## 2 The Problem  
Consider an investment universe of \(n\) assets with observations characterized by a (\(nx1\)) vector of means \(\mu\) and a (\(nxn\)) positive definite covariance matrix \(\Sigma\). The mean vector \(\mu\) and covariance matrix \(\Sigma\) are computed on time-homogeneous invariants, _i.e._, phenomena that repeat themselves identically throughout history regardless of the reference time at which an observation is made. Time-homogeneity is a key property that our observations must satisfy, so that we do not need to re-estimate \(\mu,\Sigma\) for a sufficiently long period of time. For instance, compounded returns are generally accepted as good time-homogeneous invariants for equity, commodity and foreign-exchange products. In the case of fixed income products, changes in the yield-to-maturity are typically used. In the case of derivatives, changes in the rolling forward at-the-money implied volatility are usually considered a good time-homogeneous invariant (see [14] for a comprehensive discussion of this subject). The numerical example discussed in Section 5 is based on equity products, hence the mean vector \(\mu\) and covariance matrix \(\Sigma\) are computed on compounded returns. Following [11], we solve a quadratic programming problem subject to linear constraints in inequalities and one linear constraint in equality, and so we need some nomenclature to cover those inputs:  
* \(N=\{1,2,...,n\}\) is a set of indices that number the investment universe. * \(\omega\) is the (\(nx1\)) vector of asset weights, which is our optimization variable. * \(l\) is the \((nxI)\) vector of lower bounds, with \(\omega_{i}\geq l_{i},\forall i\in N\). * \(u\) is the \((nxI)\) vector of upper bounds, with \(\omega_{i}\leq u_{i},\forall i\in N\). * \(F\subseteq N\) is the subset of _free_ assets, where \(l_{i}<\omega_{i}<u_{i}\). In words, free assets are those that do not lie on their respective boundaries. \(F\) has length \(1\leq k\leq n\). * \(B\subset N\) is the subset of weights that lie on one of the bounds. By definition, \(B\cup F=N\). Accordingly, we can construct \(\mu\) and \(\Sigma\) as a block array and matrix. \[\Sigma= \begin{bmatrix}\Sigma_{F}&\Sigma_{FB}\\ \Sigma_{BF}&\Sigma_{B}\end{bmatrix} \text{, }\mu= \begin{bmatrix}\mu_{F}\\ \mu_{B}\end{bmatrix} \text{, }\omega= \begin{bmatrix}\omega_{F}\\ \omega_{B}\end{bmatrix}  \tag{1}\]  
where \(\Sigma_{F}\) denotes the \((kxk)\) covariance matrix among free assets, \(\Sigma_{B}\) the \(((n-k)x(n-k))\) covariance matrix among assets lying on a boundary condition, and \(\Sigma_{FB}\) the \((kx(n-k))\) covariance between elements of \(F\) and \(B\), which obviously is equal to \(\Sigma_{BF}\)' (the transpose of \(\Sigma_{BF}\)) since \(\Sigma\) is symmetric. Similarly, \(\mu_{F}\) is the \((kxI)\) vector of means associated with \(F,\,\mu_{B}\) is the \(((n-k)xI)\) vector of means associated with \(B\), \(\omega_{F}\) is the \((kxI)\) vector of weights associated with \(F\), \(\omega_{B}\) is the \(((n-k)xI)\) vector of weights associated with \(B\). The solution to the unconstrained problem (\"Unconstrained problem\" is a bit of a misnomer, because this problem indeed contains two linear equality constraints: Full investment (the weights add up to one) and target portfolio mean. What is meant is to indicate that no specific constraints are imposed on individual weights) consists in minimizing the Lagrange function with respect to the vector of weights \(\omega\) and the multipliers \(\gamma\) and \(\lambda\):  
\[L[\omega,\gamma,\lambda]=\frac{1}{2}\,\omega^{\prime}\Sigma\omega-\gamma( \omega^{\prime}1_{n}-1)-\lambda\big{(}\omega^{\prime}\mu-\mu_{p}\big{)} \tag{2}\]  
where \(1_{n}\) is the \((nxI)\) vector of ones and \(\mu_{p}\) is the targeted excess return. The method of Lagrange multipliers applies first order necessary conditions on each weight and Lagrange multiplier, leading to a linear system of \(n+2\) conditions. See [14] for an analytical solution to this problem.