A New Multi-Objective Algorithm, Pareto Archived DDS Masoud Asadzadeh
Bryan A. Tolson
Department of Civil & Environmental Engineering University of Waterloo 200 University Avenue West Waterloo, Ontario, N2L 3G1, CANADA (+1) 519-781-2144
Department of Civil & Environmental Engineering University of Waterloo 200 University Avenue West
[email protected]
[email protected]
Waterloo, Ontario, N2L 3G1, CANADA (+1) 519 888 4567 ext. 33377
ABSTRACT
1. INTRODUCTION Engineering design problems can often be cast as optimization problems that usually have more than one objective. The goal of solving such a multi-objective problem (MOP) is to estimate the tradeoff between objectives. High quality estimates of the tradeoff include points that are as close as possible to the actual tradeoff and as broadly spread out as possible. In this study a minimization unconstrained problem with n objectives is considered F(x) = (f1(x), f2(x), … , fn(x)). Multi-objective evolutionary algorithms (MOEAs) that also called ideal approaches [2] for solving MOPs search for multiple tradeoff solutions simultaneously and are not very sensitive to the shape of the tradeoff [1]. Since 1984 when the first MOEA based on genetic algorithm (GA) was introduced, several other versions of MOEAs based on GAs have been introduced such as SPEA2 [9], NSGAII [3]. Other MOEAs include PAES [4] and AMALGAM [7]. Almost all MOEAs are modified versions of single objective evolutionary algorithms that basically archive the non-dominated solutions during the search.
The dynamically Dimensioned Search (DDS) continuous global optimization algorithm [5] is modified to solve continuous multiobjective unconstrained optimization problems. Inspired by Pareto Archived Evolution Strategy (PAES), the proposed multiobjective optimization, PA-DDS uses DDS as a search engine and archives all the non-dominated solutions during the search. In order to maintain the diversity of solutions, PA-DDS, which is single solution based, samples from less crowded parts of the external set of non-dominated solutions in each iteration. This tool inherits the parsimonious characteristic of DDS, so it has only one algorithm parameter from DDS, which does not need tuning, and one new parameter that defines the portion of computational budget for finding individual minima. PA-DDS uses crowding distance measure to sample from less populated parts of the tradeoff. The performance of the proposed tool is assessed in solving two test problems ZDT4 and ZDT6 [8] that have multiple local Pareto fronts. Results show that PA-DDS is promising relative to two high quality benchmark algorithms NSGA-II [3, 7] and AMALGAM [7].
This paper introduces a new multi-objective algorithm based on a relatively new and efficient single objective optimizer called the Dynamically Dimensioned Search (DDS) algorithm [5]. The DDS continuous global optimization algorithm is a neighbourhood search strategy that is modified to solve continuous multiobjective unconstrained optimization problems. Inspired by the Pareto Archived Evolution Strategy or PAES (see [4]), the proposed multi-objective optimization, Pareto Archived DDS (PA-DDS) uses DDS as a search engine and archives all the nondominated solutions during the search.
Categories and Subject Descriptors I.2.8 [Artifcial Intelligence]: Problem Solving, Control Methods, and Search – Heuristic methods, Multi-objective Optimization.
General Terms Algorithms, Design, Performance.
2. Background on DDS The DDS algorithm is extremely simple and described in full in [5]. In general, the DDS algorithm searches for the solution both globally and locally because it starts the search in all dimensions of the decision space and reduces the number of dimensions as the search goes on. This is the key aspect of DDS – dynamically reducing the dimensionality of the neighbourhood as a function of the relative computation budget that has been consumed. The dimensions (decision variables) considered in the neighborhood are determined probabilistically. Decision variables in the neighborhood are each perturbed with an independent normal random variable, that is reflected at decision variable bounds if necessary, in order to generate a candidate solution.
Keywords Convergence, Crowding Distance, Dynamically Dimensioned Search, Multi-Objective optimization, Pareto Archive, Parsimony, Test Problems.
Several characteristics of DDS make it desirable as a multiobjective search engine. First of all, DDS is a parsimonious algorithm as it has only one algorithm parameter and this parameter is shown to have a robust default setting such that tuning it for each problem is not recommended [5]. Second, the
Copyright is held by the author/owner(s). GECCO’09, July 8–12, 2009, Montréal Québec, Canada. ACM 978-1-60558-505-5/09/07.
