SAA method based on modified Newton method for stochastic variational inequality with second-order cone constraints and application in portfolio optimization

Abstract

In this paper we apply modified Newton method based on sample average approximation (SAA) to solve stochastic variational inequality with stochastic second-order cone constraints (SSOCCVI). Under some moderate conditions, the SAA solution converges to its true counterpart with probability approaching one at exponential rate as sample size increases. We apply the theoretical results for solving a class of stochastic second order cone complementarity problems and stochastic programming problems with stochastic second order cone constraints. Some illustrative examples are given to show how the globally convergent method works and the comparison results between our method and other methods. Furthermore, we apply this method to portfolio optimization with loss risk constraints problems.

Appendix

Definition 6.1

A $\sigma$-algebra F of subsets of $\Omega$ is a collection F of subsets of $\Omega$ satisfying the following conditions:

• (a) $\emptyset \in F$

• (b) if $B\in F$ then its complement $B^{c}_{}$ is also in F

• (c) if $B^{}_{1}, B^{}_{2},\ldots$ is a countable collection of sets in F then their union $\bigcup ^{\infty }_{n=1}B^{}_{n}$.

Definition 6.2

The probability function P , must satisfy several basic axioms:

• (a) $P(B)\ge 0$ for all $B\in F$

• (b) $P(\Omega )=1$

• (c) $P(B\cup C)=P(B)+P(C)$ if $B\cap C=\emptyset$, for all $B,\ C\in F$.

Proposition 6.1

(Facchinei and Pang 2003) Let $f=g\circ G$, where $G:\ R^{n}_{}\rightarrow R^{m}_{}$ is locally Lipschitz continuous at x and where $g:\ R^{m}_{}\rightarrow R^{}_{}$ is locally Lipschitz at G(x). Then f is locally Lipschitz continuous at x and

$$\begin{aligned} \partial f(x)\subseteq \text {conv}\{\xi =H^{T}_{}\zeta :\ H\in \partial G(x),\ \zeta \in \partial g(G(x))\}. \end{aligned}$$

• (a)  g is continuously differentiable at G(x);

• (b)  g is C-regular at G(x) and G is continuous differentiable at x (in this case, f is C-regular at x).

Proposition 6.2

(Facchinei and Pang 2003) Let $G:\ R^{n}_{}\rightarrow R^{m}_{}$ be a locally Lipschitz on an open set $\Omega$. If $x\in \Omega$, then

$$\begin{aligned} \partial G(x)\subseteq (\partial G^{}_{1}(x)\times \partial G^{}_{2}(x)\times \cdots \times \partial G^{}_{m}(x))^{T}_{}. \end{aligned}$$