optimization

The optimal portfolio construction problem is one of the most extensively studied problems in quantitative finance. The Modern Portfolio Theory (MPT) as formulated by Markowitz [@markowitz1952portfolio] has laid the foundation for highly influential mean-variance portfolio optimisation approach. According to the MPT, a typical portfolio optimisation problem can be formulated as follows. Let $N$ be the number of assets, $\mu_i$ be the expected return of asset $i$, $\sigma_{ij}$ be the covariance between the returns for assets $i$ and $j$, and $R$ be the target portfolio return. Then the decision variables are the weights $w_i$, i.e. the investment associated with asset $i$ ($w_i \in \mathbb{R}$). The standard Markowitz mean-variance approach consists in the constrained, quadratic optimisation problem: $$\begin{aligned}
 \label{eq:MarObjFun}
\min \sum_{i=1}^{N} \sum_{j=1}^{N} w_i w_j \sigma_{ij}\;\;;&\sum_{i=1}^{N} 
w_i = 1\;\;;&\sum_{i=1}^{N} w_i \mu_i = R\;.\end{aligned}$$

Quadratic problems of the form (\[eq:MarObjFun\]) are efficiently solvable by standard computational means (e.g., quadratic programming with linear constraints [@fernando2000practical]) if the covariance matrix is positive definite. Interestingly, the problem can also be cast into an unconstrained quadratic optimization problem (QUBO) which is a suitable formulation for quantum annealers. This observation spurred in the last few years a couple of proof-of-principle papers that were performing runs on the first 