigma^{-1}\mathbf{1}/(\mathbf{1}^\top \Sigma^{-1}\mathbf{1})$$5, where

$$\Sigma^{-1}\mathbf{1}/(\mathbf{1}^\top \Sigma^{-1}\mathbf{1})$$6

with

$$\Sigma^{-1}\mathbf{1}/(\mathbf{1}^\top \Sigma^{-1}\mathbf{1})$$7

This produces a computable threshold rule: the long-only minimum variance portfolio consists of the lowest-beta assets up to the last index for which $$\Sigma^{-1}\mathbf{1}/(\mathbf{1}^\top \Sigma^{-1}\mathbf{1})$$8 remains positive. The same analysis yields a beta-threshold form, $$\Sigma^{-1}\mathbf{1}/(\mathbf{1}^\top \Sigma^{-1}\mathbf{1})$$9 if and only if $$X = \left\{ \bm{x} = (x_1,\dots,x_p)^\top : \sum_{i=1}^p x_i = 1,\; x_i \ge 0,\; i=1,\dots,p \right\},$$0, with $$X = \left\{ \bm{x} = (x_1,\dots,x_p)^\top : \sum_{i=1}^p x_i = 1,\; x_i \ge 0,\; i=1,\dots,p \right\},$$1 determined by the active set (Kercheval et al., 11 Apr 2026).

The multi-factor case replaces scalar beta ordering by factor-space geometry. For

$$X = \left\{ \bm{x} = (x_1,\dots,x_p)^\top : \sum_{i=1}^p x_i = 1,\; x_i \ge 0,\; i=1,\dots,p \right\},$$2

with row $$X = \left\{ \bm{x} = (x_1,\dots,x_p)^\top : \sum_{i=1}^p x_i = 1,\; x_i \ge 0,\; i=1,\dots,p \right\},$$3 the exposure vector of asset $$X = \left\{ \bm{x} = (x_1,\dots,x_p)^\top : \sum_{i=1}^p x_i = 1,\; x_i \ge 0,\; i=1,\dots,p \right\},$$4, the active set is characterized by a separating hyperplane: $$X = \left\{ \bm{x} = (x_1,\dots,x_p)^\top : \sum_{i=1}^p x_i = 1,\; x_i \ge 0,\; i=1,\dots,p \right\},$$5 for a vector $$X = \left\{ \bm{x} = (x_1,\dots,x_p)^\top : \sum_{i=1}^p x_i = 1,\; x_i \ge 0,\; i=1,\dots,p \right\},$$6 determined by the active set and the factor covariance. In this representation, active assets are exactly those whose exposure vectors lie in a particular half-space, while inactive assets lie on the other side of the hyperplane (Gunther et al., 8 Mar 2026).

These structural results also imply asymptotic sparsity statements. In the one-factor model with iid betas drawn from a distribution $$X = \left\{ \bm{x} = (x_1,\dots,x_p)^\top : \sum_{i=1}^p x_i = 1,\; x_i \ge 0,\; i=1,\dots,p \right\},$$7, the active ratio $$X = \left\{ \bm{x} = (x_1,\dots,x_p)^\top : \sum_{i=1}^p x_i = 1,\; x_i \ge 0,\; i=1,\dots,p \right\},$$8 converges, in almost all cases, to $$X = \left\{ \bm{x} = (x_1,\dots,x_p)^\top : \sum_{i=1}^p x_i = 1,\; x_i \ge 0,\; i=1,\dots,p \right\},$$9, where $$\min_{\bm{x}\in X} \; \bm{x}^\top \bm{\Sigma} \bm{x}.$$0 is the root of an explicit integral equation. In particular, when $$\min_{\bm{x}\in X} \; \bm{x}^\top \bm{\Sigma} \bm{x}.$$1 is continuous and all betas are positive, the active ratio converges to zero. This formalizes why long-only minimum variance portfolios can become highly concentrated in large universes even without any explicit sparsity penalty (Kercheval et al., 11 Apr 2026).

