Cutting plane algorithms for mean-CVaR portfolio optimization with nonconvex transaction costs

Abstract

This paper studies the mean-risk portfolio optimization problem with nonconvex transaction costs. We employ the conditional value-at-risk (CVaR) as a risk measure. There are a number of studies that aim at efficiently solving large-scale CVaR minimization problems. None of these studies, however, take into account nonconvex transaction costs, which are present in practical situations. To make a piecewise linear approximation of the transaction cost function, we utilized special ordered set type two constraints. Moreover, we devised a subgradient-based cutting plane algorithm to handle a large number of scenarios. This cutting plane algorithm needs to solve a mixed integer linear programming problem in each iteration, and this requires a substantial computation time. Thus, we also devised a two-phase cutting plane algorithm that is even more efficient. Numerical experiments demonstrated that our algorithms can attain near-optimal solutions to large-scale problems in a reasonable amount of time. Especially when rebalancing a current portfolio that is close to an optimal one, our algorithms considerably outperform other solution methods.

Appendix

We show that the complementary conditions, $x^{+}_i x^{-}_i = 0,~\forall i = 1,2,\ldots ,I$, can be eliminated from problem (6).

Proposition 1

Suppose that $(\hat{a},\hat{u},\hat{{\varvec{x}}},\hat{{\varvec{x}}}^{+},\hat{{\varvec{x}}}^{-})$ is an optimal solution to problem (6) without the complementary conditions. Define $\dot{x}^{+}_i := [\hat{x}_i - x^0_i]_{+}$ and $\dot{x}^{-}_i := [x^0_i - \hat{x}_i]_{+}$ for $i=1,2,\ldots ,I$. Then $(\hat{a},\hat{u},\hat{{\varvec{x}}},\dot{{\varvec{x}}}^{+},\dot{{\varvec{x}}}^{-})$ is an optimal solution to problem (6).

Proof

First, we show that $(\hat{a},\hat{u},\hat{{\varvec{x}}},\dot{{\varvec{x}}}^{+},\dot{{\varvec{x}}}^{-})$ is a feasible solution to problem (6). It follows from the definition that

$$\begin{aligned} \hat{x}_i - x^0_i = \dot{x}^{+}_i - \dot{x}^{-}_i,~\dot{x}^{+}_i \ge 0,~\dot{x}^{-}_i \ge 0,~\dot{x}^{+}_i \dot{x}^{-}_i = 0, \quad \forall i = 1,2,\ldots ,I. \end{aligned}$$

In addition, since $0 \le \hat{x}^{+}_i,~\hat{x}_i - x^0_i = \hat{x}^{+}_i - \hat{x}^{-}_i \le \hat{x}^{+}_i$ and $0 \le \hat{x}^{-}_i,~x^0_i - \hat{x}_i = \hat{x}^{-}_i - \hat{x}^{+}_i \le \hat{x}^{-}_i$, we have

$$\begin{aligned} \dot{x}^{+}_i = \max \{\hat{x}_i - x^0_i, 0\} \le \hat{x}^{+}_i,~\dot{x}^{-}_i = \max \{x^0_i - \hat{x}_i, 0\} \le \hat{x}^{-}_i, \quad \forall i = 1,2,\ldots ,I. \end{aligned}$$

Considering that $\mathcal{C}^{+}_i$ and $\mathcal{C}^{-}_i$ are nondecreasing functions, we can see that

$$\begin{aligned} \mathcal{C}^{+}_i(\dot{x}^{+}_i) + \mathcal{C}^{-}_i(\dot{x}^{-}_i) \le \mathcal{C}^{+}_i(\hat{x}^{+}_i) + \mathcal{C}^{-}_i(\hat{x}^{-}_i), \quad \forall i = 1,2,\ldots ,I. \end{aligned}$$

(17)

Thus, $(\hat{a},\hat{u},\hat{{\varvec{x}}},\dot{{\varvec{x}}}^{+},\dot{{\varvec{x}}}^{-})$ satisfies all the constraints of problem (6).

Now we show that $(\hat{a},\hat{u},\hat{{\varvec{x}}},\dot{{\varvec{x}}}^{+},\dot{{\varvec{x}}}^{-})$ is an optimal solution to problem (6). It follows from (17) that the objective function value of $(\hat{a},\hat{u},\hat{{\varvec{x}}},\dot{{\varvec{x}}}^{+},\dot{{\varvec{x}}}^{-})$ is not greater than that of $(\hat{a},\hat{u},\hat{{\varvec{x}}},\hat{{\varvec{x}}}^{+},\hat{{\varvec{x}}}^{-})$. Recall that $(\hat{a},\hat{u},\hat{{\varvec{x}}},\hat{{\varvec{x}}}^{+},\hat{{\varvec{x}}}^{-})$ is an optimal solution to a relaxed problem, i.e., problem (6) without complementary conditions. This completes the proof. $\square$

Remark 1

When $\mathcal{C}^{+}_i$ and $\mathcal{C}^{-}_i$ are strictly increasing functions, $(\hat{a},\hat{u},\hat{{\varvec{x}}},\hat{{\varvec{x}}}^{+},\hat{{\varvec{x}}}^{-})$ satisfies the complementary conditions by itself. This is because if $i$ exists such that $\hat{x}^{+}_i \hat{x}^{-}_i \not = 0$, the objective function value of $(\hat{a},\hat{u},\hat{{\varvec{x}}},\dot{{\varvec{x}}}^{+},\dot{{\varvec{x}}}^{-})$ is smaller than that of $(\hat{a},\hat{u},\hat{{\varvec{x}}},\hat{{\varvec{x}}}^{+},\hat{{\varvec{x}}}^{-})$, which contradicts the optimality of $(\hat{a},\hat{u},\hat{{\varvec{x}}},\hat{{\varvec{x}}}^{+},\hat{{\varvec{x}}}^{-})$.