The use of kriging in cost-effective single-objective optimization is well established, and a wide variety of different criteria now exist for selecting design vectors to evaluate in the search for the global minimum. Additionly, a large number of methods exist for transforming a multi-objective optimization problem to a single-objective problem. With these two facts in mind, this paper discusses the range of kriging assisted algorithms which are possible (and which remain to be explored) for cost-effective multi-objective optimization

Hawe, G.I.

Sykulski, J.K.

English

Southampton (e-Prints Soton)

ISEF 2007 - XIII International Symposium on Electromagnetic Fields in Mechatronics,  Electrical and Electronic Engineering Prague, Czech Republic, September 13-15, 2007     SCALARIZING COST-EFFECTIVE MULTIOBJECTIVE OPTIMIZATION ALGORITHMS MADE POSSIBLE WITH KRIGING    G. I. Hawe1,2 and J. K. Sykulski2 1Vector Fields Ltd., 24 Bankside, Kidlington, Oxford, U. K.  2School of ECS, University of Southampton, U. K.  Abstract – The use of kriging in cost-effective single-objective optimization is well established, and a wide variety of different criteria now exist for selecting design vectors to evaluate in the search for the global minimum.  Additionly, a large number of methods exist for transforming a multi-objective optimization problem to a single-objective problem.  With these two facts in mind, this paper discusses the range of kriging assisted algorithms which are possible (and which remain to be explored) for cost-effective multi-objective optimization.   Selection Criteria with Kriging Models for Single-Objective Optimization The use of kriging surrogate models in single-objective optimization is now well established (including within the electromagnetic design community [1]) and a wide variety of methods now exist for exploiting statistical information from them for the purpose of selecting a design vector to evaluate in the search for the global minimum [2].  These include (but are not limited to) evaluating the design vectors which maximize: the probability of improvement (POI) [2]; the expectation value of the improvement (EI) [3]; the generalized expected improvement (GEI) [4]; the weighted expected improvement (WEI) [5]; the credibility of a hypothesis (CH) about the location of the minimum [2] (also known as the one-stage approach); and most recently the ‘minimizer entropy’ (ME) criterion [6] .  Due to lack of space, their exact descriptions are omitted; suffice to say several allow the delicate balance between exploration and exploitation to be controlled through ‘cooling’ schemes; furthermore many offer the facility for selecting multiple design vectors for evaluation each iteration.  Scalarization of Multi-Objective Optimization Problems A popular method for solving a multi-objective optimization problem (MOOP) is to transform it to a single-objective optimization problem (SOOP), and then solve the SOOP using a single-objective optimization algorithm.  A large number of methods exist for transforming a MOOP to a SOOP including (but again not limited to): ε-constraint (ε-C);  weighting method (W); weighted metrics (WM) (including the Tchbycheff metric) method; achievement scalarizing function approach (AF); lexicographic ordering approach (LO); and the value function method (VF), descriptions of all of which may be found in [7].  This wide variety of scalarizing methods, coupled with the large range of selection crietria from single-objective optimization mentioned in the previous paragraph, leads to a plethora of (scalarizing) multi-objective algorithms being possible with kriging; a selection of these are shown in Table 1.  Despite this fact, surprisingly few have been explored in the literature:  two which have been explored are labelled with their reference.  One of the potential algorithms, marked X in Table 1, was tested on an electromagnetic design problem.  The voltage on, and position of, the focus electrode of an electron gun was varied so as to achieve two objectives: to focus the beam of electrons on the center of the anode as much as possible, and to make the electrons hit the anode face as perpendicular as possible.  Formally, denoting the voltage on the focus electrode by V Volts, and Table 1. The family of scalarizing multi-objective optimization algorithms made possible with kriging. Scalarizing Method Selection Criteria GC W  ε-C WM AF LO VF POI      EI     [3] [8]  GEI      WEI      CH      ME      Hybrids    X   its perpendicular distance from the emitting surface by d cm, the objective functions to be minimized are  and ∫ =anode21 d ) ( ) , ( S r r J d V f ∫ + ++=anode2 2 22 22 d) () () , ( Sv v vv vd V fz y xy x with  ] 1000 , 0 [ ∈ V and , where r is the radial distance from the center of the anode surface, J(r) is the current density at r, and the  integrals are taken over the surface of the anode. v] 10 , 4 [ ∈ dx, vy and vz are the components of the electron velocities as  they hit the surface of the anode, which lies in the xy plane. Each analysis was carried out using the Vector Fields space charge solver, SCALA. The Pareto optimal points found, along with one of the solutions, are shown in Fig.1.  (a)    (b)  Fig.1 (a) Pareto optimal front for electron gun problem, and (b) the left-most Pareto-optimal solution (which has the most parallel beam, as measured by objective f2), as found using algorithm X from Table 1.  Conclusions Many advances have been made in recent years in kriging-assisted single-objective optimization.  Using scalarizing methods, it is possible to also use each in multi-objective optimization.  Despite this, relatively few of the possibilities have been explored in the literature.  This paper has made it explicit the range of multi-objective algorithms possible with kriging, and demonstrated the use of one of the algorithms on an electromagnetic design problem.  Other algorithms from the family of scalarizing algorithms will be explored in the full paper. References                                                                                                                                                           [1] L. Lebensztain, C.A.R. Marretto, M.C. Costa and J-L. Coulomb, Kriging: a useful tool for electromagnetic devices optimization, IEEE Transactions on Magnetics, Vol. 40, No. 2, pp. 1196-1199, 2004. [2] D. R. Jones, A taxonomy of Global Optimization Methods Based on Response Surfaces, Journal of Global Optimization, Vol. 21, pp. 345-383, 2001. [3] D. R. Jones, M. Schonlau and W. J. Welch, Efficient Global Optimization of Expensive Black-Box Functions, Journal of Global Optimization, Vol. 13, pp. 455-492, 1998. [4] M. J. Sasena, P. Y. Papalambros and P. Goovaerts, Exploration of metamodelling sampling criteria for constrained global optimization, Engineering Optimization, Vol. 34, No. 3, pp. 263-278, 2002. [5] A. Sobester, S. J. Leary and A. J. Keane, “On the design of optimization strategies based on global response surface approximation models”, Journal of Global Optimization, vol. 33, pp. 31–59, 2005. [6] J. Villemonteix, E. Vazquez, and E.Walter, An informational approach to the global optimization of expensive-to-evaluate functions, to appear in the Journal of Global Optimization, 2007. [7] K. M. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, 1999. [8] J. Knowles, ParEGO: A hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems, IEEE Transactions on Evolutionary Computation, vol. 10, no. 1, pp. 50 66, 2006. 

