Scalable benchmark problems
Figure 3.16 shows the average number of evaluations (over 100 independent runs) spent by MO-GOMEA and NSGA-II on scalable benchmark problems until the whole Pareto-optimal fronts are obtained. MO-GOMEA variants clearly outperform NSGA-II variants on solving the Trap-Inverse Trap problem. Trap functions can only be efficiently solved by optimizers that have linkage learning abilities, which NSGA-II does not employ. For the Zeromax-Onemax problem, where all variables are independent and linkage learning is not necessary, MO-GOMEA variants still have a better scalability than NSGA-II variants. NSGA-II variants perform better than MO-GOMEA variants, however, when solving the LOTZ problem due to the mutation operator as discussed in Section 3.5.2. Section 5 above shows that if MO-GOMEA is coupled with a mutation operator, it can also solve LOTZ easily and does so more efficiently than NSGA-II. For these three scalable benchmark problems, the termination criterion of small populations shows no influence on MO-GOMEA while it does result in small improvements for NSGA-II (the differences are found to be statistically significant).
Section 3.5.2 shows that NSGA-II with a population of size 4 can solve all Zeromax-Onemax and LOTZ problem instances more efficiently compared to MO-GOMEA. In fact, for these two problems, NSGA-II does not need to employ any solution recombination but only a mutation operator. NSGA-II simply needs to run many generations and wait for the right bits to be flipped at the right time
3.8. Comparison with the NSGA-II and the influence of stopping (inefficient) small populations
1e+02 1e+03 1e+04 1e+05 1e+06 1e+07
25 50 100 200 400
ZEROMAX - ONEMAX
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25 50 100 200 400
TRAP - INVERSE TRAP
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25 50 100 200 400
LOTZ
NSGA-II base 2 NSGA-II base 4
NSGA-II stop base 2 NSGA-II stop base 4
MO-GOMEA no mut.
MO-GOMEA stop no mut.
Figure 3.16: Performance of MO-GOMEA and NSGA-II when terminating small populations on scalable benchmark problems. Horizontal axis: Problem size. Vertical axis: The number of evalu-ations until the whole Pareto-optimal front is obtained.
to obtain a Pareto-optimal solution. The population-sizing-free NSGA-II variant, however, cannot solve Zeromax-Onemax nor LOTZ as efficiently as the original NSGA-II. The population-sizing-free scheme that we employ here introduces too many large populations for NSGA-II too quickly and all fitness evaluations are used up before the meaningful mutation events occur.
MAXCUT & Knapsack
Figures 3.17 and 3.18 show the performance of MO-GOMEA and NSGA-II with and without terminating inefficient populations on MAXCUT and knapsack respectively.
In almost all cases, the termination criterion improves (statistically) significantly the performance of NSGA-II. This confirms the fact that when small populations are inefficient for NSGA-II, they should be terminated as soon as possible so that fitness
0
0e+00 1e+04 2e+04 3e+04 4e+04 5e+04
MAXCUT 12
0e+00 1e+05 2e+05 3e+05 4e+05 5e+05
MAXCUT 25
0e+00 1e+06 2e+06 3e+06 4e+06 5e+06
MAXCUT 50
0e+00 4e+06 8e+06 1e+07 2e+07 2e+07
MAXCUT 100
Figure 3.17: Average convergence performance of MO-GOMEA and NSGA-II when terminating small populations on MAXCUT. Horizontal axis: number of evaluations (both objectives per evaluation). Vertical axis: DPF→S.
evaluations would not be wasted on them. Additionally, Figures 3.17 and 3.18 show again that base 2 is a better setting for NSGA-II than base 4. It was suggested that smaller base values are more suitable for the parameter-less GA if it suffers from the effects of genetic drift [29]. Diversity preservation is an important task in multi-objective optimization, and the base-2 parameter-less scheme introduces larger population sizes with more diverse candidate solutions at a faster rate than the base-4 scheme. NSGA-II with base 2 coupled with the termination criterion of small populations can get rid of small and inefficient populations and move to sufficiently larger populations more quickly.
