Long-Short Portfolio Optimization Under Cardinality Constraints by Difference of Convex Functions Algorithm

Abstract

In the matter of Portfolio selection, we consider an extended version of the Mean-Absolute Deviation (MAD) model, which includes discrete asset choice constraints (threshold and cardinality constraints) and one is allowed to sell assets short if it leads to a better risk-return tradeoff. Cardinality constraints limit the number of assets in the optimal portfolio and threshold constraints limit the amount of capital to be invested in (or sold short from) each asset and prevent very small investments in (or short selling from) any asset. The problem is formulated as a mixed 0–1 programming problem, which is known to be NP-hard. Attempting to use DC (Difference of Convex functions) programming and DCA (DC Algorithms), an efficient approach in non-convex programming framework, we reformulate the problem in terms of a DC program, and investigate a DCA scheme to solve it. Some computational results carried out on benchmark data sets show that DCA has a better performance in comparison to the standard solver IBM CPLEX.

Appendix: DCA for Solving (44) When the Transaction Cost Functions Are Concave

Suppose that the functions c(⋅) and d(⋅) are concave and the DC representation of the problem (44) is as follows:

$$ \min \bigl\{ g\bigl( \mathbf{u} , \mathbf{y} , \mathbf{y}^{\prime}, \mathbf{z} , \mathbf{z}^{\prime} \bigr)-h\bigl( \mathbf{u} , \mathbf{y} , \mathbf{y}^{\prime}, \mathbf{z} , \mathbf{z}^{\prime}\bigr) : \bigl( \mathbf{u} , \mathbf{y} , \mathbf{y}^{\prime}, \mathbf{z} , \mathbf{z}^{\prime}\bigr)\in \mathbb{R}^{4n+T} \bigr\}, $$

where

$$ g\bigl( \mathbf{u} , \mathbf{y} , \mathbf{y}^{\prime}, \mathbf{z} , \mathbf{z}^{\prime}\bigr):=\sum_{i=1}^{n}r_{i} \bigl(y^{\prime}_{i} - y_{i}\bigr)+\chi_{\mathcal{A}} \bigl( \mathbf{u} , \mathbf{y} , \mathbf{y}^{\prime}, \mathbf{z} , \mathbf{z}^{\prime}\bigr), $$

and

$$ h\bigl( \mathbf{u} , \mathbf{y} , \mathbf{y}^{\prime}, \mathbf{z} , \mathbf{z}^{\prime}\bigr):=\theta \Biggl(\sum_{i=1}^{n} \bigl(z_{i} -z^{\prime }_{i}\bigr)^{2} - \sum _{i=1}^{n}\bigl(z_{i}+z^{\prime}_{i} \bigr) \Biggr) - \sum_{i=1}^{n} \bigl(c(y_{i}) + d\bigl(y^{\prime}_{i}\bigr) \bigr). $$

$\chi_{\mathcal{A}}$ is the indicator function on $\mathcal{A}$.

In a similar way to the preceding DCA algorithm, we compute a sub-gradient of the function h defined by

$$ h\bigl( \mathbf{u} , \mathbf{y} , \mathbf{y}^{\prime}, \mathbf{z} , \mathbf{z}^{\prime}\bigr):=\theta \Biggl(\sum_{i=1}^{n} \bigl(z_{i} -z^{\prime }_{i}\bigr)^{2} - \sum _{i=1}^{n}\bigl(z_{i}+z^{\prime}_{i} \bigr) \Biggr) - \sum_{i=1}^{n} \bigl(c(y_{i}) + d\bigl(y^{\prime}_{i}\bigr) \bigr). $$