3. Covariance estimation as the central problem

Because the long-only GMV objective depends only on $$\min_{\bm{x}\in X} \; \bm{x}^\top \bm{\Sigma} \bm{x}.$$2, most of the modern literature can be read as covariance engineering for a constrained quadratic program. One line of work uses nonlinear latent-variable models. The Student’s $$\min_{\bm{x}\in X} \; \bm{x}^\top \bm{\Sigma} \bm{x}.$$3-process latent variable model (TPLVM) replaces the Gaussian process in GPLVM by a Student’s $$\min_{\bm{x}\in X} \; \bm{x}^\top \bm{\Sigma} \bm{x}.$$4-process, yielding a predictive covariance

$$\min_{\bm{x}\in X} \; \bm{x}^\top \bm{\Sigma} \bm{x}.$$5

That covariance can then be inserted directly into the long-only program

$$\min_{\bm{x}\in X} \; \bm{x}^\top \bm{\Sigma} \bm{x}.$$6

In the global-index application, the TPLVM-based minimum-variance portfolio outperformed the GPLVM-based alternative, with lower realized volatility across all subperiods and especially better behavior in crisis conditions (Uchiyama et al., 2020).

A second line of work targets high-dimensional $$\min_{\bm{x}\in X} \; \bm{x}^\top \bm{\Sigma} \bm{x}.$$7 settings through sparse precision estimation. The Bayesian adaptive graphical LASSO framework estimates the precision matrix $$\min_{\bm{x}\in X} \; \bm{x}^\top \bm{\Sigma} \bm{x}.$$8 under a multivariate normal likelihood with adaptive Laplace shrinkage on off-diagonals and an MCMC scheme that guarantees positive definiteness at each draw. The unconstrained GMV then uses $$\min_{\bm{x}\in X} \; \bm{x}^\top \bm{\Sigma} \bm{x}.$$9; the long-only extension is entirely at the optimization stage, where one first obtains $\bm{\mu}$0 and then solves the standard long-only quadratic program (Oya, 2021).

A third line emphasizes robust or structured covariance cleaning. Tyler’s M-estimator with Toeplitz projection, whitening, and Random Matrix Theory eigenvalue cleaning is proposed in one study on the Maximum Variety portfolio, with the authors explicitly stating that “the same improvements apply also in the other optimisation problems such as the Minimum Variance Portfolio.” In that pipeline, a robust scatter estimate is whitened, de-noised via the Marčenko–Pastur threshold, and mapped back to the original scale before entering the portfolio optimizer (1804.00191). Under a spiked covariance model, the SCRGMVP approach instead constructs a spectrally corrected covariance and a regularized inverse surrogate

$\bm{\mu}$1

then uses it in the GMV formula; the same estimator can be inserted into a long-only QP even though the paper itself treats the unconstrained case (Li et al., 2023).

Further developments pursue sparsity and dimension reduction more directly. A post-processed posterior based on inverse-Wishart samples and hard-thresholding attains optimal minimax rates for sparse covariance estimation and for the unconstrained GMV weights, while still producing positive-definite matrices after eigenvalue adjustment (Lee et al., 2021). A clustering-based “divide and conquer” estimator constructs covariance in two stages—within clusters and between clusters—and bounded clustering is proposed precisely to control maximum cluster size; the reported best cluster-based GMVP achieves out-of-sample standard deviation around $\bm{\mu}$2, versus $\bm{\mu}$3–$\bm{\mu}$4 for stock-based GMVP in the thesis experiments (Park, 2020).

What unifies these strands is that long-only constraints are typically imposed after covariance estimation, not inside it. The estimator produces $\bm{\mu}$5 or $\bm{\mu}$6; the portfolio layer then solves

$\bm{\mu}$7

This separation is methodologically convenient, but it also means that the quality of $\bm{\mu}$8 remains the dominant determinant of long-only GMV behavior.

4. Heavy tails, robustness, and decision geometry

Recent theory clarifies that covariance error should not be assessed only by matrix norms, because the GMV decision depends on a much smaller geometry. For the unconstrained GMVP, with $\bm{\mu}$9 and $$w^{\star} = \frac{\widehat{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}^\top \widehat{\Sigma}^{-1}\mathbf{1}},$$0, the exact regret identity is

$$w^{\star} = \frac{\widehat{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}^\top \widehat{\Sigma}^{-1}\mathbf{1}},$$1

and the first-order displacement is

$$w^{\star} = \frac{\widehat{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}^\top \widehat{\Sigma}^{-1}\mathbf{1}},$$2

The same analysis shows that regret is invariant to covariance-scaling errors, depends on estimation error only through its action on the portfolio weights, and is bounded by a term proportional to $$w^{\star} = \frac{\widehat{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}^\top \widehat{\Sigma}^{-1}\mathbf{1}},$$3. Under heavy tails with tail index $$w^{\star} = \frac{\widehat{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}^\top \widehat{\Sigma}^{-1}\mathbf{1}},$$4, the paper derives

$$w^{\star} = \frac{\widehat{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}^\top \widehat{\Sigma}^{-1}\mathbf{1}},$$5

These results are exact for the unconstrained GMVP, and the paper explicitly states that they “do not immediately carry over, in exact form, to the long-only GMVP,” but they still provide design principles for constrained portfolios: concentration and conditioning amplify decision error, whereas some covariance perturbations are decision-irrelevant (Fonseca, 25 Jun 2026).