The termination criterion of small populations, however, shows little or insignifi-cant influence on the performance of MO-GOMEA. This suggests that MO-GOMEA can operate effectively with small populations and that indeed the shared use of the elitist archive in all populations make terminating smaller, inefficient populations
3.9. Conclusions
0e+00 1e+06 2e+06 3e+06 4e+06 5e+06
KNAPSACK 100
0e+00 2e+06 4e+06 6e+06 8e+06 1e+07
KNAPSACK 250
0e+00 3e+06 6e+06 9e+06 1e+07 2e+07
KNAPSACK 500
0e+00 4e+06 8e+06 1e+07 2e+07 2e+07
KNAPSACK 750
Figure 3.18: Average convergence performance of MO-GOMEA and NSGA-II when terminating small populations on Knapsack. Horizontal axis: number of evaluations (both objectives per evaluation). Vertical axis: DPF→S.
unnecessary. What we observe here conforms with previous research on the scalabil-ity of GOMEAs in single-objective optimization, in which GOMEAs generally have minimally required population sizes that are much smaller than other population-based EAs [12, 26].
3.9. Conclusions
We have presented the multi-objective GOMEA (MO-GOMEA). We have shown that for the combination with the linkage tree model, superior scalability for solv-ing different classes of MO optimization problems can be achieved as compared to classic GAs (i.e. NSGA-II) and even state-of-the-art EDAs (i.e. mohBOA). Our ex-perimental results further support that the key features of scalable MO optimizers that we identified and incorporated into MO-GOMEA are indeed responsible for
the observed performance. These features are: an elitist archive to keep track of the non-dominated front, clustering to process different regions of the front differ-ently, linkage learning and an efficient mechanism for exploiting the learned linkage relations to generate offspring solutions.
Population clustering ensures that MO-GOMEA allocates an equal amount of search effort to every region and the whole Pareto-optimal front can thus be evenly approached. Especially the cluster-based operating mechanism of MO-GOMEA is convenient for dedicated adaptations if different regions of the Pareto-optimal front have different characteristics and thus require different strategies to exploit problem structure effectively and efficiently. In the multi-objective knapsack benchmark (see Sections 3.7 and 3.8), by clustering the working population, it is straightforward to assign the multi-objective repair mechanism to the middle-region clusters and the suitable single-objective repair mechanism to the corresponding extreme-region cluster. Population clustering helps MO-GOMEA score on the diversity part of the DPF→S performance indicator.
As each cluster of MO-GOMEA approaches a specific region of the Pareto-optimal front, linkage learning captures problem-variable dependencies that are relevant to that region. Following the structure of the linkage tree dedicatedly learned from a cluster, the Gene-pool Optimal Mixing operator creates new can-didate solutions by juxtaposing currently existing building blocks in a way that is specifically suitable to that cluster. The genetic local search nature of Gene-pool Optimal Mixing also ensures that an offspring is better or at least as good as its parent solution. Linkage learning and Gene-pool Optimal Mixing together ensure that the building blocks relevant to each cluster are detected and propagated to ensure effective convergence toward the Pareto-optimal front, helping MO-GOMEA score on the proximity part of the DPF→S performance indicator.
The combined effect of clustering the population and exploiting linkage infor-mation results in the better performance for MO-GOMEA over other MOEAs.
We then made MO-GOMEA an easy-to-use solver by placing MO-GOMEA in a population-sizing-free framework that eliminates the required setting of the popu-lation size parameter, which is notoriously difficult for any popupopu-lation-based EA, and the number-of-clusters parameter. As a consequence, users now only specify how long the algorithm is allowed to run. Alternatively, MO-GOMEA can be used as an anytime algorithm, i.e., the more time it runs, the better solutions would be found, and it can be terminated when a satisfying solution is obtained. The pa-rameter setting-free MO-GOMEA was shown to retain the scalability of the original MO-GOMEA and to have excellent performance on a wide range of benchmark prob-lems. The scalability and usability of MO-GOMEA suggest that MO-GOMEA is a promising solver for tackling complicated (real-world) multi-objective optimization problems. We look at the application of MO-GOMEA to solving the multi-objective DNEP problem later in Chapter 6.
