Sequential Approximate Multiobjective Optimization using Computational Intelligence

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1 The Ninth Internationa Symposium on Operations Research and Its Appications (ISORA 10) Chengdu-Jiuzhaigou, China, August 19 23, 2010 Copyright 2010 ORSC & APORC, pp Sequentia Approximate Mutiobjective Optimization using Computationa Inteigence Hirotaka Nakayama Department of Inteigence and Informatics, Konan University, Kobe Abstract In many practica probems, in particuar in engineering design, the function form of criteria is not given expicity in terms of design variabes. Given the vaue of design variabes, under this circumstance, the vaue of objective functions is obtained by rea/computationa experiments such as structura anaysis, fuidmechanic anaysis, thermodynamic anaysis, and so on. Usuay, these experiments are time consuming and expensive. One of recent trends in optimization is how to treat these expensive criteria. In order to make the number of these experiments as few as possibe, optimization is performed in parae with predicting the form of objective functions. This is caed sequentia approximate optimization with meta-modeing. It has been observed that techniques of computationa inteigence can be effectivey appied for this purpose. This tak wi discuss severa issues in sequentia approximate mutiobjective optimization using computationa inteigence. Keywords Sequentia Approximate Optimization; Mutiobjective Optimization; Metamodeing; Goba Optimizaiton 1 Introduction In order to sove optimization probems with expensive objective/constraint functions, we can appy effectivey sequentia approximate optimization technniques in which optimization is performed in parae with predicting those expensive functions. Sequentia approximate optimization consists of two phases: 1) construction of a metamode, and 2) optimization for the metamode. For the phase 1), there have been deveoped many kinds of methods which are caed metamodeing (in the sense of making mode of the mode"), surrogate modeing, response surface methods, etc. depending on research communities [8, 11, 21]. For the phase 2), a kinds of optimization techniques may be avaiabe in genera. In many practica enginering design probems, however, due to compex noninearity and mui-modaity metaheuristic methods such as genetic agorithms, partice swarm optimization methods, ant coony optimization methods, and differentia evoution methods may be effectivey appied. In the foowing, we discuss metamodeing in more detai. 2 Metamodeing The identification of function forms of objective functions and constraints functions is referred to as modeing" in practica probems. For the sake of simpicity, we consider

2 2 The 9th Internationa Symposium on Operations Research and Its Appications throughout this section the foowing simpe probem: minimize f (x) over x X R n. Then, the function f is aso considered as a mode" of objectives. When it is difficut to identify the function form of f, but when we can observe the vaue of f (x) for samped point x, we try to get an approximate function (or mode) ˆf for f on the basis of observations (x i,y i ) where y i = f (x i ) and i = 1,...,. This approximate function ˆf is caed a metamode. Now, our aim is to construct a good metamode in the sense that i) we can obtain an approximate optima soution through the metamode with the property ˆf (ˆx ) f (x ) ε 1, where ˆx and x minimize ˆf and f, respectivey, and ε 1 is a given sma positive number, ii) the tota number of observations is as sma as possibe, iii) the metamode ˆf approximates we f entirey, if possibe. Namey ˆf f ε 2, where ε 2 is a given sma positive number. If our aim is merey to find the optima soution minimizing f (x), then the metamode ˆf does not necessariy approximate we f entirey, but sufficienty we at east in a neighborhood of the optima soution x. Depending on practica probems, however, one may want to see the goba behavior of the mode f. Therefore, the priority of iii) above is behind the criteria i) and ii) which are crucia in genera. Metamodes can be constructed by pynomia regression, ogistic regression, thin pate spines, radia basis function networks, support vector regression (SVR) and so on. In this paper, we review SVR briefy. Support Vector Machine (SVM) proposed by Vapnik et a. in the midde of 90 s [1] is now recognized as an effective too for machine earning [3, 17, 18]. SVR is a version of SVM appied to regression probems, in which the ε insensitive oss function is introduced [18]. It shoud be noted that the sparseness of support vectors resuts from this ε insensitive oss function. Denote the given sampe data by (x i,y i ), i = 1,...,. Define a metamode ˆf on the feature space Z mapped from the data space by some noninear map Φ as foows: ˆf (z) = w T z + b, and the inear ε insensitive oss function is defined by L ε (z,y, ˆf ) = y ˆf (z) ε = max(0, y ˆf (z) ε). Taking into account this inear ε insensitive oss function, C-SVR aowing an error ξ i ( ξ i ) for each data to some extent is defined as foows: For a given insensitivity parameter

