a hybrid algorithm for many-objective optimization€¦ · solve challenging optimization problems,...

Inverse Problems, Design and Optimization Symposium IPDO2019Tianjin, China, September 24-26, 2019

A HYBRID ALGORITHM FOR MANY-OBJECTIVE OPTIMIZATION

Sohail R. ReddySchool of Biomedical, Materials and Mechanical Eng.

MAIDROC Lab., Florida International UniversityMiami, Florida, U.S.A.

[email protected]

George S. DulikravichSchool of Biomedical, Materials and Mechanical Eng.

MAIDROC Lab., Florida International UniversityMiami, Florida, U.S.A.

[email protected]

Abstract—An increase in computing speeds available frommodern processor and GPUs, has allowed more complexdesign optimization problems to be solved. This increase incomplexity typically results in a difficult problem with a highlynon-linear objective function topology. This work develops arobust optimization algorithm capable of traversing higher-dimensional objective function space to find Pareto optimaldesigns. The developed algorithm hybridizes several existingalgorithms into a single suite and actively switches between thealgorithms to accelerate the convergence, avoid local minimaand arrive at a diverse set of optimal designs. The optimizationalgorithm is able to robustly solve problems with up to 15objectives. Due to its ability solve problems with large numberof objective and its hybrid nature, the developed algorithmis named Many-Objective Hybrid Optimizer (MOHO). TheMOHO algorithm was tested on 20 analytical test problemsand outperformed the other algorithms in 65% of the testproblems.

1. INTRODUCTIONBefore the advent of multi-objective optimization al-

gorithms, multi-objective design problems were typicallysolved by converting it to a single objective problem usinga linear combination of multiple objectives. The develop-ment of non-dominated approach for optimization madeit possible to solve multi-objective problems in their trueessence. The optimization problems faced today require thesolution of five to 10+ objectives and are referred to asmany-objective optimization problems. [1], [2].

Many-objective optimization problems pose a difficultyfor algorithms due to the increased dimensionality of theobjective function space. Several algorithms suffer fromclustering of solutions to certain regions of the objectivefunctions space and fail to preserve diversity in the Paretosolutions. This higher dimensional objective function spacealso gives rise to several local Pareto fronts, any of which analgorithm might prematurely converge to. These challengeshave given rise to a new set of many-objective optimizationalgorithm.

The typical optimization problem with two or moreobjectives can be stated as

min ~f (~x)

~f = {f1 (~x) , . . . , fl (~x)}~x = {x1, . . . , xm}subject to : x ∈ [ai, bi] , i = 1, . . . ,m

(1)

subject to certain equality and inequality constraints. Thealgorithms that solve this problem within the many-objective framework typically use various niching tech-niques to preserve diversity. Many-objective optimizationalgorithms can typically be separated into categories: De-composition based and Indicator based. Decompositionbased algorithms [3], [4], [5], [6] make use of referencepoints or reference vectors to segment the objective func-tion space and guide the search to the scarce areas of thePareto fronts. Indicator based algorithms [7], [8] use per-formance metrics to quantify the superiority of one designover another to control the recombination and selectionoperators.

Although several algorithms have been developed tosolve challenging optimization problems, no one algorithmhas shown the superior over another in every single case.This is clearly stated in the No-Free-Lunch Theorem forOptimization [9]. Therefore, it makes sense to combineseveral algorithms into a single suite and actively switchbetween them. This type of hybrization is known as a relay-hybridization [10]. The single-objective version of relay-hybrid algorithms have been extensively developed over theyears [11], [12], [13], [14]. The relay-hybrid algorithm wasalso developed for multi-objective optimization problems[15]. Their approach monitored five metrics of performanceand randomly switched to an alternative algorithm is thecurrently operating algorithm under performs. This leads toa chaotic switching where the hybrid algorithm is not ableto deterministically select the best performing algorithmfor the current problem. This issue is addressed in thecurrent switching strategy. To the authors knowledge, amany-objective hybrid optimization algorithm has not yetbeen developed.


2. MANY-OBJECTIVE HYBRID OPTI-MIZER (MOHO)

As previously mentioned, the MOHO algorithm ac-tively switching between constitutive algorithms in its suite.The current version of MOHO contains the NSGA-III[3], MOEA/DD [5], SPEA-R [4], NSDE-R1B [6], NSDE-D3 [6] algorithms. These algorithms were selected fortheir ability to preserve diversity and increase convergence.The inner workings of each algorithm is not stated herefor brevity but the readers are referred to their originalmanuscripts.

