I am going to formulate an optimization problem for finding a feasible portfolio with the largest value of Sortino ratio. It will defined as follows:
\begin{align*} \max_{x} \quad & STR(x) = \frac{\mu^T x - R}{\sqrt{\mathbb{E}[\min\{0, (r-\mu)^T x\}]^2}} \\ \text{s.t.} \quad & x \in X \end{align*} where:
- $x \in \mathbb{R}^n$ is the vector representing the portfolio weights.
- $R$ is the target return rate on the expected return of the portfolio.
- $\mu \in \mathbb{R}^n$ is the vector of expected returns for each asset in the portfolio.
- $r \in \mathbb{R}^n$ is a random vector representing the actual returns of each asset.
The numerator is linear, and hence both convex and concave.The complexity arises with the denominator, which is the square root of an expected value. Furthermore, I knew the squared minimum function can be non-convex due to the min operation involving a linear term and a constant (zero). The expectation does not alter the convexity properties of the function it is applied to. Furthermore, the square root function is concave and non-decreasing over its domain.
But how do we in general tell if this objective function is convex, concave, or neither?