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4
Static Distribution Network Expansion Planning
Meet driemaal eer gij eens snijdt.
Measure thrice, cut once.
Dutch proverb
This chapter tackles the Distribution Network Expansion Planning (DNEP) problem that has to be solved by distribution network operators to decide which enhancements to electricity networks should be introduced to satisfy the future power demands.
We consider three types of evolutionary algorithms (EAs) for optimizing expansion plans: the classic Genetic Algorithm (GA), the Estimation-of-Distribution Algo-rithm (EDA), and the Gene-pool Optimal Mixing Evolutionary AlgoAlgo-rithm (GOMEA).
Not fully knowing the structure of the problem, we study the effect of linkage learning through the use of three linkage models: univariate, marginal product, and linkage tree. We furthermore experiment with the impact of incorporating different levels of problem-specific knowledge in the variation operators. Based on experimental results, we suggest that when selecting optimization algorithms for real-world ap-plications like DNEP, EAs that have the ability to effectively model and efficiently exploit problem structures, such as GOMEA, should be given priority, especially in the case of black-box or grey-box optimization. The best performance is obtained when both linkage information and problem-specific knowledge can be exploited.
Parts of this chapter have been presented at EA ’13 [1], PSCC ’14 [2], GECCO ’15 [3] and published in [4].
4.1. Introduction
Peak loads on distribution networks normally increase every year due to devel-opments in residential and industrial electricity consumption. Consequently, the magnitude of the power flows that are carried through network components (e.g., cables, transformers) to satisfy customers’ power demands will at some point exceed the currently existing network capacity. In order for distribution networks to work properly, distribution network operators (DNOs) have to ensure that the capacities of network assets are sufficient to handle the magnitude of the required power flows.
Otherwise, bottlenecks can cause overloads, which heat up the cable wires. This is detrimental to the normal operation and safety of the networks, and may cause blackouts or earlier asset replacements. Therefore, DNOs need to perform distri-bution network expansion planning (DNEP) to determine where on the networks asset reinforcements should be made and what types of devices should be installed there. The dynamic DNEP formulation also involves the question when those en-hancement activities should be started during the planning period while in the static DNEP formulation this time-dependent decision making issue is omitted. The static DNEP problem is the focus of this chapter, and its dynamic version will be tackled in Chapter 5. The goal of DNEP is to find the most economical expansion plan, in terms of investment and/or operation costs, for which the network satisfies the power demand over the planning period.
Evolutionary algorithms (EAs) have been widely applied and achieved practical results in DNEP, see e.g., [5–8]. This is mostly due to the straightforward imple-mentation and broad applicability of EAs. However, most DNEP studies in liter-ature overlook several important issues when employing EAs. First, experiments are usually conducted by using only one, arbitrarily chosen, EA with a customized problem-specific variation operator (VO), omitting both questions why that spe-cific EA should be chosen over other available EAs and what the advantages that VO has compared to other alternatives. Second, the comparison of how effective various constraint-handling mechanisms help the solvers traverse the search space is often disregarded. In this chapter, while aiming to solve a formulation of the DNEP problem that captures many important real-world considerations, we also address these issues. We employ three EA solvers: a classic Genetic Algorithm (GA), a Estimation-of-Distribution Algorithm (EDA), and a Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) [9, 10]. The GA is arguably the most popular EA in DNEP literature, but it is rarely used out of the box in practice.
Practitioners often customize its VOs (i.e., crossover and mutation) with expert and problem-specific knowledge (PSK) so that important problem structures are re-spected during variation, e.g., cables in the same feeder group in the network should be treated together when constructing new networks. Taking the perspective of black-box optimization, where such PSK is assumed to be hardly available, linkage learning (LL) can be performed to identify, during optimization, which variables are inter-dependent and should thus be jointly considered when generating new so-lutions. EDAs, such as BOA [11] or ECGA [12], are well-known examples of EAs that build probabilistic models that exhibit a degree of variable dependency that is aligned with variable linkage to effectively generate high-quality solutions.