In order to control the switching mechanism in MOHO,the superiority of one Pareto solution set over another mustbe quantified. This metric should be able quantify boththe convergence and diversity of the solution set withoutrequiring the analytical solution to be known. Once suchmetric is the hypervolume [16]. Hypervolume measures thesize of the objective function space dominated by solutionsand a reference point dominated by all Pareto optimalsolutions, Fig. 1. It can be said that one algorithm outperforms another if the hypervolume of its Pareto solutionsis larger than the hypervolume of the Pareto solutionsobtained using another algorithm. The switching procedureis performed when an algorithm fails to make progress andenlarge the hypervolume.

Figure 1: Hypervolume of Pareto front

In previous works [15], the switching was random.This could lead to an under-performing algorithm to beselected. This issue can be addressed using a mechanismwhich monitors the performance history of the algorithmand selects the best performing algorithm. The MOHOalgorithm uses the Probability-of-Success (POS) to controlthe switching between algorithms. During the optimizationprocess, the number of attempts made by each algorithmand the number of successful attempts of each algorithm is

monitored and updated at each generation. The POS valueis then computed as

POS (·) = Nsuccess (·)Ntotal (·)

(2)

where · is a particular algorithm, Nsuccess are the num-ber of successful attempts, and Ntotal are the total attemptsmade by the algorithm. At each generation, MOHO selectsthe algorithm with the largest POS to perform the searchat the next generation.

3. ANALYTICAL TEST PROBLEMS FORVALIDATING MOHO

The performance of MOHO was investigated using a setof test problem for which the analytical Pareto optimal frontis known. The test problems are taken from the DTLZ testsuite [17]. These test problems can be scaled to any numberof variables and objectives. Test problems with three, five,eight, 10 and 15 objectives are investigated. The problemstaken from the DTLZ suite and their characteristics areshown in Table 1. The values in the DTLZ test problemswere set as the default values suggested by the authors [17].The algorithm properties were held constants and are takendirectly from [6], [5].

TABLE 1: Properties of DTLZ test problems used to validate the opti-mization algorithms

Test problem Properties Analytical solutionDTLZ1 Linear, multi-modal Half-unit hyperplaneDTLZ2 Concave Unit hypersphereDTLZ3 Concave, multi-modal Unit hypersphereDTLZ4 Concave, biased Unit hypersphere

The performance of the MOHO algorithm is comparedto that of each algorithm contained within the suite. Theperformance of the algorithms are quantified using theIGD metric [18]. The IGD metric measure the distancebetween two sets of points. The two sets the points on theanalytical Pareto front, and the Pareto front obtained usingthe algorithms. A small value of IGD indicates convergenceto the analytical Pareto front.

4. PERFORMANCE OF MOHO ONANALYTICAL TEST PROBLEMS

Figure 2 shows the results of the optimization algo-rithm on a three-objective DTLZ1 problem. It shows theanalytical Pareto front in grey and the Pareto solutionconverged to by the six algorithms in blue. The three-objective DTLZ1 problem is a linear, multi-modal problemand features 161,000 local Pareto fronts. It can be seen thatthe SPEA-R algorithm under-performs for the DTLZ1 testproblem, thereby proving the validity of the No-Free-Lunchtheorem. Despite having an under-performing algorithm inits suite, MOHO is able to both converge to the Pareto front


and preserve diversity. This is also seen in the remainingfour algorithms, but MOHO was able to converge faster.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: The Pareto front for a three objective DTLZ1 problem, obtainedusing: a) NSGA-III, b) MOEA-DD, c) SPEA-R, d) NSDE-R1B, e) NSDE-D3 and f) MOHO

In higher dimensional problems, the performance of thealgorithms is expected to decrease. Figure 3 shows resultsof the 15 objective DTLZ1 problem in parallel coordinates.Since the analytical Pareto front is a half-unit hyperplane,a well converge solution will have values of all objectivesbetween 0 and 0.5. A well diversified solution set willhave a uniformly distributed parallel coordinate graph. Thisproblem also features 161,000 local Pareto fronts, each ofwhich can cause premature convergence of each algorithm.It can be seen that in this case, both the SPEA-R algorithmand the MOEA/DD algorithm fail to converge. Despitethis, since the MOHO suite contains other well-performingalgorithms, it is able to converge to the Pareto front.


(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: The Pareto front for a 15 objective DTLZ1 problem, obtainedusing: a) NSGA-III, b) MOEA-DD, c) SPEA-R, d) NSDE-R1B, e) NSDE-D3 and f) MOHO

Figure 4 shows the results of the optimization algo-rithms on a three-objective DTLZ2 problem. This problemis concave whose Pareto optimal solutions lie on the unithypersphere. In this case, it can be seen that all algorithmshave converged to the analytical Pareto front. However, theMOEA/DD algorithm is unable to preserve diversity as canbe seen by the scarcity of Pareto solution in the centralregion of the front.

