## 6 Discussion of Results  
### CEF-ARPO with Certain Inputs  
We evaluate the quality of our ARPO algorithm by means of the so called _average percentage loss_ (APL). APL can be defined as follows. Let \(R_{h}\) be the minimum required rate of return, and \(V_{CEF}(R_{h})\) and \(V_{UEF}(R_{h})\) ( \(\forall h\in\{1,2,\ldots,H\}\), where \(h\) denotes a point of the frontier, and \(H\) denotes the number of points on the frontier) the values of \(f(x)\) evaluated in the solutions to the constrained and unconstrained optimization problem, henceforth CEF and UEF, respectively. The latter is provided in ORlib, whereas the former is returned by ARPO. Then, \(APL=\frac{100}{H}\sum_{h=1}^{H}\frac{V_{CEF}(R_{h})-V_{UEF}(R_{h})}{V_{UEF}(R_ {h})}\). Table 1 shows the values of APL and associated computational times. Notice that, in terms of the minimum APL, our ARPO algorithm outperforms on Instances 2-5 the hybrid solvers proposed by [41], which comprise combinations of first descent and steepest descent with quadratic programming (FD + QP and SD + QP, respectively). With regard to the first instance, our APL is greater, but this result may emanate from rounding errors. In terms of computational time, ARPO shows a superior performance relative to that of the solver's SD + QP and is comparable or better than the solver's FP + QP performance. We next contrast our results with the results reported by [39; 40]. Although the minimum APL provided by ARPO is slightly superior to the hybrid solver combining a genetic algorithm (GA) and quadratic programming (QP) in [40], on the remaining instances the minimum APL accomplished by ARPO is lower. Furthermore, our computational times are considerably lower than those reported by the tabu search (TS) in [39], and by GA + QP in [40]. The UEF (as provided in the ORlib) and CEF (as provided by ARPO) for the five stock market indices are compared in Panels A-E of Figure 1. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{2}{c}{**[PERSON] et al. [41]**} & \multicolumn{2}{c}{**[PERSON] et al. [40]**} & \multicolumn{2}{c}{**[PERSON] [39]**} & \multicolumn{2}{c}{**ARPO**} \\ \cline{2-10}  & \multicolumn{2}{c}{**FD + QP**} & \multicolumn{2}{c}{**SD + QP**} & \multicolumn{2}{c}{**GA + QP**} & \multicolumn{2}{c}{**TS**} & \multicolumn{2}{c}{**LS + QP**} \\ \cline{2-10}  & **APL** & **T(s)** & **APL** & **T(s)** & **APL** & **T(s)** & **APL** & **T(s)** & **APL** & **T(Std) (s)** \\ \hline HS & 0.00366 & 1.5 & 0.00321 & 3.1 & 0.00321 & 415.1 & 0.00409 & 251 & 0.00399 & 2.0 (0.87) \\ DAX 100 & 2.66104 & 9.6 & 2.53139 & 14.1 & 2.53180 & 552.7 & 2.53617 & 531 & 2.45403 & 1.0 (1.0) \\ FTSE 100 & 2.00146 & 10.1 & 1.92146 & 16.1 & 1.92150 & 886.3 & 1.92597 & 583 & 1.88340 & 13.7 (9.28) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of Results. Panel A of Figure 1 depicts the CEF for the Hang Seng (Hong Kong) stock market. A visual inspection suggests that for the Hang Seng stock index the CEF is hardly distinguishable from the UEF. However, as the rate of expected return increases, along with increasing risk of investment, the CEF tends to diverge relatively less from the UEF. In particular, at the higher end of the CEF that features rewarding but risky portfolios, the expected rate of return can be attained with fewer assets. Panel B of Figure 1 depicts the CEF for the DAX 100 (Germany) stock market. Visual inspection indicates that for the DAX 100 stock index the CEF diverges from the UEF at the lower end of expected return, more specifically, for \(R<0.006\). As the rate of expected return increases, the CEF becomes indistinguishable from the UEF. Notably, portfolios with an expected return at the lower end of the CEF tend to be riskier (i.e., with higher portfolio variance) than portfolios on the UEF.