A complementary robustness program avoids estimating and inverting the entire covariance matrix. “Robustifying Markowitz” replaces plug-in covariance inversion by projected gradient descent with a robust estimator of the gradient increment $$w^{\star} = \frac{\widehat{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}^\top \widehat{\Sigma}^{-1}\mathbf{1}},$$6, built from median-of-means ideas and a spectral-center procedure. The exposition of that method specializes the update to long-only GMV by replacing the hyperplane projection with projection onto the simplex. In the empirical comparisons summarized there, the long-only sample-based benchmark appears as $$w^{\star} = \frac{\widehat{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}^\top \widehat{\Sigma}^{-1}\mathbf{1}},$$7, and the paper’s discussion notes that $$w^{\star} = \frac{\widehat{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}^\top \widehat{\Sigma}^{-1}\mathbf{1}},$$8 is more stable than unconstrained GMV but remains sensitive: which assets are heavily weighted or dropped changes substantially over time (Härdle et al., 2022).

From the long-only perspective, these two strands converge on the same point. Heavy-tailed data make covariance estimation slow and unstable; simplex constraints do not remove that instability, they only reshape it. Robust scatter estimation, robust gradient estimation, and careful conditioning are therefore not auxiliary refinements but central elements of long-only minimum-variance construction.

5. Decision-focused and end-to-end learning

A recent machine-learning literature replaces prediction-focused covariance estimation by direct optimization of portfolio decisions. One decision-focused learning paper derives the analytic gradient of GMVP regret using the unconstrained solution

$$w^{\star} = \frac{\widehat{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}^\top \widehat{\Sigma}^{-1}\mathbf{1}},$$9

and defines loss as the true-variance regret of the portfolio induced by $$\Omega \mathbf{1}/(\mathbf{1}^\top \Omega \mathbf{1})$$0. The paper argues that prediction-focused methods minimizing covariance MSE can be suboptimal for GMV construction, whereas DFL-based methods optimize decision quality directly. Its theoretical analysis concentrates on the principal directions of the Jacobian $$\Omega \mathbf{1}/(\mathbf{1}^\top \Omega \mathbf{1})$$1, and its empirical study reports lower realized volatility than historical, LW-D, LW-CC, and OAS estimators across the tested universes and horizons (Kim et al., 14 Aug 2025).

A more elaborate end-to-end system is the rotation-invariant neural architecture that mirrors the algebra of GMV through three modules: a lag-transformation module, an eigenvalue-cleaning BiLSTM, and a marginal-volatility MLP. The network outputs

$$\Omega \mathbf{1}/(\mathbf{1}^\top \Omega \mathbf{1})$$2

then forms unconstrained GMV weights in closed form during training. For long-only portfolios, the learned covariance is passed to an external QP solver. In frictionless long-only tests with $$\Omega \mathbf{1}/(\mathbf{1}^\top \Omega \mathbf{1})$$3, the paper reports for the neural estimator $$\Omega \mathbf{1}/(\mathbf{1}^\top \Omega \mathbf{1})$$4, annualized volatility $$\Omega \mathbf{1}/(\mathbf{1}^\top \Omega \mathbf{1})$$5, mean return $$\Omega \mathbf{1}/(\mathbf{1}^\top \Omega \mathbf{1})$$6, effective $$\Omega \mathbf{1}/(\mathbf{1}^\top \Omega \mathbf{1})$$7, and turnover $$\Omega \mathbf{1}/(\mathbf{1}^\top \Omega \mathbf{1})$$8. In realistic long-only tests on $$\Omega \mathbf{1}/(\mathbf{1}^\top \Omega \mathbf{1})$$9 equities from January 2000 to December 2024, it reports $\Omega=\widehat{\Sigma}^{-1}$0, mean return $\Omega=\widehat{\Sigma}^{-1}$1, volatility $\Omega=\widehat{\Sigma}^{-1}$2, maximum drawdown $\Omega=\widehat{\Sigma}^{-1}$3, turnover $\Omega=\widehat{\Sigma}^{-1}$4, and effective $\Omega=\widehat{\Sigma}^{-1}$5, outperforming AO, PM, QIS, MLE, IVP, and MCAP on the reported metrics (Bongiorno et al., 2 Jul 2025).