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algorithm adjusts the search strategy to the user input computational budget. Thirdly, DDS has shown excellent relative performance in solving computationally intensive environmental simulation model automatic calibration problems [5]. Lastly, a discrete version of DDS has been developed and therefore the PADDS proposed here can also apply to discrete optimization problems. This discrete version of DDS is used for solving single objective optimization water distribution network (WDN) design problems [6]. Discrete DDS outperformed the state-of-the-art existing single objective optimization algorithms for WDN in terms of searching ability and computational efficiency. For example, in one case study the discrete DDS outperformed a GA with only 1% of GA computational effort. This clearly suggests that DDS is fast and efficient enough to be considered for solving MOPs. Given that WDN design problems are sometimes solved as MOPs, it is hoped that using PA-DDS results in a more efficient multi-objective optimization tool for solving WDN MOPs.
3.1 Finding Individual Minima In STEP2 of figure 1, PA-DDS spends some of the computational budget for finding individual minima of the problem. In fact, finding these endpoints of the tradeoff is critical to achieving good performance of PA-DDS. The reason is that, during the search for these individual minima, it is highly likely that several nondominated solutions with respect to the endpoints become identified. Testing has shown that these solutions definitely represent a better estimation of the tradeoff than a randomly generated set of solutions at the beginning of the search. The rest of the search in PA-DDS then will try to refine these nondominated solutions.
3.2 Fitness Assignment The objective function in DDS is used to define the current solution and thus the centre of the neighborhood from which candidate solutions will be sampled. In the original DDS, the best known solution is always defined as the current solution. In PADDS there is no explicit objective function and instead, any solution that is non-dominated can be selected as the current solution and thus perturbed to generate new solutions.
3. Solution Methodology In this paper, a new multi-objective optimization tool called PADDS is introduced. The idea of this tool comes from PAES that in its simplest form uses a local, single solution based optimization strategy to explore the search space and archives all the nondominated solutions during the search [4]. The PA-DDS pseudo code is given in figure 1.
After PA-DDS generates an initial set of non-dominated solutions based on the search history from optimizing each objective individually, one non-dominated point is selected as the current solution based on crowding distance measure as in [2]. PA-DDS samples new solutions by perturbing the current solution in the exact same way as the original DDS algorithm. The current solution is updated as the new solution if the new solution is nondominated. Otherwise, if the new solution is dominated, a new current solution is sampled from the external set of nondominated solutions. The new current solution is sampled by the roulette wheel to guide the search to another area of the objective space that is less crowded (crowding distance measure same as in [2]).
STEP0: Define the MOP with n objectives STEP1: Find individual minima by a portion of computational budget • Perform n optimization trials with DDS to minimize each objective • Archive non-dominated solutions in the external set STEP2: Select a current solution to sample from • Calculate crowding distance of solutions in the external set • Select a non-dominated solution based on crowding distance STEP3: Sample one new solution and evaluate • Generate new solution by perturbing the current solution xcur to xnew (note that neighbourhood defined as in original DDS algorithm) • Check the dominance of xnew against the external set of solutions • If computational budget is not exceeded o If xnew is non-dominated Set xcur = xnew o Else Repeat STEP3 • Else o Stop; external set of solutions is the tradeoff estimate
Similar to MOEAs this tool is not sensitive to the shape of the tradeoff, convex or concave, continuous or discontinuous, and it can be used for solving MOPs with any arbitrary number of the objectives. Therefore, the shape of the tradeoff and the number of objective functions increases the complexity of the problem but not the complexity of the PA-DDS.
3.3 Results
Figure 1. PA-DDS pseudo code.
Preliminary testing of PA-DDS on the ZDT test suite [8] of MO problems showed the algorithm was generally capable of closely approximating the actual tradeoff for the majority of test problems. Therefore, two of the more difficult test problems were selected for a comparison with published results on these problems using the NSGA-II algorithm [3] and a new algorithm called AMALGAM [7].
Although both PA-DDS and the simplest version of PAES sample from one solution at the time, they differ in the way that they update the current solution to sample from. If the offspring of PAES is dominated, the parent is perturbed again for the new generation. But in this case, PA-DDS selects another parent from the set of non-dominated solutions based on the crowding distance measure. Also, from the non-dominated offspring and parent, PAES chose the one that is in less populated part of the objective space as the parent of the next generation, while PA-DDS always accepts non-dominated offspring as the parent of the next generation since it can help refine the shape of the tradeoff.
PA-DDS results were generated using either 10 or 30 optimization trials and budget of 15,000 or 25,000 evaluations. These settings enabled a more direct comparison of published NSGA-II [3] and AMALGAM [7] results. The PA-DDS algorithm as described in Section 3 was applied with the default parameter setting. The algorithm results are compared using the convergence metric proposed in [2] and all convergence metric calculations used a sample of 500 actual tradeoff solutions that were evenly
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distributed along the Pareto optimal front. However, the convergence metric calculations for PA-DDS were based on a set of 500 tradeoff solutions that were not necessarily the same as those used in [3] or [7]. As such, these preliminary comparison metric values are not perfect and comparisons of PA-DDS convergence metrics with the other algorithms should be taken to be approximate. Again, the purpose of this paper is to establish whether the PA-DDS algorithm is promising relative to some high quality benchmark algorithms.