3 Sequentia Approximate Mutiobjective Optimization 3 ε, minimize w,b,ξ, ξ subject to ( ) w 2 2 +C (ξ i + ξ i ) ( w T z i + b ) y i ε + ξ i, i = 1,...,, (C SVR) P y i ( w T z i + b ) ε + ξ i, i = 1,...,, ε, ξ i, ξi 0, where C is a trade-off parameter between the norm of w and ξ i ( ξ i ). On the other hand, the authors proposed another version of SVR, caed µ SVR, minimizing the maximum of error ξ i as foows [15]: For a given insensitivity parameter ε, minimize w,b,ξ, ξ subject to 1 2 w µ(ξ + ξ ) (µ SVR) P ( w T z i + b ) y i ε + ξ, i = 1,...,, y i ( w T z i + b ) ε + ξ, i = 1,...,, ε, ξ, ξ 0, where µ is a trade-off parameter between the norm of w and ξ ( ξ ). The dua formuation to the probem (µ SVR) P is given by maximize α,άα subject to i, j=1 (ά i α i )( α j α j )K (x i, x j ) (ά i α i )y i ε (ά i α i ) = 0, ά i µ, α i µ, ά i 0, α i 0, i = 1,...,. (ά i + α i ) (µ SVR) where K(x i, x j ) is a kerne satisfying K(x i, x j ) =< Φ(x i ),Φ(x j ) >. The optima formua of metamode is given by ˆf (x) = ( ) α i αi K(x, x i ) + b where α i, α i are the soutions to (µ SVR) and b is the Lagrange mutipier to the equaity constraint in (µ SVR). It has been observed that µ SVR provides a good performance robust against outiers with ess support vectors [15].

4 4 The 9th Internationa Symposium on Operations Research and Its Appications 3 Incrementa Design of Experiments In sequentia approximate optimization, one of most major issues is how to obtain an approximate optima soution by as ess number of experiments as possibe. To this aim, we usuay start with a reativey sma number of experiments and add experiments sequentiay, if necessary. For design of incrementa experiments, severa optimaity criteria in the thoery of experimenta design may be appied. However, it has been observed that those optimaity criteria such as the so-caed aphabetica optimaity (D-optimaity, E-optimaity and etc.) depend on mode formua. For exampe. if we adopt inear poynomia modes and D-optimaity, then candidate sampes to be added tend to be concentrated in the area of edgepoints. On the other hand, Jones et a. proposed to decide additiona sampes on the basis of expected improvement in the efficient goba optimization (EGO) [7]. The EGO is a kind of goba optimization methods using Bayesian approach (Bayesian goba optimization) [10, 23], and uses the Kriging mode as a prediction method for unknown function. 3.1 Expected Improvement The expected improvement is evauated on the basis of the Kriging mode as foows: Consider the response y(x) as a reaization of a random function, Y (x) such that Y (x) = µ(x) + Z(x). (1) Here, µ(x) is a goba mode and Z(x) refecting a deviation from the goba mode is a random function with zero mean and nonzero covariance given by cov[z(x),z(x )] = σ 2 R(x, x ) (2) where R is the correation between Z(x) and Z(x ). Usuay, the stochastic process is supposed to be stationary, which impies that the correation R(x, x ) depends ony on x x, namey R(x, x ) = R(x x ). (3) A commony used exampe of such correation functions is R(x, x ) = exp[ n θ i x i x i 2 ], (4) where x i and x i are i-th component of x and x, respectivey. Athough a inear regressrion mode k j=1 µ j f j (x) can be appied as a goba mode in (1) (universa Kriging), µ(x) = µ in which µ is unknown but constant is commony used in many cases (ordinary Kriging). In the ordinary Kriging, the best inear unbiased predictor of y at an untried x can be given by ŷ(x) = ˆµ + r T (x)r 1 (y 1 ˆµ), (5) where ˆµ = (1 T R 1 1) 1 1 T R 1 y is the generaized east squares estimator of µ, r(x) is the n 1 vector of correations R(x, x i ) between Z at x and samped points x i (i = 1,..., p), R is an n n correation matrix with (i, j)-eement defined by R(x i, x j ) and 1 is a unity vector whose components are a 1.

