Quadratic Programming (QP)

In portfolio optimization, the covariance matrix is positive semi-definite, making the QP problem convex. This property ensures that any local minimum is also the global minimum, eliminating the risk of settling on a suboptimal allocation. Convexity is critical for production systems because it guarantees deterministic, repeatable solutions regardless of the starting point of the solver.

Constraints in QP are strictly linear equalities and inequalities, forming a convex polytope of feasible solutions. Common portfolio constraints include:

• Full investment: Sum of weights = 1
• Long-only: All weights ≥ 0
• Sector limits: Sum of weights in a sector ≤ 40%
• Turnover: Sum of absolute weight changes ≤ 20% These linear boundaries ensure the feasible region remains convex and computationally tractable.

The objective function takes the form minimize wᵀΣw − λμᵀw, where w is the weight vector, Σ is the covariance matrix, and μ is expected returns. The quadratic term wᵀΣw represents portfolio variance—the primary risk measure. The linear term −λμᵀw incorporates return maximization, with λ controlling the risk-aversion trade-off. This structure directly maps the Mean-Variance Optimization framework into a solvable mathematical program.

Every QP problem has a corresponding dual problem whose variables are Lagrange multipliers. In portfolio optimization, these dual variables represent shadow prices—the marginal cost of tightening a constraint. For example, the dual variable for the full-investment constraint equals the expected return of the minimum-variance portfolio. This duality allows portfolio managers to quantify the economic impact of each constraint on performance.

Real-world covariance matrices often exhibit high condition numbers due to correlated assets, making QP solvers numerically unstable. Best practices include:

• Shrinkage estimators to regularize Σ
• Factor model decomposition to reduce dimensionality
• Quadratic normalization to scale variables Without these techniques, solvers may return infeasible or inaccurate weight vectors, especially in large universes with thousands of assets.

Two algorithmic families dominate QP solving:

• Active-set methods: Iteratively identify which constraints are binding, ideal for problems where the optimal solution activates few constraints
• Interior-point methods: Traverse the interior of the feasible region, scaling efficiently to large-scale problems with thousands of variables Modern libraries like OSQP and CVXOPT implement these methods, often exploiting sparsity in the covariance matrix for speed.

The foundational framework that QP solves. MVO seeks to maximize expected return for a given level of variance, or equivalently, minimize variance for a target return. Harry Markowitz formalized this in 1952, framing portfolio selection as a trade-off between reward (mean return) and risk (variance).

• The objective function is quadratic: minimize wᵀΣw
• The constraints are linear: weights sum to 1, target return constraint
• QP algorithms compute the exact Efficient Frontier by varying the target return

The broader mathematical class to which QP belongs. A problem is convex if its objective function and feasible set are both convex. For portfolio optimization, this means the covariance matrix Σ must be positive semi-definite.

• Guarantees that any local minimum is the global minimum
• Enables the use of highly efficient interior-point solvers
• If Σ is not positive semi-definite (due to high correlation or missing data), the problem becomes non-convex and QP solvers fail

A statistical technique used to denoise the covariance matrix before it enters the QP solver. Empirical covariance matrices are dominated by sampling noise, especially when the number of assets N is large relative to the number of observations T.

• RMT separates signal eigenvalues from the Marchenko-Pastur bulk of noise eigenvalues
• Cleaning Σ with RMT dramatically improves the stability of QP solutions
• Prevents the optimizer from chasing spurious correlations that vanish out-of-sample

A machine learning alternative that avoids QP entirely. Marcos López de Prado introduced HRP to bypass the need to invert the covariance matrix, which is unstable in high dimensions.

• Uses hierarchical clustering on the correlation matrix to build a tree of assets
• Allocates capital recursively based on inverse-variance weighting within clusters
• Produces more robust out-of-sample portfolios than QP-based MVO when N is large
• Does not require a return forecast, only the covariance structure

A risk measure that can be optimized using linear programming rather than QP. CVaR quantifies the expected loss in the worst α% of scenarios, making it a coherent risk measure.

• Rockafellar and Uryasev showed CVaR minimization can be formulated as a linear program
• Unlike variance, CVaR focuses exclusively on downside tail risk
• Often used as an alternative objective when the covariance matrix is unreliable
• Can be combined with QP by adding CVaR as a linear constraint

A Bayesian framework that generates stable expected return vectors to feed into the QP solver. Raw historical mean returns are notoriously noisy and cause extreme, unintuitive portfolio weights.

• Starts with equilibrium returns implied by the market portfolio via reverse optimization
• Blends in investor views with specified confidence levels
• The output is a posterior return vector E[R] that produces well-behaved QP solutions
• Addresses the estimation error maximization problem inherent in MVO