Fast gradient descent method for Mean-CVaR optimization

Abstract

We propose an iterative gradient descent algorithm for solving scenario-based Mean-CVaR portfolio selection problem. The algorithm is fast and does not require any LP solver. It also has efficiency advantage over the LP approach for large scenario size.

Appendices

Appendix A: Computation of constants

For completeness, we compute the constants here and they can be used for computing the number of iterations required given in (34). Please refer to Nesterov (2005) for the details. First,

(35)

To solve min _q∈Q d ₂(q), we find that

(36)

(37)

Setting $\frac{\partial L}{\partial q_{i}} = 0$ gives

(38)

Therefore q _i =p _i and

(39)

Note that max _q∈Q d ₂(q) occurs at extreme points where there are K of q _i equal to p _i β ⁻¹ and N−K of them equal to 0, this gives max _q∈Q d ₂(q)=∑ _i p _i β ⁻¹log(p _i β ⁻¹). So we have

(40)

As in (Nesterov 2005), σ ₂ is defined as the constant such that $d_{2}(\mathbf{q}) \geq\frac{1}{2} \sigma_{2} \|\mathbf{q} - \mathbf{p}\| _{1}^{2}$. By Cauchy-Schwartz inequality,

(41)

Consider

(42)

(43)

we have $q_{i} = \frac{(1 - \gamma)\beta^{-1}}{2} p_{i} \propto p_{i}$, and thus q _i =p _i . As a result,

(44)

Finally, by (11)

(45)

By setting $\mu= \frac{\varepsilon}{2 D_{2}}$, we can guarantee that our algorithm stops by at most K iterations as defined in (34).

Appendix B: Algorithm