Effects of Temporal Memory

The Rsize determines the extent in which information about past PS∗t is stored. A large Rsize allows a higher degree of information exploitation at the expense of a more diverse repertoire of past PS∗_t. On the other hand, very limited information regarding each past P S∗

t is avaliable when Rsize is small. The relationship between Rsize = {0, 5, 10, 20} with various settings of nT and τT for FDA1 are shown in Figure. 6.16. Note that no memory is retained at Rsize = 0. These relationships are similarly investigated for dMOP1, dMOP2 and dMOP3 and illustrated in Figure. 6.18-6.20.

CHAPTER 6. 171 0 5 10 20 0.1 0.15 0.2 0.25 0.3 0.35 VD offline SC_ratio (a) 0 5 10 20 0.99 0.992 0.994 0.996 0.998 MS offline SC_ratio (b) 0 5 10 20 0 0.1 0.2 0.3 0.4 0.5 VD offline SC_ratio (c) 0 5 10 20 0.985 0.99 0.995 MS offline SC_ratio (d)

Figure 6.19: Performance metrics of (a) VDof f lineand (b) MSof f lineat nt=1.0 (4), nt=10.0 (◦), and nt=20.0 (2) and (c) VDof f line and (d) MSof f line at τT=5.0 (4), τT=10.0 (◦), and τT=25.0 (2) for dMOP2 over different settings of Rsize

0 5 10 20 0.01 0.02 0.03 0.04 0.05 0.06 VD offline SC_ratio (a) 0 5 10 20 0.93 0.94 0.95 0.96 0.97 0.98 0.99 MS offline SC_ratio (b) 0 5 10 20 0 0.02 0.04 0.06 0.08 0.1 0.12 VD offline SC_ratio (c) 0 5 10 20 0.88 0.9 0.92 0.94 0.96 0.98 1 MS offline SC_ratio (d)

Figure 6.20: Performance metrics of (a) VDof f lineand (b) MSof f lineat nt=1.0 (4), nt=10.0 (◦), and nt=20.0 (2) and (c) VDof f line and (d) MSof f line at τT=5.0 (4), τT=10.0 (◦), and τT=25.0 (2) for dMOP3 over different settings of Rsize

mances at higher τT and at nT = 1.0 for FDA1, dMOP2 and dMOP3 can also be observed over the different Rsize settings. Considering the contribution of temporal memory to the tracking capability of dCOEA, Figure. 6.18-6.20 show that the incorporation of appropri- ately sized memory tends to improve convergence as indicated by the metric of VDof f line. The only exception occurs for the case of dMOP1 at the setting of nT = 1.0 and τT = 5.0. The tradeoff between exploration and exploitation of information is also evident from the performance trend with increasing Rsize. For instance, when repetition of similar P St∗ is very frequent as in the case of nT = 1.0, a large Rsize can be used to mine information from past P S_t∗ since the number of different PS∗_t that needs to be represented in the memory is

small. Vice versa, only a small Rsize should be applied when number of different PS∗t over

time is higher.

6.6

Conclusion

This chapter presented a new coevolutionary paradigm that incorporates both competitive and cooperative mechanisms observed in nature to solve MO optimization problems and to track the Pareto front in a dynamic environment. The proposed competitive-cooperation coevolution is capable of overcoming the limitations of conventional coevolutionary mod- els by allowing the decomposition process of the optimization problem to emerge based on problem requirements as well as exploiting the high speed of convergence to allow the algo- rithm to adapt quickly to the changing environment. Based on this coevolutionary model, a competitive-cooperation coevolutionary algorithm (COEA) is proposed for multi-objective optimization. Subsequently, this algorithm is extended as a dynamic COEA (dCOEA) and incorporated the features of stochastic competitors that allows the algorithm to track the changing solution set and temporal memory that allows the algorithm to exploit past in- formation. Extensive studies upon three benchmark problems demonstrates that COEA is capable of evolving near-optimal, diverse and uniformly distribution Pareto fronts even for problems with severe parameter interactions. The parameter settings and working dynamics of the competitive mechanism as well as different competitive schemes are also examined, illustrating the robustness and importance of both competitive and cooperative elements in a common framework. Likewise, extensive studies are performed to investigate the perfor- mances of dCOEA over different settings of change severity and change frequency. Simula- tion results shows that dCOEA is capable of tracking the different environmental changes in the test functions employed effectively and efficiently. In addition, the contribution and parameter settings of the diversity scheme and the temporal memory are also analyzed over various problem settings.