A taxonomy of Global Optimization Methods Based on Response Surfaces,

and E.Walter, An informational approach to the global optimization of expensive-to-evaluate functions,

Efficient Global Optimization of Expensive Black-Box Functions,

Exploration of metamodelling sampling criteria for constrained global optimization,

Kriging: a useful tool for electromagnetic devices optimization,

Nonlinear Multiobjective Optimization,

On the design of optimization strategies based on global response surface approximation models”,

ParEGO: A hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems,

Scalarizing cost-effective multiobjective optimization algorithms made possible with kriging

Glenn Hawe

Jan Sykulski

Crossref

Scalarizing cost‐effective multi‐objective optimization algorithms made possible with kriging

Purpose – The purpose of this paper is threefold: to make explicitly clear the range of efficient multi-objective optimization algorithms which are available with kriging; to demonstrate a previously uninvestigated algorithm on an electromagnetic design problem; and to identify algorithms particularly worthy of investigation in this field. Design/methodology/approach – The paper concentrates exclusively on scalarizing multi-objective optimization algorithms. By reviewing the range of selection criteria based on kriging models for single-objective optimization along with the range of methods available for transforming a multi-objective optimization problem to a single-objective problem, the family of scalarizing multi-objective optimization algorithms is made explicitly clear. Findings – One of the proposed algorithms is demonstrated on the multi-objective design of an electron gun. It is able to identify efficiently an approximation to the Pareto-optimal front. Research limitations/implications – The algorithms proposed are applicable to unconstrained problems only. One future development is to incorporate constraint-handling techniques from single-objective optimization into the scalarizing algorithms. Originality/value – A family of algorithms, most of which have not been explored before in the literature, is proposed. Algorithms of particular potential (utilizing the most promising developments in single-objective optimization) are identified

Hawe, G.

A hybrid one-then-two stage algorithm for computationally expensive electromagnetic design optimisation”,

A taxonomy of global optimization methods based on response surfaces”,

An enhanced probability of improvement utility function for locating Pareto-optimal solutions”,

Efﬁcient global optimization of expensive black-box functions”,

Exploration of metamodelling sampling criteria for constrained global optimisation”,

Kriging: a useful tool for electromagnetic devices optimisation”,

ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multi-objective optimization problems”,

Scalarizing cost-effective multi-objective optimization algorithms made possible with kriging

http://eprints.soton.ac.uk/264525/1/ISEF2007_paper_short.pdf

Scalarizing cost-effective multiobjective optimization algorithms made possible with kriging

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