These learning-based approaches alter the conventional division between estimation and optimization. In classical GMV research, $\Omega=\widehat{\Sigma}^{-1}$6 is estimated first and only then supplied to a portfolio solver. In DFL and end-to-end systems, the covariance representation is itself trained against portfolio variance. For long-only GMV, the decisive practical point is that both papers still use a standard quadratic program at deployment time once a positive-definite covariance estimate has been learned.

6. Empirical behavior, implementation, and open issues

Across heterogeneous datasets and estimators, a stable empirical pattern is that better covariance regularization translates into lower realized risk, but often at the cost of concentration or turnover. In the TPLVM global-index study, the whole-period comparison reports Return $\Omega=\widehat{\Sigma}^{-1}$7, Risk $\Omega=\widehat{\Sigma}^{-1}$8, and $\Omega=\widehat{\Sigma}^{-1}$9 for the GPLVM-based portfolio, against Return $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta, \qquad \Delta = \mathrm{diag}(\delta_1^2,\dots,\delta_p^2),$$0, Risk $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta, \qquad \Delta = \mathrm{diag}(\delta_1^2,\dots,\delta_p^2),$$1, and $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta, \qquad \Delta = \mathrm{diag}(\delta_1^2,\dots,\delta_p^2),$$2 for the TPLVM-based portfolio; the improvement is strongest in the crisis subperiod (Uchiyama et al., 2020). In the Bayesian adaptive graphical LASSO study with $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta, \qquad \Delta = \mathrm{diag}(\delta_1^2,\dots,\delta_p^2),$$3, the extreme $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta, \qquad \Delta = \mathrm{diag}(\delta_1^2,\dots,\delta_p^2),$$4 case $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta, \qquad \Delta = \mathrm{diag}(\delta_1^2,\dots,\delta_p^2),$$5 yields a Sharpe ratio of $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta, \qquad \Delta = \mathrm{diag}(\delta_1^2,\dots,\delta_p^2),$$6 for Bada-PD while the non-Bayesian graphical LASSO fails to estimate $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta, \qquad \Delta = \mathrm{diag}(\delta_1^2,\dots,\delta_p^2),$$7 (Oya, 2021).

Other studies reinforce the same message from different angles. The $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta, \qquad \Delta = \mathrm{diag}(\delta_1^2,\dots,\delta_p^2),$$8-BAHC paper reports better Sharpe ratios after transaction costs than competing filtering methods, “despite requiring a larger turnover,” and explicitly evaluates both long-short and long-only GMV portfolios (Bongiorno et al., 2020). The broad shrinkage comparison across six datasets concludes that “in most scenarios, the GMV model combined with the Ledoit Wolf two parameter shrinkage covariance estimator (COV2) represents the optimal selection for a broad spectrum of investors,” with the optimization carried out directly on the long-only feasible set (Yadav et al., 28 Jan 2026). The bounded-clustering thesis reports that limiting maximum cluster size decreases the gap between in-sample and out-of-sample volatility and reduces out-of-sample volatility itself, again underscoring that the primary challenge is covariance estimation rather than the portfolio solver alone (Park, 2020).

Several practical issues remain recurrent. One is the gap between unconstrained theory and long-only execution: many exact formulas, regret identities, and asymptotic characterizations are derived for the unconstrained GMVP and then adapted to long-only only at the optimization stage. Another is that many backtests ignore transaction costs, financing costs, or turnover penalties, whereas the few studies that model implementation frictions explicitly show that turnover can materially alter rankings. A third is model risk: kernel choice in latent-variable models, sparsity priors in graphical methods, shrinkage targets, active-set instability near factor thresholds, and hyperparameter tuning in neural systems all materially affect the resulting simplex-constrained allocation.

A persistent misconception is that banning short sales largely solves the GMV instability problem. The literature does not support that view. Long-only constraints remove negative positions, but they do not eliminate covariance estimation error, ill-conditioning, heavy-tail sensitivity, or active-set instability. The long-only GMV portfolio is therefore best understood not as a simpler variant of GMV, but as a covariance-driven quadratic program whose distinctive properties arise from the interaction of simplex geometry with regularized, sparse, robust, or learned estimates of $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta, \qquad \Delta = \mathrm{diag}(\delta_1^2,\dots,\delta_p^2),$$9.