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The Lyxor white paper Regularization of Portfolio Allocation contains a lot on this topic. The head of quant research there, Thierry Roncalli, also held a talk about this recently.


With respect to issue one, it can be simpler to consider the case where the constraint on the expected return is an equality. In that case, the first problem can be transformed to Minimize with respect to $\left\{ x,\lambda_{1},\lambda_{2}\right\} $: $x'\Sigma x + \lambda_{1} (\mu'x - r) + \lambda_{2} (1'x - 1)$ by the technique of Lagrangian multipliers, ...


Sure, the variance of the total wealth can be expressed in terms of the variances and covariances of the prices of the assets. If $$ W = \sum_{i} \pi_i P_i $$ where $\pi_i$ is the total dollar amount invested in asset $i$ with price $P_i$. The variance of total wealth is then $$ Var(W) = \sum_i \pi_i Var(P_i) + \sum_i \sum_{j, j\neq i} \pi_i \pi_j Cov(P_i, ...


* For a given period t and a set of securities and cash denoted with index i which individually have returns r and weights w in a portfolio the portfolio return could be computed as $$ R = \sum_i w^s _i r^s _i + w_i^l r_i^l $$ where the sups l and s mean short and and long respectively. Note that the weights need to sum up to unity $$ \sum_i (w^s_i + ...

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