5 Sequentia Approximate Mutiobjective Optimization 5 The estimated vaue of the mean of the stochastic process, ˆµ, is given by In this event, the variation σ 2 is estimated by The mean squared error of the predictor is estimated by ˆµ = 1T R 1 y 1 T R 1 1. (6) ˆσ 2 = (y 1 ˆµ) T R 1 (y 1 ˆµ). (7) n s 2 (x) = σ 2 [1 r T R 1 r + (1 1T R 1 r) 2 1 T R 1 ]. (8) 1 In the foowing s = s 2 (x) is caed a standard error. Using the above predictor on the basis of stochastic process mode, Jones et a. appied the expected improvemnet for adding a new sampe point. Let f p min = min{y 1,...,y p } be the current best function vaue. They mode the uncertainty at y(x) by treating it as the reaization of a normay distributed random variabe Y with mean and standard deviation given by the above predictor and its standard error. For minimization cases, the improvement at x is I = [max( f p min Y, 0). Therefore, the expected improvement is given by E[I(x)] = E[max( f p min Y, 0)]. It has been shown that the above formua can be expanded as foows: { E(I) = ( f p min ŷ)φ( f p min ŷ s ) + sφ( f p min ŷ s ) if s < 0 0 if s = 0, (9) where φ is the standard norma density and Φ is the distribution function. We can add a new sampe point which maximizes the expected improvement. Athough Jones et a. proposed a method for maximizing the expected improvement by using the branch and bound method, it is possibe to seect the best one among severa candidates which are generated randomy in the design variabe space. 3.2 Distance-based Loca and Goba Information One of most important tasks in optimization is to baance between exporation and expoitation. The exporation is to search an optima soution from a goba viewpoint, whie the expoitation is to search an optima soution in a good precision (namey, from a oca viewpoint). Therefore, it is important to design additiona exepriments taking into account goba information and oca information on the metamode. It shoud be noted that EGO using EI decides an additiona sampe by controing the weight on goba information and oca information depnding on the situation. Nakayama et a. [13] have suggested the method which gives both goba information for predicting the entire objective function and oca information near the optima point