Define $\mathcal{C}$ and $\mathcal{D}$ as the gradient vectors of c(⋅) and d(⋅), respectively. From the definition of h(⋅) we have

$$ \everymath{\displaystyle} \begin{array}{c} \bigl(\nu^{k}, \mathbf{v}^{k}, \mathbf{v}^{\prime k}, \mathbf{w}^{k}, \mathbf{w}^{\prime k}\bigr)\in\partial h( \mathbf{u}^{k}, \mathbf{y}^{k}, \mathbf{y}^{\prime k}, \mathbf{z}^{k}, \mathbf{z}^{\prime k}) \\[2pt] \Updownarrow \\[2pt] \nu^{k}_{t}=0,\quad v^{k}_{i}=\mathcal{C}_i|_{y=y^{k}},\quad v^{\prime k}_{i}=\mathcal{D}_i|_{y^{\prime}=y^{\prime k}},\\[4pt] w^{k}_{i}=\theta\bigl(2\bigl(z^{k}_{i}-z^{\prime k}_{i}\bigr)-1\bigr),\quad w^{\prime k}_{i}=\theta\bigl(2\bigl(z^{\prime k}_{i}-z^{k}_{i}\bigr)-1\bigr), \end{array} $$

for all i=1,…,n and t=1,…,T. Secondly, we must compute an optimal solution of the following linear program:

$$ \min\Biggl\{\sum_{i=1}^{n}r_{i} \bigl(y^{\prime}_{i} - y_{i}\bigr)- \bigl\langle\bigl( \mathbf{u} , \mathbf{y} , \mathbf{y}^{\prime}, \mathbf{z} , \mathbf{z}^{\prime k}, \mathbf{v}^{k}, \mathbf{v}^{\prime k}, \mathbf{w}^{k}, \mathbf{w}^{\prime k}\bigr)\bigr\rangle: \bigl( \mathbf{u} , \mathbf{y} , \mathbf{y}^{\prime}, \mathbf{z} , \mathbf{z}^{\prime}\bigr)\in A\Biggr\}, $$

(66)

which will be (u ^k+1,y ^k+1,y ^′k+1,z ^k+1,z ^′k+1). Any standard LP solver can be used to solve (66).

To sum up, the DCA algorithm will be as follows:

Algorithm

(Algorithm DCA for the Program (44) When the Transaction Cost Functions Are Concave)

• Step 1: Let ϵ be a sufficiently small positive number, choose an initial point such as (u ⁰,y ⁰,y ^′0,z ⁰,z ^′0)∈ℝ^4n+T, define k as the iteration indicator and set k=0;

• Step 2:

  For i=1,…,n and t=1,…,T, set

  $\nu^{k}_{t}=0$,

  $v^{k}_{i}=\mathcal{C}_{i}|_{y=y^{k}}$,

  $v^{\prime k}_{i}=\mathcal{D}_{i}|_{y^{\prime}=y^{\prime k}}$,

  $w^{k}_{i}=\theta(2(z^{k}_{i}-z^{\prime k}_{i})-1)$,

  $w^{\prime k}_{i}=\theta(2(z^{\prime k}_{i}-z^{k}_{i})-1)$,

  and solve the following linear program:

  $$ \min\Biggl\{\sum_{i=1}^{n} r_{i} \bigl(y^{\prime}_{i} - y_{i}\bigr)- \bigl\langle\bigl( \mathbf{u} , \mathbf{y} , \mathbf{y}^{\prime}, \mathbf{z} , \mathbf{z}^{\prime k}, \mathbf{v}^{k}, \mathbf{v}^{\prime k}, \mathbf{w}^{k}, \mathbf{w}^{\prime k}\bigr)\bigr\rangle: \bigl( \mathbf{u} , \mathbf{y} , \mathbf{y}^{\prime}, \mathbf{z} , \mathbf{z}^{\prime}\bigr)\in A\Biggr\}, $$

  to obtain (u ^k+1,y ^k+1,y ^′k+1,z ^k+1,z ^′k+1).

• Step 3: If ∥u ^k+1−u ^k∥+∥y ^k+1−y ^k∥+∥y ^′k+1−y ^′k∥+∥z ^k+1−z ^k∥+∥z ^′k+1−z ^′k∥≤ϵ, then stop, the stopping criterion is met and (u ^k+1,y ^k+1,y ^′k+1,z ^k+1,z ^′k+1) is an optimal solution, otherwise set k=k+1 and go to Step 2.