A Solver for Problems with Second-Order Stochastic Dominance Constraints ======================================================================== Victor Zverovich, Gautam Mitra, Csaba Fábián
Second-Order Stochastic Dominance ---------------------------------

Let $$R$$ and $$R'$$ be random variables defined on the probability space $$(\Omega, \mathcal{F}, P)$$.

$$R$$ dominates $$R'$$ with respect to SSD if and only if $$\textrm{E}[U(R)] \ge \textrm{E}[U(R')]$$ for any nondecreasing and concave utility function $$U$$.

This sets out the use of SSD relation to determine preferences of a risk-averse decision maker.

Denoted as $$R \succeq_{_{SSD}} R'$$.

Strict relation: $$R \succ_{_{SSD}} R' \Leftrightarrow R \succeq_{_{SSD}} R' \mbox{ and } R' \not{\succeq_{_{SSD}}} R.$$

Alternative Definitions of SSD ------------------------------
• Definition using the performance function (Fishburn and Vickson, 1978): $F^{(2)}_R(t) \leq F^{(2)}_{R'}(t) \mbox{ for all t \in \mathbb{R},}$ where the performance function $$F^{(2)}_R(t) = \int_{-\infty}^{t} F_R(u) \mathrm{d}u$$ represents the area under the graph of the cumulative distribution function $$F_R(t) = P(R \leq t)$$ of a real-valued random variable $$R$$.
• Definition using the $$\textrm{Tail}$$ function (Ogryczak and Ruszczyński, 2002): $\textrm{Tail}_{\alpha}(R) \geq \textrm{Tail}_{\alpha}(R') \mbox{ for all 0 \lt \alpha \leq 1,}$ where $$\textrm{Tail}_{\alpha}(R)$$ denotes the unconditional expectation of the smallest $$\alpha \cdot 100\%$$ of the outcomes of $$R$$.
Illustration of Second-Order Stochastic Dominance -------------------------------------------------

CDF

Performance Functions
Portfolio Problem/Constraints -----------------------------
There are $$n$$ assets and at the beginning of a time period an investor has to decide what proportion $$x_i$$ of the initial wealth to invest in asset $$i$$. So a portfolio is represented by a vector $$\textbf{x} = (x_1, x_2, \dots, x_n) \in X \subset \mathbb{R}^n$$, where $$X$$ is a bounded convex polytope representing the set of feasible portfolios; in particular it can be defined as $X = \{\textbf{x} \in \mathbb{R}_+^n: \sum_{i=1}^n{x_i} = 1\},$ if short positions are not allowed and there are no other modelling restrictions. Let $$\textbf{R}$$ denote the $$n$$-dimensional random vector of asset returns at the end of the time period. Then the real-valued random variable $$R_{\textbf{x}} = \textbf{R}^T \textbf{x}$$ is the random return of portfolio $$\textbf{x}$$.
Model of Dentcheva and Ruszczynski ----------------------------------

Dentcheva and Ruszczyński (2006) proposed the following model with an SSD constraint:

$\begin{array}{ll} \mathrm{maximize} & f(x) \\ \mathrm{s.t.} & x \in X, \\ & R_{\textbf{x}} \succeq_{_{SSD}} \widehat{R}, \\ \end{array}$

where $$f$$ is a concave continuous function, $$\widehat{R}$$ is a reference random return such as the return of a stock market index.

Special case: $$f(x) = \textrm{E}[{R_{\textbf{x}}}]$$

Model of Roman, Darby-Dowman, and Mitra ---------------------------------------

Roman et al. (2006) formulated a multiobjective LP model, the Pareto efficient solutions of which are SSD efficient portfolios.

Assuming finite discrete distributions of returns with equiprobable outcomes, Fábián et al. (2009) converted it into a more efficient computational model with single objective and a finite system of inequalities representing an SSD constraint:

$\begin{array}{ll} \mathrm{maximize} & \vartheta \\ \mathrm{s.t.} & \vartheta \in \mathbb{R}, \textbf{x} \in X \\ & \mathrm{Tail}_{\frac{i}{S}}(R_{\textbf{x}}) \geq \mathrm{Tail}_{\frac{i}{S}}(\widehat{R}) + \vartheta, \quad i = 1, 2, \ldots, S. \\ \end{array}$

Here one seeks a portfolio with a distribution which dominates the reference one or comes close to it uniformly (the smallest tail difference $$\vartheta$$ is maximized).