(a)


(b)

(c)

(d)

(e)

(f)


Figure 5 shows results of the 15 objective DTLZ2 prob-lem in parallel coordinates. Since the analytical Pareto frontis a unit hypersphere, a well converged solution will havevalues of all objectives between 0 and 1.0. It can be seenthat, unlike for a three-objective problem, the MOEA/DDalgorithm performs well for the 15 objective problem. TheSPEA-R algorithm, however, continues to under-perform.The MOHO algorithm is again able to converge to thePareto front automatically.

(a)

(b)


(c)

(d)

(e)

(f)


Figure 6 shows the results of the optimization algorithmon a three-objective DTLZ3 problem. The three-objective

DTLZ3 problem is a concave, multi-modal problem andfeatures 59,000 local Pareto fronts. Its characteristics aresimilar to the combination of the DTLZ1 and DTLZ2problems. It can be seen that both the SPEA-R algorithmand the NSDE-D3 algorithm are unable to converge for theDTLZ3 test problem. The MOEA/DD algorithm is ableto converge but does not produce uniformly distributedsolution. The MOHO algorithm, however, is able to bothconverge to the Pareto front and preserve diversity despitehaving three under-performing algorithms in its suite.

(a)

(b)

(c)


(d)

(e)

(f)


Figure 7 shows results of the 15 objective DTLZ3problem in parallel coordinates. Since the analytical Paretofront is a unit hypersphere, a well converge solution willhave values of all objectives between 0 and 1.0. It canbe seen that, unlike for a three-objective problem, theMOEA/DD algorithm again performs well for the 15 ob-jective problem. The SPEA-R and the NSDE-D3 algorithm,however, continue to under-perform. It can be seen that theSPEA-R algorithm is neither able to converge or produceuniformly distributed solution, whereas the NSDE-D3 isnot able to converge but does produce uniformly distributedPareto solution. The convergence of MOHO algorithm isslightly affected for higher dimensional problems due to the

large number of under-performing algorithms in the suite.Nonetheless, it is able to produce satisfactory results.

(a)

(b)

(c)

(d)


(e)

(f)


Figure 8 shows the results of the optimization algorithmon a three-objective DTLZ4 problem. The three-objectiveDTLZ4 problem is a concave biased problem that draws thesearch to a particular region of the Pareto front. It can beseen that for the lower dimensional problems all algorithmsare able to both converge to the Pareto front and preservediversity. The MOEA/DD algorithm slightly struggles topreserve diversity in the central region of the front.

(a)

(b)

(c)

(d)

(e)


(f)


Figure 9 shows results of the 15 objective DTLZ4problem in parallel coordinates. Since the analytical Paretofront is a unit hypersphere, a well converge solution willhave values of all objectives between 0 and 1.0. Forhigher dimensional problems, the SPEA-R algorithm under-performs. The remaining algorithms are able to convergeand preserve diversity in the Pareto front despite the biasednature of the DTLZ4 problem.

(a)

(b)

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(f)



Table 2. shows the mean and standard deviation of theIGD values obtained from 20 runs of each optimizationalgorithm on each test problem, where the algorithm withthe lowest mean is highlighted in red. It can be seen thatMOHO consistently performs best for 65% of the testproblems.

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5. CONCLUSION

A robust many-objective hybrid optimization algorithmwas developed by combining five constitutive algorithms.The developed algorithm actively and deterministicallyswitches between the constitutive algorithms to avoid localminima and accelerate convergence. The performance ofthe algorithm was investigated on the 20 test problemsfrom the DTLZ test suite. It was shown that the hybridalgorithm consistently produced reliable results despitehaving under performing algorithms within its suite. Itshowed that the switching criteria is able to select analgorithm that performs best for the current problem.The MOHO algorithm performed better in 65% of thetest cases. Whereas certain algorithms successfully solvelow-dimensional problem but fail for high-dimensionalproblems, MOHO is able to perform robustly for both highand low dimensional problems. It was demonstrated thatMOHO can cope with problems with up to 15 objectiveswith varying degree of difficulty in the objective functiontopology.

ACKNOWLEDGMENTSThe first author gratefully acknowledges the financial

support from Florida International University in the form ofan FIU Presidential Fellowship and FIU Dissertation YearFellowship. The authors are grateful for the research fund-ing provided by NASA grant NNX17AJ96A via TAMU.

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a hybrid algorithm for many-objective optimization€¦ · solve challenging optimization problems,...

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