6 6 The 9th Internationa Symposium on Operations Research and Its Appications at the same time. In this method, two kinds of additiona data are taken for reearning the form of the objective function. One of them is seected from a neighborhood of the current optima point in order to add oca information near the (estimated) optima point. The size of this neighborhood is controed during the convergence process. The other one is seected far away from the current optima vaue in order to give a better prediction of the form of the objective function. The former additiona data gives more detaied information near the current optima point. The atter data prevents from converging to oca optimum point. The neighborhood of the current optima point is given by a square S, whose center is the current optima point, with the ength of a side. Let S 0 be a square, whose center is the current optima point, with the fixed ength of a side 0. The square S is shrinked according to the number C x of optima points appeared continuousy in S 0 in the past. That is, = 0 1 C x + 1. (10) The first additiona data is seected inside the square S randomy. The second additiona data is seected in an area, in which the existing earning data are sparse, outside the square S. An area with sparsey existing data may be found as foows: i) First, a certain number (N rand ) of data are generated randomy outside the square S. ii) Denote d i j the distance from this random data p i (i = 1,...,N rand ) to the existing earning data q j ( j = 1,...,N). Seect the shortest k distances d i j ( j = 1,...,k) for each p i, and sum up these k distances, i.e., D i = k j=1 d i j. iii) Take p t which maximizes {D i } (,...,Nrand ) as an additiona data outside S. The agorithm using distance-based oca and goba information by [13] can be summarized as foows: Step 1. Predict the form of the objective function by some regression method on the basis of the given training data. Step 2. Estimate an optima point for the predicted objective function by some optimization method. Step 3. Count the number of optima points appeared continuousy in the past in S 0. This number is represented by C x. Step 4. Terminate the iteration, if C x is arger than or equa to the given Cx 0 a priori, or if the best vaue of the objective function obtained so far is identica during the ast certain number (C 0 f ) of iterations. Otherwise cacuate by the equation (10), and go to the next step. Step 5. Seect an additiona data near the current optima vaue, i.e., inside S. Step 6. Seect another additiona data outside S in a pace in which the density of the training data is ow as stated above. Step 7. Go to Step 1.

7 Sequentia Approximate Mutiobjective Optimization 7 Exampe 1. Consider a simpe exampe given by maximize x 1,x 2 f (x) = 10exp ( (x 1 10) 2 + (x 2 15) 2 ) sinx 1 (11) 100 subject to 0 x 1 15, 0 x This probem has an optima (maxima) vaue f = at x 1 = and x 2 = 15. Fig.1 shows the resut at the 63 sampes by EGO using EI. On the other hand, Fig.2 shows the resut at the 61 sampes by RBF using distance-based goba and oca information. One may see that both resuts are amost the same: athough EI seems sighty better than by distance-based goba and oca information, the cacuation of EI takes more time. y x2 x 2 x 1 x 1 ˆx 1 = , ˆx 2 = , ˆf = Figure 1: #data = 63 (fina iteration) by EGO using EI y x2 x 2 x 1 x 1 ˆx 1 = , ˆx 2 = , ˆf = Figure 2: #data = 61 (fina iteration) by distance-based goba and oca information 4 Sequentia Approximate Mutiobjective Optimization In muti-objective optimization, it is one of main issues how to obtain Pareto optima soutions, and how to choose one soution from many Pareto optima soutions. To this

8 8 The 9th Internationa Symposium on Operations Research and Its Appications end, the interactive optimization method [5, 9, 22], for exampe, aspiration eve method [16, 12], have been deveoped. Aspiration eve method searches a soution by processing the foowing two stages repeatedy: 1) soving auxiiary optimization probem to obtain the cosest Pareto optima soution to a given aspiration eve of decision maker, and 2) revising her/his aspiration eve by making the trade-off anaysis. Conventiona interactive optimization methods are usefu in particuar in cases with many objective functions, in which it is difficut to visuaize Pareto frontier, and aso to depict the tradeoff among many objective functions. In cases with two or three objective functions, on the other hand, it may be the best way to depict Pareto frontier, because visuaizing Pareto frontier heps to grasp trade-off among objective functions. For that purpose, evoutionary methods such as genetic agorithm (GA) have been effectivey appied for soving a muti-objective optimization probem: so caed evoutionary muti-objective optimization (EMO) methods have been proposed for generating Pareto optima soutions [4, 2]. However, EMO has some probems: i) it is difficut to visuaize Pareto frontiers in cases with many objective functions, ii) many function evauations are usuay needed for generating the whoe Pareto frontier. Considering the number of function evauations, it woud be rather reasonabe to generate not the whoe Pareto frontier, but a necessary part of it in which the decision maker may be interested. To this aim, we introduce some methods combining aspiration eve approach and computationa inteigence method in this section. A muti-objective optimization probem (MOP) can be formuated as foows: minimize f (x) = ( f 1 (x),..., f r (x)) T x subject to x X = { x R n g j (x) 0, j = 1,...,m }, (MOP) where X denotes the set of a feasibe soutions in the design variabe space. To begin with, we summarize the method for sequentia approximate muti-objective optimization (shorty, SAMO) using satisficing trade-off method proposed by [16] as foows: Step 1. Cacuate the rea vaues of objective functions f (x 1 ),..., f (x ) for given sampe points x 1,..., x. Step 2. Approximate each objective function f k (x), k = 1,...,r, by using some regression method on the basis of training data set (x i, f k (x i )), i = 1,...,. An optima soution/vaue to approximate objective function ˆf (x) is caed an approximate optima soution/vaue. Step 3. Find an approximate optima soution x a cosest to the given aspiration eve f by soving the foowing probem (AP) of satisficing trade-off method: minimize x,z subject to z + α r w i ˆf i (x) w i ( ˆf i (x) f i ) z, i = 1,...,r, x X, where α is a sufficienty sma positive number, for exampe 10 6, w i = 1/( f i fi ) and f i is an idea vaue. (AP)

