We have considered a basic TEV (tracking error variance) minimization model in Section 15.4.1, where we are given the weights of a benchmark (target)

portfolio that must be tracked, the covariance matrix between assets returns

(assumed known), and the maximum cardinality of the tracking portfolio (number of assets included). Short-selling was not allowed and we did not consider transaction costs. Extend the model as follows:

Short-selling is allowed.
We hold a current portfolio, with given weights, which should be rebalanced in order to improve tracking. However, we want to trade off tracking against turnover, i.e., changes with respect to the current portfolio.
We do not want to include explicit transaction costs, but we do not want to change the portfolio too much. Thus, in the objective function, we want to include the L1 norm distance between the new and the current portfolio, suitably penalized.
We do not really trust our estimate of the covariance matrix, and we consider a finite uncertainty set consisting ofmcovariance matrices (thus, we consider a simple form of distributional ambiguity). The model should be robust and minimize the worst-case performance over this set of alternative matrices.
Formulate the model in such a way that it is solvable by a commercial software tool implementing a branch-and-bound approach for mixed-integer problems with linear or quadratic objective and constraints.