This paper describes a novel utility function for choosing design vectors to evaluate in multi-objective optimization problems which are statistically most probable to be Pareto-optimal, given the points already evaluated. The method is tunable to the number of existing Pareto-optimal solutions that an unevaluated design vector is sought to dominate, is naturally parallelized, and removes any need for combining the multiple objectives into a single objective with a scalarizing function

Hawe, G.I.

Sykulski, J.K.

English

Southampton (e-Prints Soton)

6. OPTIMIZATION, INVERSE PROBLEMSAn Enhanced Probability of Improvement Utility Function for LocatingPareto Optimal SolutionsGlenn I. Hawe∗,† and Jan K. Sykulski†∗Vector Fields, Ltd., 24 Bankside, Kidlington, Oxford, U.K. OX5 1JE†School of Electronics and Computer Science, University of Southampton, Southampton, U.K. SO17 1BJEmail: glenn.hawe@vectorﬁelds.co.uk, jks@soton.ac.ukAbstract—This paper describes a novel utility function forchoosing design vectors to evaluate in multi-objective optimiza-tion problems which are statistically most probable to be Pareto-optimal, given the points already evaluated. The method istunable to the number of existing Pareto-optimal solutions thatan unevaluated design vector is sought to dominate, is naturallyparallelized, and removes any need for combining the multipleobjectives into a single objective with a scalarizing function.I. IIn [1], the probability of improvement criteria was identiﬁedas an eﬀective utility function to use with kriging surrogatemodels [2] for locating design vectors to evaluate in single-objective optimization. By modeling the prediction of a krigingmodel as the realization of a Gaussian distribution, with meanˆ f(x) and standard error s(x) as given by the kriging model,the probability of an unevaluated design vector x having anobjective function value less than T isP(f(x) < T) = ΦT − ˆ f(x)s(x)(1)where Φ is the normal cumulative distribution function. This isillustrated below in Fig. 1; the probability of the unevaluateddesign vector x∗ having an objective function value less thanT is represented by the shaded region.0Tfmin2461 1.5 2 2.5 3 3.5 4 4.5 5012345Krigingpredictionˆf(x)StandarderrorxSampled pointsUnevaluated point x∗Kriging predictionStandard errorP(f(x∗) < T)Fig. 1. Uncertainty in objective function value for an unevaluated designvector as predicted by a kriging model.This method of selecting design vectors to evaluate suﬀeredone drawback however, in that it was sensitive to the level ofimprovement sought (i.e. the value of T). Two methods wereproposed in [1] for overcoming this sensitivity:1) Evaluate the design vector which maximizes the expec-tation of the improvement, or2) Evaluate several design vectors per iteration, each cor-responding to a diﬀerent level of improvement (i.e. adiﬀerent value of T).The ﬁrst method led to the development of the EGOalgorithm [3], which has subsequently received signiﬁcantattention in the literature. However, the second method, re-ferred to as ‘enhanced probability of improvement’ is theapproach recommended in [1]. It has the advantage that it isvery robust and easily parallelized, although it suﬀers in thatthe diﬀerent levels of improvement sought in each iteration(i.e. the diﬀerent values of T used) are arbitrary. This paperextends this enhanced method to the multi-objective case,where it shall be seen that natural levels of improvement exist.II. P  I  E I M-O OThe concept of improvement in multi-objective optimizationhas only appeared recently in the literature. First, recall that asolution S a (with design vector xa) is said to dominate anothersolution S b (with design vector xb) if and only if S a is strictlybetter than S b in at least one objective, and no worse in allother objectives. The set of Pareto-optimal solutions are thenthose solutions which are not dominated by any other existingsolution. Suppose then, that a set S of Npar Pareto-optimalsolutions exists, S = {S 1,S 2,...,S Npar}, after performing anexperimental design. Then a design vector x is said to yieldan improvement if it is non-dominated by the solutions inS [4]. This may happen in one of two ways:1) x dominates at least one of the solutions in S, or2) x does not dominate any solution in S, nor does anysolution in S dominate x.This is shown in Fig. 2 for the two-objective case, whenNpar = 5 Pareto solutions exist. Design vectors which yieldan improvement map to either of the shaded regions; designvectors which dominate at least one solution in S map to theregion labeled ‘Dominating Designs’, whilst design vectorswhich do not dominate any solution in S (but which stillconstitute an improvement) map to the region labeled ‘Equiva-lent Designs’. Equations for the probability of an unevaluateddesign vector yielding an improvement are given in [4] and [5],and