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: \(R \succ_{_{SSD}} R' \Leftrightarrow R \succeq_{_{SSD}} R' \mbox{ and } R' \not{\succeq_{_{SSD}}} R.\)

• 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 \(R\).

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

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.

Expression Trees

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

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;

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;