ZDT6 (15000 iterations) 1 0.9 0.8 0.7 0.6 f 2 0.5 0.4 0.3 0.2 0.1 0
ZDT4 (15000 iterations) 1.1 1 0.9 0.8 0.7 0.6 f 2 0.5 0.4 0.3 0.2 0.1 0
Actual tradeoff Median of 30 trials
Median of 30 trials Best of 30 trials 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Best of 30 trials
f1 Figure 4. The actual, the best and the median converged approximated tradeoffs among 30 independent trials regarding the results in table1.
ZDT6 (25000 iterations) 1 0.9 0.8 0.7 0.6 f 2 0.5 0.4 0.3 0.2 0.1 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
f1 Figure 2. The actual, the best and the median converged approximated tradeoffs among 30 independent trials regarding the results in table1.
1.1 1 0.9 0.8 0.7 0.6 f 2 0.5 0.4 0.3 0.2 0.1 0
Actual tradeoff
ZDT4 (25000 iterations) Actual tradeoff Median of 10 trials
Actual tradeoff Median of 10 trials Worst of 10 trials 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Best of 10 trials
f1 Figure 5. The actual, the best and the median converged approximated tradeoffs among 10 independent trials regarding the results in table2 Table1. The average of the convergence measure [2] for 30 optimization trials with 15,000 iterations. Convergence Metric Y
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PA-DDS
NSGA II*
AMALGAM*
ZDT4
0.049
0.052
0.002
ZDT6
0.002
0.050
0.001
f1 Figure 3. The actual, the best and the median converged approximated tradeoffs among 10 independent trials regarding the results in table2.
*. As in [7]
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Table2. The average of the convergence measure [2] for 10 optimization trials with 25,000 iterations.
6. REFERENCES [1] Coello, C. 2001. A short tutorial on evolutionary multiobjective optimization. In: Evolutionary multicriterion optimization. Lecture Notes in Computer Science. Springer. 21–40.
Convergence Metric Y PA-DDS
NSGA II*
ZDT4
0.024
0.091
ZDT6
0.001
0.053
[2] Deb, K. 2001. Multi-objective optimization using evolutionary algorithms. John Wiley & Sons. [3] Deb, K., Pratap, A. and Agarwal, S. 2002. A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6(2), 182-197,
*. As in [3]
[4] Knowles, J. D. and Corne, D. W. 2000. Approximating the nondominated front using the Pareto archived evolution strategy. Evol. Comput. 8(2), 149-172.
4. Conclusion A new, simple and parsimonious MO algorithm called PADDS was introduced. Overall, comparative results with NSGA-II and AMALGAM shows PA-DDS has promise. For example, PA-DDS outperforms NSGA-II on the two test problems considered here (ZDT4 and ZDT6 in [8]). Importantly, these promising PA-DDS findings were achieved without any algorithm parameter tuning experiments. Work is therefore ongoing to consider more test problems (especially in three or more objectives as well as real engineering design problems) and also expand the algorithm comparison to include the most recent variant of NSGA-II as well as SPEA2. Experiments to test PA-DDS sensitivity to the portion of computational budget allocated to find individual minima are also ongoing.
[5] Tolson, B. A. and Shoemaker C. A. 2007. Dynamically Dimensioned Search algorithm for computationally efficient watershed model calibration. Water Resour.Res. 43(1), W01413 1-16. [6] Tolson, B. A., Asadzadeh, M., Zecchin, A. C. and Maier H. R. 2008. A New Algorithm for Water Distribution System Optimization: Discrete Dynamically Dimensioned Search, World Environmental & Water Resources Congress, Honolulu, Hawaii. [7] Vrugt, J. A. and Robinson B. A. 2007. Improved evolutionary optimization from genetically adaptive multimethod search. Proc.Natl.Acad.Sci.U.S.A. 104(3), 708711. [8] Zitzler, E. 1999. Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications. PhD dissertation. ETH Zurich. Switzerland. [9] Zitzler, E., Laumanns, M. and Thiele L. 2001. SPEA2: Improving the Strength Pareto Evolutionary Algorithm for Multiobjective Optimization. Proc. EUROGEN 2001. Evolutionary Methods for Design, Optimization and Control with Applications to Industrial. Athens. Greece.
5. ACKNOWLEDGMENTS We would like to acknowledge our funding source for this project - NSERC Discovery Grant.
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