By Gabriele Eichfelder

ISBN-10: 3540791574

ISBN-13: 9783540791577

This booklet provides adaptive answer equipment for multiobjective optimization difficulties according to parameter established scalarization ways. With assistance from sensitivity effects an adaptive parameter keep watch over is built such that high quality approximations of the effective set are generated. those examinations are according to a different scalarization process, however the software of those effects to many different recognized scalarization equipment is usually awarded. Thereby very normal multiobjective optimization difficulties are thought of with an arbitrary partial ordering outlined via a closed pointed convex cone within the aim house. The effectiveness of those new tools is tested with a number of try difficulties in addition to with a up to date challenge in intensity-modulated radiotherapy. The e-book concludes with an additional program: a method for fixing multiobjective bilevel optimization difficulties is given and is utilized to a bicriteria bilevel challenge in clinical engineering.

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22) t ∈ R, x ∈ Ω, s ∈ Rm−1 for j ∈ {1, . . , m − 1} with minimal solution (tmax,j , xmax,j , smax,j ) and minimal value −smax,j . We get j m−1 H 0 := y ∈ Rm si v i , si ∈ [smin,i , smax,i ], i = 1, . . , m − 1 i i y= i=1 44 2 Scalarization Approaches ˜ ⊂ H 0 . This is a suitable restriction of the parameter set H as with H the following lemma shows. 20. Let x ¯ be a K-minimal solution of the multiobjective optimization problem (MOP). Let r ∈ K \ {0m }. Then there is a pa¯) is a minimal solution of rameter a ¯ ∈ H 0 and some t¯ ∈ R so that (t¯, x (SP(¯ a, r)).

We will discuss these important properties among others in the following section. Problem (SP(a, r)) is also discussed by Helbig in [104]. He interprets the point a as a reference point and the parameter r as a direction. For r ∈ int(Rm + ) this corresponds to the interpretation of r as a weighting of the objective functions with the weights wi := r1i , i = 1, . . 2 Properties of the Pascoletti-Seraﬁni Scalarization 25 (compare with the weighted Chebyshev norm). 1,c)) which is closest to the reference point.

5 Relations Between Scalarizations 53 (i. e. b = (0, 1) , β = 0). 7) are equal to min f1 (x) and min f2 (x). x∈Ω For x ¯1 and x ¯2 x∈Ω respectively minimal solutions of these problems we get a1 + (1 − λ)¯ a2 , λ ∈ [0, 1]} H a := {y ∈ H | y = λ¯ with xi ) − a ¯i := f (¯ b f (¯ xi ) − β r= b r xi ) f1 (¯ 0 , i = 1, 2, and hence H a = {y = (ε, 0) | ε = λ f1 (¯ x1 ) + (1 − λ) f1 (¯ x2 ), λ ∈ [0, 1]} x1 ) ≤ ε ≤ f1 (¯ x2 )}. 31. Let x ¯ be an EP-minimal solution of the multiobjective optimization problem (MOP) with m = 2.

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