Model with SSD Constraints --------------------------

Fábián et al. (2010) proposed an enhanced version of the model of Roman et al. which is expressed in the following SSD constrained form:

$\begin{array}{ll} \textrm{maximize} & \vartheta \\ \textrm{s.t.} & \vartheta \in \mathbb{R}, \textbf{x} \in X, \\ & R_{\textbf{x}} \succeq_{_{SSD}} \widehat{R} + \vartheta. \\ \end{array}$

In this model one computes a portfolio that dominates a sum of the reference return and a riskless return $$\vartheta$$.

Formulation Using Tails

Let $$S$$ denote the number of equiprobable outcomes,

$$\textbf{r}^{(1)}, \textbf{r}^{(2)}, \ldots, \textbf{r}^{(S)}$$ - the realisations of $$\textbf{R}$$,

$$\widehat{r}^{(1)}, \widehat{r}^{(2)}, \ldots, \widehat{r}^{(S)}$$ - the realisations of $$\widehat{R}$$.

The enhanced model can be formulated as follows:

$\begin{array}{ll} \mathrm{maximize} & \vartheta \\ \mathrm{s.t.} & \vartheta \in \mathbb{R}, \textbf{x} \in X, \\ & \mathrm{Tail}_{\frac{i}{S}}(R_{\textbf{x}}) \geq \mathrm{Tail}_{\frac{i}{S}}(\widehat{R}) + \frac{i}{S} \vartheta, \\ & \quad i = 1, 2, \ldots, S. \\ \end{array}$

Cutting-Plane Formulation Using Tails

Fábián et al. (2009) obtained the cutting-plane representation of the $$\textrm{Tail}$$ function: $\begin{array}{ll} \textrm{Tail}_{\frac{i}{S}}(R_{\textbf{x}}) = & \displaystyle \min \frac{1}{S} \sum_{j \in J_i} \textbf{r}^{(j) T} \textbf{x} \\ & \mbox{such that } J_i \subset \{1, 2, \ldots, S\}, \quad |J_i| = i. \\ \end{array}$

Cutting-plane representation of the enhanced model: $\begin{array}{lll} \textrm{maximize} & \vartheta & \\ \textrm{s.t.} & \vartheta \in \mathbb{R}, \textbf{x} \in X, & \\ & \displaystyle \frac{1}{S} \sum_{j \in J_i} \textbf{r}^{(j) T} \textbf{x} \geq \widehat{\tau_i} + \frac{i}{S} \vartheta, & \forall J_i \subset \{1, 2, \ldots, S\}, \\ & & |J_i| = i, \; i = 1, 2, \ldots, S, \\ \end{array}$ where $$\widehat{\tau_i} = \textrm{Tail}_{\frac{i}{S}}(\widehat{R})$$.

Cutting-Plane Method

By changing the scope of optimisation we get a problem of minimising a piecewise-linear convex function:

$\begin{array}{ll} \mathrm{minimize} & \varphi(\textbf{x}) \\ \mathrm{s.t.} & \textbf{x} \in X, \\ \end{array}$ where $\begin{array}{ll} \varphi(\textbf{x}) =& \displaystyle \max \left( -\frac{1}{i} \sum_{j \in J_i} \textbf{r}^{(j) T} \textbf{x} + \frac{S}{i} \widehat{\tau_i} \right), \\ & \mbox{such that } J_i \subset \{1, 2, \ldots, S\}, |J_i| = i, \\ & i = 1, 2, \ldots, S. \\ \end{array}$

It can be regularised by the level method.

Cut Generation

The cut $$l(x)$$ at the iteration $$k$$ is constructed as follows:

Let $$\textbf{x}^* \in X$$ denote the solution of the approximation function at iteration $$k$$ and $$\textbf{r}^{(j_1^*)} \leq \textbf{r}^{(j_2^*)} \leq \ldots \leq \textbf{r}^{(j_S^*)}$$ denote the ordered realisations of $$R_{\textbf{x}^*}$$.

Select $$\displaystyle i^* \in \textrm{argmax}_{1 \leq i \leq S} \left( -\frac{1}{i} \sum_{j \in J_i^*} \textbf{r}^{(j)T} \textbf{x}^* + \frac{S}{i} \widehat{\tau}_i \right).$$ Then $$\displaystyle l(\textbf{x}) = -\frac{1}{i^*} \sum_{j \in J_{i^*}^*} \textbf{r}^{(j)T} \textbf{x} + \frac{S}{i^*} \widehat{\tau}_{i^*}.$$

Sets $$J_i^* = (j^*_1, \ldots, j^*_i)$$ correspond to ordered realisations.

