The efficient frontier

In this section, we compare of the performance of the standard B&B and layered B&B algorithms by examining their performance on obtaining the efficient frontiers of the five problems. We plot the efficient frontiers for the five instances Port 11, Port 21, Port 31, Port 41 and Port 51. For each instance, we have computed the mean-variance frontier by setting the expected return from 0.2% to 0.8% with the step of 0.012 (50 portfolios are thus plotted for each efficient frontier). Linear interpolation is used to plot the intermediate values.

To obtain the exact efficient frontier for the constrained problem, model (7-7) needs to be solved optimally for each expected returnR. However, as we stated before, this is not achievable due to the introduction of the integer constraints. We therefore use the CPLEX 10.0 solver to compute an approximate efficient frontier for model (7-7) by running the algorithm for a long computational time. For each point on the efficient frontier, the running time limit is set as 60 seconds. We denote this as the default efficient frontier.

Fig. 7.4 illustrates the default efficient frontiers calculated by the default CPLEX solver with a long running time, and the efficient frontiers obtained from the layered B&B.

Here, in the layered B&B, we solve the subproblem at the top layer to optimal before going down to the second layer. The time spent on each point on the efficient frontier is given in Table 7.4, i.e. around 0.02 to 2 seconds for each point. This is much less than the time limit for the default CPLEX solver which is 60 seconds.

Fig. 7.4 Efficient frontiers from the default B&B and layered B&B

From Fig. 7.4 we can see that the quality of solutions from the layered B&B is extremely good. For instance Hang Seng, the average gap between the default efficient frontier and our layered B&B efficient frontier is under 0.01%. For instance DAX, the

gap for portfolios with lower risk is larger than those of with higher risk, but the overall gap is still quite small. The gaps for instances FTSE and S&P are quite small as well. However, for large instances Nikkei, the gap is relatively larger.

Due to the additional constraints introduced in our problem formulations, it is difficult to compare the results against those from other approaches in the literature. Even for approaches that are tested on the same OR Library instances, a fair comparison is still difficult to conduct as, to the best of our knowledge, all the existing approaches in the literature analyze the results of problems with only a single additional constraint. Our approach considers the model with all three constraints simultaneously for the first time.

In order to more accurately compare our results against the default frontier obtained by B&B with a long running time, we compute the percentage deviations of each portfolio from the default frontier by calculating the distance between the risk obtained by our layered B&B algorithm and that on the default frontier. This measure represents the deviation of the obtained solution from an approximation of the exact solution, and is also used in [109] [112] to evaluate the quality of results and provide an indication of the performance of the algorithm. Table 7.5 presents the comparisons between heuristic approaches.

In Table 7.5, the genetic algorithm, tabu search and simulated annealing methods investigated in [109] are pure metaheuristic methods. Pooled (GA, TS, SA) method, also investigated in [109], combines the three sets of non-dominated points given by the three algorithms into one set. Those points which are dominated are eliminated from this new set. In [112], the integer restart method applies the previous integer solution of QP as the first feasible solution and the upper bound of the following QP.

In Table 7.5, the percentage deviations of our method are quite small compared with other approaches. One thing to note is that the problems solved by the integer restart method are formulated in three independent models; each consisting of one of the three additional constraints investigated in this chapter, while our model consists of all the

three additional constraints. Compared with these simpler models, the percentage deviations of our solutions are still highly competitive.

Table 7.5 Comparisons of the layered B&B with existing approaches in the literature

Instance Solution method Percentage

deviation Hang Seng GA 0.94570 TS 0.99080 SA 0.98920 Pooled (GA,TS,SA) 0.93320 Integer restart 0.01415 Layered B&B 0.00008 DAX GA 1.9515 TS 3.06350 SA 2.42990 Pooled (GA,TS,SA) 2.19270 Integer restart 0.01399 Layered B&B 0.00992 FTSE GA 0.87840 TS 1.39080 SA 1.13440 Pooled (GA,TS,SA) 0.77900 Integer restart 0.01141 Layered B&B 0.02074 S&P GA 1.71570 TS 3.16780 SA 2.69700 Pooled (GA,TS,SA) 1.31060 Integer restart 0.01586 Layered B&B 0.03079 Nikkei GA 0.6431 TS 0.9891 SA 0.6370 Pooled (GA,TS,SA) 0.5690 Integer restart 0.00618 Layered B&B 0.03901

7.5 Conclusions

In this chapter, we extend the Markowitz mean-variance portfolio selection problems with three additional real-world trading constraints simultaneously in a single model. The resulting formulation, which is a Mixed Integer Quadratic Programming problem, thus has different features corresponding to different groups of integer variables (i.e. binary variable and general integer variable). These features motivate the development of a decomposition approach, layered Branch-and-Bound (layered B&B) algorithm, for solving the problem which we are concerned with. The work has been submitted to Journal of Heuristics, see List of Publications.

In the B&B search tree, sets of variables are layered (decomposed) according to their different features, and search is performed on one layer before another in sequence. Two tailored branching heuristics and one node selection heuristic are applied to individual layers of the B&B tree in order to speed up the search for optimal solutions. The performance of the layered B&B is analyzed and compared based on the extended instances in the OR Library with all three additional constraints. The efficient frontiers are plotted for each instance to provide a graphic illustration of the results. It can be seen that the quality of solutions from the layered B&B algorithm is extremely good, with a much less computational time, compared with the default B&B.

The layered B&B algorithm can be seen as firstly searching on the top layer of the tree (subproblem of a set of variables) thendivinginto a particular region of the search space in order to explore it intensively. In this chapter, both layers are searched by the B&B, and with different tailored branching rules to the corresponding layer, i.e. these branching rules are tailored explicitly to the features of the variables.

The layered B&B for the portfolio selection problems with integer constraints is proposed based on the features of different variables in the problem, i.e. the B&B tree is layered to multi-levels accordingly to the binary variables and general integer variables.

For problems without this specific feature, the layered B&B technique can still be generalized and applied to solve the problem accordingly.

One possible generalization of the layered B&B is to apply it in solving problems represented as decomposed constraint graphs. Constraint graph has been widely used in the literature, especially in constraint satisfaction, where nodes represent variables and edges represent constraints. By decomposing the constraint graph (e.g. using the clique partition), the problem can be partitioned into subproblems (e.g. cliques). Search can then be applied on each layer corresponding to each subproblem. This generalised framework of layered B&B provides the possibility of applying different search methods on each layer. It is not necessary to perform the B&B on all layers. In the next chapter, different algorithms including local search will be introduced into the layered B&B algorithm, with the exact B&B search, to solve different decomposed subproblems in the layered tree.

Chapter 8 A hybrid local search and Branch-and-