9 Sequentia Approximate Mutiobjective Optimization 9 Furthermore, generate approximate Pareto optima soutions x 1,..., x p to ˆf by using EMO for approximating the whoe set of Pareto soutions and for deciding the second additiona sampes which wi be stated in Step 5 beow. Step 4. Stop the iteration if a certain stop condition is satisfied. Otherwise, go to the next step. The stop condition is given by, for exampe, the imitation of the number of sampe points, the count of no-changing approximate soution obtained in the Step 3, and so on. Step 5. Choose additiona sampe points for reearning, and go to Step 1. It is important to improve the prediction abiity for function approximation in order to find an approximate soution coser to the exact one with as sma number of sampe data as possibe. To this aim, starting with reativey few initia sampes, we add new sampes step by step, if necessary. Here, we introduce one of the ways how to choose additiona sampe points proposed by Yun-Yoon-Nakayama [20]. (i) First, one additiona sampe point is added as the soution x a cosest to the given aspiration eve which is found in step 3. This is for approximating we a neighborhood of Pareto optima soution cosest to the aspiration eve, which enabes to make easiy the trade-off anaysis among objective functions. Here, the additiona point x a is considered as a oca information of Pareto frontier, because x a can provide the information around the cosest Pareto optima soution to the aspiration eve. (ii) Another additiona sampe point is for depicting the configuration of Pareto frontier. This is for giving a rough information of the whoe Pareto frontier, and we ca this a goba information of Pareto frontier in contrast with the above oca information. Stage 1. Evauate the rank R i for each sampe point x i, i = 1,..., by the ranking method of Godberg [6]. Stage 2. Approximate a ranking function ˆR(x) on the basis of training data set (x i,r i ), i = 1,..., by some regression method. Stage 3. Cacuate the vaues of ranking function ˆR(x) for the approximate Pareto optima soutions x j, j = 1,..., p generated in step 3. Stage 4. Among them, seect a point with the best rank x b = arg min ˆR(x j ). j=1,...,p Exampe 2. (Case 1) Consider the foowing probem with two design variabes and two objective functions: minimize x 1,x 2 f 1 (x) = x 1 (Ex-1) f 2 (x) = 1 + x 2 2 x 1 0.1sin(5πx 1 ) subject to 0 x 1 1, 2 x 2 2.