the equation for the probability of an unevaluated designvector dominating at least one solution in S for the two-objective case is given in [5]. Both [4] and [5] report problems965*PC6-186. OPTIMIZATION, INVERSE PROBLEMSfS11 fS21 fS31 fS41 fS51fS12fS22fS32fS42fS52Equivalent DesignsDominating Designs •S1•S2•S3•S4•S5 = SNparf1 (to be minimized)f2(tobeminimized)Fig. 2. Regions of dominance and equivalence for Npar = 5 Pareto solutions.with the probability of improvement criteria yielding searcheswhich are not very global, and so only small improvementsare made. Both overcome this by instead maximizing the ﬁrstmoment of the probability of improvement around the Pareto-optimal front, which is the multi-objective equivalent of theexpected improvement utility function, i.e. they use the ﬁrstmethod suggested in Section 1 for overcoming the equivalentsensitivity problem in the single-objective case. The followingSection shows how to overcome this sensitivity problem inthe multi-objective case using the second method proposedin Section 1, that is, it proposes an enhanced probability ofimprovement criteria for multi-objective optimization.III. E P  I  M-OOIn the multi-objective case, k = Npar natural levels of im-provement may be deﬁned, where the kth level of improvementyields a solution which dominates exactly k of the existingPareto-optimal solutions. In addition, a level of equivalencemay also be deﬁned (k = 0), which yields an additional Pareto-optimal solution which does not dominate any of the existingPareto-optimal solutions (i.e a design vector which maps to theregion labeled ‘Equivalent Designs’ in Fig. 2). These levels ofimprovement are shown in Fig. 3 for the Npar = 5 Paretosolutions considered earlier.Levels of ImprovementLevel 0Level 1Level 2Level 3Level 4Level 5•S1•S2•S3•S4•S5 = SNparP0[I0]P1[I0]P2[I0]P3[I0]P4[I0]P5[I0]P0[I1]P1[I1]P2[I1]P3[I1]P4[I1]P0[I2]P1[I2]P2[I2]P3[I2]P0[I3]P1[I3]P2[I3]P0[I4]P1[I4] P0[I5]fS11 fS21 fS31 fS41 fS51fS12fS22fS32fS42fS52f1 (to be minimized)f2(tobeminimized)Fig. 3. Probability of improvement levels for Npar = 5 Pareto solutions.As can be seen, in the two-objective case, for an improve-ment level k, a design vector may map to Npar−k+1 regions ofobjective function space. Denoting P(Ik(x)) as the probabilitythat an unknown design vector x yields a level of improvementk (i.e. it dominates exactly k existing Pareto-optimal solutions),Pi(Ik(x)) as the probability that design vector x will dominatethe k Pareto solutions S i+1,S i+2,...,S i+k (these sub-regionsare labeled in Fig. 3), and deﬁningΦi1(x) B Φ fS i1 − ˆ f1(x)s1(x)(2)Φi2(x) B Φ fS i2 − ˆ f2(x)s2(x)(3)withΦ01(x) B 0 (4)ΦNpar+11 (x) B 1 (5)Φ02(x) B 1 (6)ΦNpar+12 (x) B 0 (7)where ˆ f1(·) and s1(·) are the kriging predictions and standarderrors for the ﬁrst objective function respectively (similarly forthe second objective function), and fS i1 is the ﬁrst objectivefunction value of the ith Pareto solution (similarly for thesecond objective function), then:P(Ik(x)) =Npar−k Xi=0Pi(Ik(x)) (8)=Npar−k Xi=0 Φi+11 (x) − Φi1(x) Φk+i2 (x) − Φk+i+12 (x). (9)Furthermore, denoting by P∗(Ik(x)) the probability that xwill dominate at least k existing Pareto-optimal solutions, thenP∗(Ik(x)) =Npar Xj=kNpar−j Xi=0 Φi+11 (x)−Φi1(x) Φj+i2 (x)−Φj+i+12 (x). (10)Equations (9) and (10) are two multi-objective equiva-lents of the ‘enhanced probability of improvement’ in single-objective optimization, for the case of two objectives. Themethod is extensible to higher numbers of objectives.IV. CA novel utility function, which is easily parallelized andwhich does not require normalization of objective functions,has been proposed for use in computationally expensive multi-objective optimization. Its performance in optimal electromag-netic design will be discussed in the full paper.R[1] D. R. Jones, “A Taxonomy of Global Optimization Methods Based onResponse Surfaces”, Journal of Global Optimization, vol. 21, pp. 345–383, 2001.[2] L. Lebensztajn, C. A. R. Marretto, M. C. Costa and J-L. Coulomb,“Kriging: a useful tool for electromagnetic devices optimization”, IEEETransactions on Magnetics, vol. 40, no. 2, pp. 1196–1199, 2004.[3] D. R. Jones, M. Schonlau and W. J. Welch, “Eﬃcient Global Optimizationof Expensive Black-Box Functions”, Journal of Global Optimization,vol. 13, pp. 455–492, 1998.[4] M. T. M. Emmerich, K. C. Giannakoglou and B. Naujoks, “Single- andMultiobjective Evolutionary Optimization Assisted by Gaussian RandomField Metamodels”, IEEE Transactions on Evolutionary Computation,vol. 10, no. 4, pp. 421-439, 2006.[5] A. J. Keane, “Statistical Improvement Criteria for Use in MultiobjectiveDesign Optimization”, AIAA Journal, vol. 44, no. 4, pp. 879–891, 2006.966*PC6-18

An Enhanced Probability of Improvement Utility Function for Locating Pareto Optimal Solutions

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