Chapter 7

An Investigation on Noise-Induced

Features in Robust Evolutionary

Multi-Objective Optimization

Branke [19] considered robust optimization as a special case of dynamic optimization where solutions cannot be adapted fast enough to keep pace with environmental changes. In such cases, it would be desirable to find solutions that perform reasonably well within some range of change. Many real-world applications are susceptible to decision or environmental parameter variation which results in large or unacceptable performance variation. Robust optimization of MO problems is the third and final type of uncertainty considered in this work and it involves the optimization of a set of Pareto optimal solutions that remain satisfactory in the face of parametric variations. This chapter addresses the issue of robust MO optimization by presenting a robust continuous MO test suite with features of noise- induced solution space, fitness landscape and decision space variation. In addition, the vehicle routing problem with stochastic demand (VRPSD) is presented a practical example of robust combinatorial MO optimization problems.

7.1

Robust measures

In order to avoid any confusion in the subsequent discussions, it will be instructive to make a distinction between the notations used for deterministic MO and robust MO optimization. The terms PF∗and PS∗ refer to the desired Pareto front and solution set in the general sense, without representing any specific case. The optimal Pareto front and the corresponding Pareto solution set of a particular deterministic MO problem will be denoted as PF∗_det and PS∗_det respectively. Note that PF∗_det may not be known a priori and it is fixed for any particular MO problem. The final set of nondominated solutions evolved by MOEA will be termed as PFA_det.

In the case of robust MO optimization, the optimal robust Pareto front and solution set are also dependent on the noise model and the robust measure. This implies that, contrary to PF∗and PS∗, the optimal robust Pareto set is not fixed. Furthermore, the structure of the Pareto front, i.e. its dimensionality may change as well due to the additional optimization criteria of robustness. Therefore, the notation should reflect the noise model and the robust measure used. In this paper, the optimal robust Pareto front and optimal solution set are denoted as PF∗_rm,σ and PS∗_rm,σ respectively. The terms rm and σ refers to the robust measure and noise model in consideration. Accordingly, PFA_rm,σ refers to the final set of nondominated solutions evolved by robust MOEA based on the robust measure, rm and noise model, σ.

There are several possible notions of robustness and many different robust measures have been applied in the literature. The most popular and straight-forward measure is the optimization of the expected performance over the possible disturbances, i.e. E(fi) =

1

N ·

PN

i=1fi(~x + ~σi). Solutions that are optimized based on expected fitness are known as

effective solutions. Hence, for MO optimization, the resulting Pareto front is known as the effective Pareto front (PFA_{ef f,σ}). Other measures includes the optimization of the worst case scenario [155], as a constraint to be satisfied [40], and various forms of variances.

CHAPTER 7. 175 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A

Figure 7.1: Illustration of the different robust measures, constrained (– –), standard devia- tion (- - -), effective (-·-·) and worst case (· · ·), with respect to the deterministic landscape (—)

Each of these robust measures reflects the different aspects of robustness and Figure 7.1 illustrates the behavior of the different robust measures for an arbitrary function of varying sensitivities in the search space. The various plots are generated by sampling the values of x with uniform distribution of [−0.025, 0.025]. If the model is known with absolute certainty and the solution can be implemented exactly, then the global optimal represented by the deterministic solution at x = 0.5 is the ideal solution. However, if variable x is stochastic, then the solutions presented by the other approaches will be more viable and the location of the optimal is also different. In particular, it can be noted the expected mean approach will favor the solution at x = 0.11 while the approaches based on variance and worst case will favor the solution at x = 0.75. On the other hand, the constrained approach indicates the feasible solutions which satisfies the pre-defined criteria.