10 10 The 9th Internationa Symposium on Operations Research and Its Appications Starting with initia sampe 10 points generated randomy, we stop the iteration after 15 additiona earning. We approximate each objective function by using µ SVR with gauss kerne function. Fig. 3 shows the resuts with 40 sampe points after 15 additiona earning, in which one may see that the obtained approximate Pareto optima soutions are amost the same as the exact ones. f idea point aspiration eve f 1 (a) approximate vaue cosest to aspiration eve f idea point aspiration eve f 1 (b) approximate Pareto frontier Figure 3: # data = 40 sampe pts (after 15 additiona earning) (Ex-1) : case 1 Exampe 2. (Case 2) Note that the second additiona sampe at Step 5 seems to provide a oca information in the sense that it is for generating a better approximation of the whoe Pareto frontier. Indeed, the additiona sampes in the space of design variabes x are concentrated in the neighborhood of Pareto optima soutions. From a viewpoint of generating a better approximation of each objective function, further sampe points for goba information may be needed. As the third additiona sampe point, we recommend to add a point in a sparse area of existing sampes. This is performed by a simiar way as the distance based oca and goba information method. Fig. 4 shows the resut obtained by this method. One may see that each objective function is approximated we enough and that a we approximated Pareto frontier is generated. 5 Concuding Remarks The most prominent feature in SAMO described above is that combining the aspiration eve method and EMO, it is possibe to find the most interesting part of Pareto frontier for the decision maker as we as to grasp the configuration of the whoe Pareto frontier. In particuar in cases with many objective functions, since it is difficut and expensive to visuaize the whoe Pareto frontier, it becomes effective to restrict the search region to the most interesting part of Pareto frontier. An exampe of this approach can be seen in [15]. With further devices pecuiar to given probems, the effectiveness of SAMO has been observed in a wide range of engineering probems [15].

11 Sequentia Approximate Mutiobjective Optimization 11 f idea point aspiration eve f 1 (c) approximate vaue cosest to aspiration eve f idea point aspiration eve f 1 (d) approximate Pareto frontier References Figure 4: # data = 40 (after 10 additiona earning) : case 2 [1] Cortes, C. and Vapnik, V., Support vector networks, Machine Learning, vo. 20, pp , 1995 [2] Coeo, C. A. C., Van Vedhuizen, D. A. and Lamont, G. B., Evoutionary Agorithms for Soving Muti-Objective Probems, Kuwer Academic Pubishers, 2001 [3] Cristianini, N. and Shawe-Tayor, J., An Introduction to Support Vector Machines and Other Kerne-based Learning Methods, Cambridge University Press, [4] Deb, K., Muti-Objective Optimization using Evoutionary Agorithms, Wiey, [5] Ga, T., Stewart, T. J. and Hanne, T., JMuticriteria Decision Making Advances in MCDM Modes, Agorithms, Theory, and Appications, Kuwer Academic Pubishers, [6] Godberg, D. E., Genetic Agorithms in Search, Optimization and Machine Learning, Addison Wesey, [7] Jones, D. R., Schonau, M. and Wech, W. J., Efficient Goba Optimization of Expensive Back-Box Functions, Journa of Goba Optimization, , [8] Knowes, J. and Nakayama, H., Meta-Modeing in Muti-Objective Optimization, in Branke, J., Deb, K., Miettinen, K. and Sowinski, R. (eds.), Mutiobjective Optimization Interactive and Evoutionary Approaches, Springer, [9] Miettinen, K. M., Noninear Mutiobjective Optimization, Kuwer Academic Pubishers, [10] Mockus, J., Appication of Bayesian Approach to Numerica Methods of Goba and Stochastic Optimization, Journa of Goba Optimization, , [11] Myers, R. and Montogomery, D., Response Surface Methodoogy, Wiey, [12] Nakayama, H., Aspiration Leve Approach to Interactive Muti-objective Programming and its Appications, in Pardaos, P.M., Siskos, Y. and Zopounidis, C. (eds.) Advances in Muticriteria Anaysis, Kuwer Academic Pubishers, , [13] Nakayama, H., Arakawa, M. and Sasaki, R., Simuation-based Optimization using Computationa Inteigence, Optimization and Engineering, , [14] Nakayama, H., Yun, Y., Generating Support Vector Machines using Mutiobjective Optimization and Goa Programming, in Yaochu Jin (ed.), Muti-Objective Machine Learning, Springer, 2006.

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