Why a New Solver? ----------------- * Old implementation: * Cuts are a part of the model * Difficult to reuse * New implementation: * Cuts are added automatically by the solver * Easy to use * "Clean" model * Faster
AMPL Solver Library ------------------- AMPL Solver Library (ASL) is an open-source library for connecting solvers to AMPL. * C interface: - described in [Hooking Your Solver to AMPL](http://www.ampl.com/hooking.html) - used by most solvers * [C++ interface](https://github.com/vitaut/ampl/tree/master/solvers/util): - makes connecting new solvers super easy - type-safe: no casts needed when working with expression trees - efficient: no overhead compared to the C interface - used by several CP solvers and the SSD solver
SSD Solver Architecture -----------------------
* ASL does all the heavy lifting such as interaction with AMPL and an external solver which makes SSD solver implemenation very simple (~300 LOC!) * Function library provides the ssd_uniform function that is translated into an SSD relation by the solver. * External solver is used for subproblems. * Solver library is optional but facilitates testing.

Expression Trees

The solver extracts linear expressions from the expression trees representing arguments of ssd_uniform.

Portfolio Model in AMPL with Cuts ---------------------------------

param nASSET integer >= 0;  # number of assets
set ASSETS := 1..nASSET;    # set of assets

param nSCEN > 0;
param asset_returns{1..nASSET, 1..nSCEN};
param index_returns{1..nSCEN};

param nCUT integer >= 0 default 0; # number of cuts
set CUTS := 1..nCUT;        # set of cuts

param cut_const {CUTS};     # constant in cut
param cut {CUTS,ASSETS};    # multipliers in cut
param scaling_factor {CUTS} default 1;

# portfolio: investments into different assets
var Invest {ASSETS} >= 0 default 1 / nASSET;
var Dom;                    # dominance measure

maximize Uniform_Dominance: Dom;

subject to Dom_constraint {c in CUTS}:
scaling_factor[c] * Dom + cut_const[c]
<= sum {a in ASSETS} cut[c,a] * Invest[a];

subject to Budget: sum {a in ASSETS} Invest[a] = 1;


Portfolio Model in AMPL using SSD Solver ----------------------------------------

include ssd.ampl;

param NumScenarios;
param NumAssets;

set Scenarios = 1..NumScenarios;
set Assets = 1..NumAssets;

# Return of asset a in senario s.
param Returns{a in Assets, s in Scenarios};

# Reference return in scenario s.
param Reference{s in Scenarios};

# Fraction of the budget to invest in asset a.
var invest{a in Assets} >= 0 <= 1;

subject to ssd_constraint{s in Scenarios}:
ssd_uniform(sum{a in Assets} Returns[a, s] * invest[a], Reference[s]);

subject to budget: sum{a in Assets} invest[a] = 1;


Reference Returns -----------------
Performance -----------
* 100 scenario problem with FTSE100 used as a reference. * The new implementation is 2-3 times faster.
Performance -----------
* 30000 scenario problem with FTSE100 used as a reference. * The new implementation is 2-6.5 times faster.
Summary ------- * AMPL solver interface and ASL make implementation of high-level solvers/algorithms that use other solvers easy. The same technique can be applied to - other-cutting plane methods - decomposition methods, e.g. Bender's decomposition * New solver provides an efficient implementation of a cutting-plane algorithm for solving problems with SSD constraints. * This is in line with our approach that different types of optimisation models are matched with corresponding solvers.
References ----------
• Dentcheva, D. and Ruszczyński, A. (2006). Portfolio optimization with stochastic dominance constraints. Journal of Banking & Finance, 30 , 433–451.
• Fábián, C. I., Mitra, G., and Roman, D. (2009). Processing second-order stochastic dominance models using cutting-plane representations. Mathematical Programming, Series A. DOI: 10.1007/s10107-009-0326-1.
• Fábián, C. I., Mitra, G., Roman, D., and Zverovich, V. (2010). An enhanced model for portfolio choice with ssd criteria: a constructive approach. Quantitative Finance. First published on: 11 May 2010.
• Fishburn, P. C. and Vickson, R. G. (1978). Theoretical foundations of stochastic dominance. In Stochastic Dominance: An Approach to Decision-Making Under Risk, (pp. 37–113). D.C. Heath and Company, Lexington, Massachusetts.
• Roman, D., Darby-Dowman, K., and Mitra, G. (2006). Portfolio construction based on stochastic dominance and target return distributions. Mathematical Programming, eries B, 108, 541–569.
Thank you! ----------